signs in supergeometry

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###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
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> from [Grassmann 1844, p. 61 and 84](Ausdehnungslehre#Grassmann44)

#Contents#
* table of contents
{:toc}

## Idea

Discussion of [[superalgebra]] and [[supergeometry]] involves, by definition, crucial minus signs appearing in certain expressions. While there is a systematic rule for these, when superalgebra/supergeometry is combined with other topics that crucially involve their own signs, such as [[homological algebra]], or ordering of terms, such as in [[noncommutative algebra]], then the combined sign rule may appear intricate. This page here is meant to explicitly list two equivalent sign rules for easy reference. We follow ([Deligne-Freed 99](#DeligneFreed99)) which has the same goal.

The main point to notice is that for algebraic terms that have both a [[homological algebra|homological degree]] $n \in \mathbb{Z}$ and a super-degree $\sigma \in \mathbb{Z}_2$ the "internalization sign rule" for the sign when exchanging two such elements is

$$
  \left(
    \left(
      n_1, \sigma_1
    \right)
    ,
    \left(
      n_2, \sigma_2
    \right)
  \right)
   \;\mapsto\;
   (-1)^{n_1 n_2 + \sigma_2 \sigma_2}
   \,.
$$

(e.g. [Castellani-D'Auria-Fré 91, (II.2.109)](#CastellaniDAuriaFre91), [Bonora-Bregola-Lechner-Pasti-Tonin 87, beginning of section 2](#BonoraBregolaLechnerPastiTonin87))

This we discuss below at _[The sign rule from internalization](#TheSignRuleFromInternalization)_.

Parts of the literature however adops another sign rule, given by

$$
  \left(
    \left(
      n_1, \sigma_1
    \right)
    ,
    \left(
      n_2, \sigma_2
    \right)
  \right)
   \;\mapsto\;
   (-1)^{(n_1 + \sigma_1)(n_2 + \sigma_2)}
   \,.
$$

This we discuss below in _[The super odd sign rule](#SuperOddConvention)_.

In [Deligne-Morgan 99](#DeligneMorgan99), the two sign rules are referred to as _Deligne's convention_ (for the internalization rule) and _Bernstein's convention_ (for the super odd sign rule).

[[!include sign rules in homological superalgebra -- table]]

## The sign rule from internalization
 {#TheSignRuleFromInternalization}

Here and in the following we adopt the perpective that [[homological algebra]] in [[superalgebra]] is concerned with the study of [[chain complexes]] [[internalization|internal]] to the [[symmetric monoidal category]] of [[super vector spaces]] ([Deligne 90](#Deligne90)) or equivalently internal to the [[topos]] over [[superpoints]] ([Schwarz 84](#Schwarz84)). This means that there are two contributions to the sign when exchanging two terms, one from the cohomological grading, one from the super-grading. This is "the sign rule" also used for instance in  ([Castellani-D'Auria-Fré 91, equation (II.2.106)](#CastellaniDAuriaFre91), [Bonora-Bregola-Lechner-Pasti-Tonin 87](#BonoraBregolaLechnerPastiTonin87) [Deligne-Freed 99, section 6](#DeligneFreed99)).

There is however also a different sign rule use in the literature, the relation to which we discuss [below](#SuperOddConvention).

### Differential forms on supermanifolds
 {#DifferentialFormsOnSupermanifolds}

We discuss the sign rules for the [[de Rham algebra]] of [[differential forms]] on a [[supermanifold]] ([[superdifferential forms]]), which is a [[differential graded-commutative superalgebra]].

For $p,q \in \mathbb{N}$, let $\mathbb{R}^{p|q}$ the [[super Cartesian space]] of [[dimension]] $(p|q)$. This is the [[supermanifold]] defined by the fact that its [[algebra of functions]] is freely generated, as a [[smooth superalgebra]] by even-graded [[coordinate]]-functions $\langle \{x^a\}_{1}^p$ and odd-graded coordinate function $\{ \theta^\alpha\}_{\alpha= 1}^q$

$$
  C^\infty(\mathbb{R}^{p|q}) = \langle \{x^a\}_{1}^p, \{ \theta^\alpha\}_{\alpha= 1}^q \rangle
  \,.
$$

Accordingly, by the discussion at _[[Kähler forms]]_, the [[de Rham complex]] of [[differential forms]] on $\mathbb{R}^{p|q}$ is freely generated as a super-[[module]] over the [[smooth superalgebra]] $C^\infty(\mathbb{R}^{p|q})$ by expressions of the form $\mathbf{d}x^{a_1} \wedge \cdots \wedge \mathbf{d}x^{a_k} \wedge \mathbf{d}\theta^{\alpha_1} \wedge \cdots \wedge \mathbf{d}\theta^{\alpha_l}$.

$$
  \Omega^\bullet(\mathbb{R}^{p|q})
  =
  C^\infty(\mathbb{R}^{p|q})
  \left[
   \left\{
     \mathbf{d}x^{a_1} \wedge \cdots \wedge \mathbf{d}x^{a_k}
     \wedge
     \mathbf{d}\theta^{\alpha_1} \wedge \cdots \wedge \mathbf{d}\theta^{\alpha_l}
   \right\}
  \right]
  \,.
$$

This [[de Rham complex]] now carries the structure of a

* differential graded-commutative graded commutative superalgebra

which should be thought of as bracketet as follows

* differential graded-commutative graded (commutative superalgebra).

This means in effect that elements of $\Omega^\bullet(\mathbb{R}^{p|q})$ carry a $\mathbb{Z} \times \mathbb{Z}_2$-[[graded object|grading]], where we may say that

* $\mathbb{Z}$ corresponds to the "cohomological grading";

* $\mathbb{Z}_2$ corresponds to the super-grading.

We write

$$
  (n,\sigma)\in \mathbb{Z} \times \mathbb{Z}_2
$$

for elements in this grading group.

In this notation the grading of the elements in $\Omega^\bullet(\mathbb{R}^{p|q})$ is all induced by the fact that the de Rham differential $\mathbf{d}$ itself is a [[derivation]] of degree $(1,even)$.

| $\phantom{A}$generator$\phantom{A}$ | $\phantom{A}$bi-degree$\phantom{A}$ |
|--|--|
| $\phantom{A}$$x^a$$\phantom{A}$ | $\phantom{A}$(0,even)$\phantom{A}$ |
| $\phantom{A}$$\theta^\alpha$$\phantom{A}$ | $\phantom{A}$(0,odd)$\phantom{A}$ |
| $\phantom{A}$$\mathbf{d}$$\phantom{A}$ | $\phantom{A}$(1,even)$\phantom{A}$ |

Here the last line means that we have

| $\phantom{A}$generator$\phantom{A}$ | $\phantom{A}$bi-degree$\phantom{A}$ |
|--|--|
| $\phantom{A}$$x^a$$\phantom{A}$ | $\phantom{A}$(0,even)$\phantom{A}$ |
| $\phantom{A}$$\theta^\alpha$$\phantom{A}$ | $\phantom{A}$(0,odd)$\phantom{A}$ |
| $\phantom{A}$$\mathbf{d}x^a$$\phantom{A}$ | $\phantom{A}$(1,even)$\phantom{A}$ |
| $\phantom{A}$$\mathbf{d}\theta^\alpha$$\phantom{A}$ | $\phantom{A}$(1,odd)$\phantom{A}$ |

The formula for the "cohomologically- and super-graded commutativity" in $\Omega^\bullet(\mathbb{R}^{p|q})$ is

$$
  \alpha \wedge \beta
  =
  \;
  (- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta}
  \;
  \beta \wedge \alpha
$$

for all $\alpha,  \beta \in \Omega^\bullet(\mathbb{R}^{p|q})$ of homogeneous $\mathbb{Z}\times \mathbb{Z}_2$-degree. Hence there are _two_ contributions to the sign picked up when exchanging two super-differential forms in the wedge product:

1. there is a "cohomological sign" which for commuting an $n_1$-forms past an $n_2$-form is $(-1)^{n_1 n_2}$;

1. in addition there is a "super-grading" sich which for commuting a $\sigma_1$-graded coordinate function past a $\sigma_2$-graded coordinate function (possibly under the de Rham differential) is $(-1)^{\sigma_1 \sigma_2}$.

Some examples:

| $\phantom{A}$[[signs in supergeometry|sign rule]]$\phantom{A}$  | $\phantom{A}$[Deligne's](signs+in+supergeometry#TheSignRuleFromInternalization)$\phantom{A}$ | [Bernstein's](signs+in+supergeometry#SuperOddConvention)$\phantom{A}$ |
|---|------------------|-------------------|
| $\phantom{A}$$x^{a} \; x^{b} =$ | $\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$ | $\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$ |
| $\phantom{A}$$x^a \;\theta^\alpha =$$\phantom{A}$ | $\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$ | $\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$ |
| $\phantom{A}$$\theta^{\alpha} \; \theta^{\beta} =$$\phantom{A}$ | $\phantom{A}$$- \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$ | $\phantom{A}$$ - \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$ |
| $\phantom{A}$$x^{a} (\mathbf{d}x^{a}) =$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$ |
| $\phantom{A}$$\theta^\alpha (\mathbf{d}x^a) =$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$ | $\phantom{A}$${\color{blue}{-}} (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$ |
| $\phantom{A}$$\theta^{\alpha} (\mathbf{d}\theta^{\beta})  = $$\phantom{A}$ | $\phantom{A}$$- (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$ | $\phantom{A}$${\color{blue}{+}} (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$ |
| $\phantom{A}$$ (\mathbf{d}x^{a}) (\mathbf{d} x^{b}) =$$\phantom{A}$ |  $\phantom{A}$$- (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$ | $\phantom{A}$$ - (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$ |
| $\phantom{A}$$ (\mathbf{d}x^a) (\mathbf{d} \theta^{\alpha}) =$$\phantom{A}$ | $\phantom{A}$$ - (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a) $$\phantom{A}$ | $\phantom{A}$$ {\color{blue}{+}} (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a) $$\phantom{A}$ |
| $\phantom{A}$$(\mathbf{d}\theta^{\alpha}) (\mathbf{d} \theta^{\beta}) =$ | $\phantom{A}$$ + (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$ | $\phantom{A}$$ + (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$ |
{: style='margin:auto'}

### Cohomology of super Minkowski spacetime

The above signs rules for [[differential forms on supermanifolds]] directly translate to the signs rule for [[super Lie algebra]] [[Lie algebra cohomology]], by the identification of elements in the [[Chevalley-Eilenberg algebra]] with the [[left invariant differential forms]] on the corresponding [[super Lie group]].

We discuss now the [[Chevalley-Eilenberg algebra]] of [[super Minkowski spacetime]], regarded as a [[super translation Lie algebra]], equivalently as the [[quotient]] of the [[quotient]] of the
[[super Poincaré Lie algebra]] by the [[Lorentz Lie algebra]].

This is the sub-algebra of the [above](#DifferentialFormsOnSupermanifolds) de Rham complex of the [[left invariant differential forms]]:

$$
  CE(\mathbb{R}^{d-1,1;N})
  =
  \Omega^\bullet_{li}(\mathbb{R}^{d-1,1;N})
  \hookrightarrow
  \Omega^\bullet(\mathbb{R}^{d-1,1;N})
  \,.
$$

With the [above](#DifferentialFormsOnSupermanifolds) notation these are

$$
  \psi^\alpha \coloneqq \mathbf{d}\theta^\alpha
$$

$$
  e^a \coloneqq \mathbf{d}x^a + \bar{\theta} \Gamma^a \mathbf{d}\theta
  \,.
$$

With the [above](#DifferentialFormsOnSupermanifolds) sign rules it follows that

$$
  e^{a_1} \wedge e^{a_2} = + e^{a_2} \wedge e^{a_1}
$$

$$
  e^{a} \wedge \psi^\alpha = - \psi^\alpha \wedge e^a
$$

$$
  \psi^{\alpha_1} \wedge \psi^{\alpha_2}
  =
  +
  \psi^{\alpha_2} \wedge \psi^{\alpha_1}
  \,.
$$

### Super Lie algebra cohomology

More generally, since the super-[[Chevalley-Eilenberg algebra]] of a [[super Lie algebra]] is identified with the [[left invariant differential forms]] on the corresponding [[super Lie group]], the signs from the super de Rham complex [above](#DifferentialFormsOnSupermanifolds) induce the cotresponding sign rule for the computation of super-[[Lie algebra cohomology]].

### Presheaves on superpoints

Write $Superpoint$ for the [[category]] of [[superpoints]], the [[opposite category]] of that of real finitely-generated [[Grassmann algebras]].

We list the terms with crucial signs that appear when regarding [[superalgebra]] as algebra [[internalization|internal]] to the [[presheaf topos]] over [[superpoints]] (see at _[[super infinity-groupoid]]_ for background).

#### Super Lie algebras

Given a [[super Lie algebra]] $\mathfrak{g}$, for each [[superpoint]] $\mathbb{R}^{0|q}$ write

$$
  \mathbb{g}(\mathbb{R}^{0|q})
  \coloneqq
  \left(
   \mathfrak{g}\otimes C^\infty(\mathbb{R}^{0|q})
  \right)_{even}
  \in Vect_{\mathbb{R}}
$$

for the [[vector space]] which is the even-degree part of the [[tensor product]] of the underlying [[vector spaces]] of $\mathfrak{g}$ and of the [[Grassmann algebra]] $C^\infty(\mathbb{R}^{0|q})$.

Here if $\{e_a, \psi_\alpha\}$ is a [[basis]] for $\mathfrak{g}$ with $\{e_a\}$ a basis for $\mathfrak{g}_{even}$ and $\{\psi_\alpha\}$ a basis for $\mathfrak{g}_{odd}$, and if we write $\{\epsilon^i\}_{i = 1}^q$ for the generators of $C^\infty(\mathbb{R}^{0|q})$, then $\mathbb{g}(\mathbb{R}^{0|q})$ is generated from elements of the form

$$
  e_a, \; e_a \epsilon^{i_1} \epsilon^{i_2},\;
  e_a \epsilon^{i_1} \epsilon^{i_2} \epsilon^{i_3} \epsilon^{i_4}
  ,\;
  \cdots
$$

and

$$
  \psi_\alpha \epsilon^{i_1} ,\;
  \psi_\alpha \epsilon^{i_1} \epsilon^{i_2} \epsilon^{i_3}
  ,\;
  \cdots
  \,.
$$

The [[Lie bracket]] on the ordinary vector space $\mathfrak{g}(\mathbb{R}^{0|q})$ is defined on elements

$$
  x_i \kappa_i \in
  \left(
   \mathfrak{g}\otimes C^\infty(\mathbb{R}^{0|q})
  \right)_{even}
$$

by the formula

$$
  [ x_1 \kappa_1 , x_2 \kappa_2]_{\mathfrak{g}(\mathbb{R}^{0|q})}
  \coloneqq
  [x_1, x_2]_{\mathfrak{g}} \kappa_1 \kappa_2
  \,.
$$

Some examples:

$$
  [e_{a_1}, e_{a_2}]_{\mathfrak{g}(\mathbb{R}^{0|q})}
  =
  [e_{a_1}, e_{a_2}]_{\mathfrak{g}}
$$

$$
  [e_{a}, \psi_{\alpha} \epsilon^1]_{\mathfrak{g}(\mathbb{R}^{0|q})}
  =
  [e_{a}, \psi_{\alpha}]_{\mathfrak{g}} \epsilon^1
$$

$$
  [\psi_{\alpha} \epsilon^1, e_a]_{\mathfrak{g}(\mathbb{R}^{0|q})}
  =
  [\psi_{\alpha}, e_a]_{\mathfrak{g}} \epsilon^1
$$

From this and from the ordinary skew-symmetry of the [[Lie bracket]] in $\mathfrak{g}(\mathbb{R}^{0|q})$ follows that also in the [[super Lie algebra]] we have

$$
  [e_a, \psi_{\alpha}] = - [\psi_{\alpha}, e_a]_{\mathfrak{g}}
  \,.
$$

Next, from

$$
  [\psi_{\alpha_1} \epsilon^1, \psi_{\alpha_2} \epsilon^2]_{\mathfrak{g}(\mathbb{R}^{0|q})}
  =
  [\psi_{\alpha_1}, \psi_{\alpha_2}]_{\mathfrak{g}} \epsilon^1 \epsilon^2
$$

$$
  [\psi_{\alpha_2} \epsilon^2, \psi_{\alpha_1} \epsilon^1]_{\mathfrak{g}(\mathbb{R}^{0|q})}
  =
  [\psi_{\alpha_2}, \psi_{\alpha_1}]_{\mathfrak{g}} \epsilon^2 \epsilon^1
$$

and the fact that $\epsilon^1  \epsilon^2 = - \epsilon^2  \epsilon^1$ it follows that in the [[super Lie algebra]]

$$
  [\psi_{\alpha_1}, \psi_{\alpha_2}]_{\mathfrak{g}}
  =
  +
  [\psi_{\alpha_2}, \psi_{\alpha_1}]_{\mathfrak{g}}
  \,.
$$

#### Chevalley-Eilenberg algebra

Given a [[super Lie algebra]] $\mathfrak{g}$ as above,
the [[tensor product]] $\mathfrak{g}\otimes C^\infty(\mathbb{R}^{0|1})$ is the [[free construction|free]] $C^\infty(\mathbb{R}^{0|q})$-[[module]] generated from the [[super vector space]] underlying $\mathfrak{g}$.

(...)

$$
  CE_{C^\infty(\mathbb{R}^{0|q})}( \mathfrak{g}(C^\infty(\mathbb{R}^{0|q}) )_{even}
  \,.
$$

(...)

$$
  (\psi^{\alpha_1} \epsilon^1)
  \wedge
  (\psi^{\alpha_2} \epsilon^2)
  =
  -
  (\psi^{\alpha_2} \epsilon^2)
  \wedge
  (\psi^{\alpha_1} \epsilon^1)
  =
  \psi^{\alpha_1}
    \wedge
  \psi^{\alpha^2}
  \epsilon^1 \epsilon^2
$$

(...)

## The super odd sign rule
 {#SuperOddConvention}

In all of the above we used the sign rule induced by setting up all mathematical concepts [[internalization|internal]] to the [[symmetric monoidal category]] of [[super vector spaces]] or [[internalization|internal]] the [[topos]] over [[superpoints]].

There is another sign convention in the literature, for instance in (...). In this convention one considers the map

$$
  ((-)+(-))_{mod\, 2}
  \;\colon\;
  \mathbb{Z} \times \mathbb{Z}_2
  \longrightarrow
  \mathbb{Z}_2
$$

to induce a single $\mathbb{Z}_2$-grading and then lets all signs be induced by that.

This can motivated from regarding the $\mathbb{Z}_2$-graded [[de Rham complex]] of a [[supermanifold]] as the [[algebra of functions]] on its odd-shifted tangent bundle

$$
  \Omega^\bullet(X) \simeq_{\mathbb{Z}_2} C^\infty(T[1]X)
  \,.
$$

In this convention a term such as $\mathbf{d}\theta^\alpha$ of bidegree $(1,1)$ has single degree $0$ and hence _commutes with every other term_.

### Differential forms on a supermanifold in the super odd convention
 {#DifferentialFormsInSuperOddConvention}

In this other convention the signs in the [[de Rham complex]] of the [[super Cartesian space]] $\mathbb{R}^{p|q}$ are induced as follows:

$$
  (s.o.): \;\;\; x^{a_1} x^{a_2} = + x^{a_2} x^{a_1}
$$

$$
  (s.o.): \;\;\;x^a \theta^\alpha = + \theta^\alpha x^a
$$

$$
  (s.o.): \;\;\;\theta^{\alpha_1} \theta^{\alpha_2}
  =
  +
  \theta^{\alpha_2} \theta^{\alpha_1}
$$

$$
  (s.o.): \;\;\;x^{a_1} (\mathbf{d}x^{a_2}) = + (\mathbf{d}x^{a_2}) x^{a_1}
$$

$$
  (s.o.): \;\;\;\theta^\alpha (\mathbf{d}x^a) = + (\mathbf{d}x^a) \theta^\alpha
$$

$$
  (s.o.): \;\;\;
  \theta^{\alpha_1} (\mathbf{d}\theta^{\alpha_2})
   =
   + (\mathbf{d}\theta^{\alpha_2}) \theta^{\alpha_1}
$$

$$
  (s.o.): \;\;\;
  \mathbf{d}x^{a_1}
  \wedge
  \mathbf{d} x^{a_2}
  =
  -
  \mathbf{d} x^{a_2}
  \wedge
  \mathbf{d} x^{a_1}
$$

$$
  (s.o.): \;\;\;
  \mathbf{d}x^a
  \wedge
  \mathbf{d} \theta^{\alpha}
  =
  +
  \mathbf{d}\theta^{\alpha}
  \wedge
  \mathbf{d} x^a
$$

$$
  (s.o.): \;\;\;
  \mathbf{d}\theta^{\alpha_1}
  \wedge
  \mathbf{d} \theta^{\alpha_2}
  =
  +
  \mathbf{d}\theta^{\alpha_2}
  \wedge
  \mathbf{d} \theta^{\alpha_1}
$$

## Relation between the two sign rules

### Strong symmetric monoidal equivalence
 {#StrongSymmetricMonoidalEquialence}

The two sign rules correspond to two different [[symmetric monoidal category|symmetric braiding]]-[[structures]] on the [[monoidal category]] $Ch(SuperVect)$ of [[chain complexes]] of [[super vector spaces]]. The corresponding two kinds of [[differential graded-commutative superalgebras]] are the [[commutative monoid objects]] with respect to these two choices.

We now show that the two braidings are equivalent (Prop. \ref{EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect} below), which immediately implies that the two categories of differential graded-commutative algebras are equivalent.

$\,$

+-- {: .num_defn #ChainComplexesOfSuperVectorSpaces}
###### Definition
**([[chain complexes]] of [[super vector spaces]])**

Write $Ch(SuperVect)$ for the [[category of chain complexes]] inside the category of [[super vector spaces]].

Hence for $V_\bullet \in Ch(SuperVect)$ an [[object]], for each $n \in \mathbb{Z}$ there is a [[super vector space]]

$$
  V_n = (V_n)_{even} \oplus (V_n)_{odd} \;\in SuperVect
$$

where we write the elements of the [[group of order two]] as $\mathbb{Z}/2 =\{even, odd\}$, with $even$ being the [[neutral element]].

Hence we may regard any $V_\bullet \in Ch(SuperVect)$ equivalently as a $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded object|graded]] [[vector space]] equipped with a [[differential]] of degree $(1,even)$. For $v \in V$ an element in definite ("homogeneous") bi-degree, we denote this degree by

\[
  \label{BidegreeOnChainComplexOfSuperVectorSpaces}
  deg(v) \coloneqq (n_v, \sigma_v) \;\in\; \mathbb{Z} \times (\mathbb{Z}/2)
  \,.
\]

The category $Ch(SuperVect)$ becomes a [[monoidal category]] under the [[tensor product of chain complexes]] applied to the tensor product of [[super vector spaces]]. This means that for $V, W \in Ch(SuperVect)$, the [[differential]] on a homogeneously graded element $v \otimes w \in V \otimes W$ is

$$
  \partial(v \otimes w)
  \;=\;
  (\partial v) \otimes w
  +
  (-1)^{ n_v } v \otimes \partial w
  \,.
$$

=--

+-- {: .num_prop #SymmetricStructureOnCategoryOfChainComplexesOfSuperVectorSpaces}
###### Proposition
**([[symmetric monoidal category|symmetric monoidal]] [[structure]] on [[category of chain complexes of super vector spaces]])**

The [[monoidal category|monoidal]] [[category of chain complexes of super vector spaces]] $(Ch(SuperVect), \otimes)$ from Def. \ref{ChainComplexesOfSuperVectorSpaces} becomes a [[symmetric monoidal category|symmetric monoidal]] with each of the following two [[braiding]] [[isomorphisms]], defined on tensor products of elements in homogenous bi-degree (eq:BidegreeOnChainComplexOfSuperVectorSpaces) as follows:

1. $\tau_{Deligne} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v n_w + \sigma_v \sigma_w) } w \otimes v$;

1. $\tau_{Bernst} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v + \sigma_v) (n_w + \sigma_w) } w \otimes v$.

Here in the exponents we are using the canonical [[ring]]-[[structure]] on the [[integers]] $\mathbb{Z}$ and on the [[prime field]] $\mathbb{Z}/2 = \mathbb{F}_2$, the implicit [[ring homomorphism]] $\mathbb{Z} \to \mathbb{Z}/2$ and we understand that $(-1)^{even} = 1$ and $(-1)^{odd} = -1$.

=--

+-- {: .proof}
###### Proof

Since the expressions for both sign factors are symmetric in $v$ and $w$ in both cases, it is clear that $\tau_{w,v} \circ \tau_{v,w} = id_{v \otimes w}$ in both cases. Hence if $\tau$ is indeed a [[braiding]], then it is [[symmetric monoidal category|symmetric]].

To see that $\tau$ is indeed a [[braiding]] in each case, we need to check the [[hexagon identities]]

$$
  \array{
   (V_1 \otimes V_2) \otimes V_3
   &\overset{a_{V_1, V_2, V_3}}{\longrightarrow}&
   V_1 \otimes (V_2 \otimes V_3)
   &\overset{\tau_{V_1,V_2 \otimes V_3}}{\longrightarrow}&
   (V_2 \otimes V_3) \otimes V_1
   \\
   \Big\downarrow{}^{\tau_{V_1,V_2}\otimes Id}
   &&&&
   \Big\downarrow{}^{a_{V_2,V_3,V_1}}
   \\
   (V_2 \otimes V_1) \otimes V_3
   &\overset{a_{V_2,V_1,V_3}}{\longrightarrow}&
   V_2 \otimes (V_1 \otimes V_3)
   &\overset{Id \otimes \tau_{V_1,V_3}}{\longrightarrow}&
   V_2 \otimes (V_3 \otimes V_1)
  }
$$

and

$$
  \array{
   V_1 \otimes (V_2 \otimes V_3)
   &\overset{a^{-1}_{V_1,V_2,V_3}}{\longrightarrow}&
   (V_1 \otimes V_2) \otimes V_3
   &\overset{\tau_{V_1 \otimes V_2, V_3}}{\longrightarrow}&
   V_3 \otimes (V_1 \otimes V_2)
   \\
   \Big\downarrow{}^{id \otimes \tau_{V_2,V_3}}
   &&&&
   \Big\downarrow{}^{a^{-1}_{V_3,V_1,V_2}}
   \\
   V_1 \otimes (V_3 \otimes V_2)
   &\overset{a^{-1}_{V_1,V_3,V_2}}{\longrightarrow}&
   (V_1 \otimes V_3) \otimes V_2
   &\overset{\tau_{V_1,V_3} \otimes id}{\longrightarrow}&
   (V_3 \otimes V_1) \otimes V_2
  }
  \,.
$$

Since $\tau$ differs only by multiplication by a sign from the standard symmetric braiding on the [[category of vector spaces]], which does satisfy its hexagon identities, it just remains to check that these sign factors picked up in going both ways around these diagrams agree.

Hence for $\tau_{Deligne}$ the two hexagon identities are equivalent to the conditions

$$
  (-1)^{ n_1 (n_2 + n_3) + \sigma_1( \sigma_2 + \sigma_3 )  }
  \;=\;
  (-1)^{ n_1 n_2 + \sigma_1 \sigma_2 + n_1 n_3 + \sigma_1 \sigma_3 }
$$

and

$$
  (-1)^{ (n_1 + n_2) n_3 + (\sigma_1 + \sigma_2) \sigma_3 }
  \;=\;
  (-1)^{ n_1 n_3 + \sigma_1 \sigma_3 + n_2 n_3 + \sigma_2 \sigma_3 }
$$

while for $\tau_{Bernst}$ they are equivalent to the conditions

$$
  (-1)^{ (n_1 + \sigma_1)( n_2 + \sigma_2 + n_3 + \sigma_3  ) }
  \;=\;
  (-1)^{ (n_1 + \sigma_1)(n_2 + \sigma_2) + (n_1 + \sigma_1)(n_3 + \sigma_3)  }
$$

and

$$
  (-1)^{ (n_1 + \sigma_1 + n_2 + \sigma_2)(n_3 + \sigma_3) }
  \;=\;
  (-1)^{ (n_1 + \sigma_1)(n_3 + \sigma_3) + (n_2 + \sigma_2)(n_3 + \sigma_3) }
$$

for all triples of bi-degrees $(n_i, \sigma_i) \in \mathbb{Z} \times (\mathbb{Z}/2)$.

In both cases this holds because already the relevant [[exponents]] are equal in each case, by the [[distributive law]] for [[multiplication]] and [[addition]] in $\mathbb{Z}/2 = \mathbb{F}_2$.

=--

+-- {: .num_prop #EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect}
###### Proposition
**(the two [[symmetric monoidal category|symmetric monoidal]] [[structures]] on the [[category of chain complexes of super vector spaces]] are equivalent)**

The two [[symmetric monoidal category]] [[structures]] $\tau_{Deligne}$ and $\tau_{Bernst}$ on the [[monoidal category|monoidal]] [[category of chain complexes of super vector spaces]] $(Ch(SuperVect), \otimes)$ from Prop. \ref{SymmetricStructureOnCategoryOfChainComplexesOfSuperVectorSpaces} are equivalent

$$
  \tau_{Deligne}
  \;\simeq\;
  \tau_{Bernst}
$$

in that the [[identity functor]] $Ch(SuperVect) \overset{\mu}{\longrightarrow} Ch(SuperVect)$ equipped with the following monoidal [[natural isomorphism]]

$$
  \array{
    id(V) \otimes id(W) \overset{\mu}{\longrightarrow} id(V \otimes W)
    \\
    v \otimes w
    \;\mapsto\;
    (-1)^{ n_v \sigma_w  } \, v \otimes w
  }
$$

(the second line shows its action on elements of homogeneous bidegree $(n,\sigma) \in \mathbb{Z} \times (\mathbb{Z}/2)$)

becomes a [[nLab:strong monoidal functor|strong]] [[symmetric monoidal functor]]

$$
  (Ch(SuperVect), \otimes, \tau_{Deligne})
  \underoverset{\simeq}{(id,\mu)}{\longrightarrow}
  (Ch(SuperVect), \otimes, \tau_{Bernst})
  \,.
$$

=--

+-- {: .proof}
###### Proof

First to see that we have a [[strong monoidal functor]], we need to check the [[associativity]] condition

$$
  \array{
    (id(V_1) \otimes id(V_2)) \otimes id(V_3)
      &\underoverset{\simeq}{a_{id(V_1),id(V_2),id(V_3)}}{\longrightarrow}&
    id(V_1) \otimes( id(V_2)\otimes id(V_3) )
    \\
    {}^{\mathllap{\mu_{V_1,V_2} \otimes id}}\downarrow
      &&
    \Big\downarrow{}^{\mathrlap{id\otimes \mu_{V_2,V_3}}}
    \\
    id(V_1 \otimes V_2) \otimes id(V_3)
     &&
    id(V_1) \otimes id(V_2 \otimes V_3)
    \\
    {}^{\mathllap{\mu_{V_1 \otimes V_2 , V_3} } }\Big\downarrow
      &&
    \Big\downarrow^{\mathrlap{\mu_{ V_1, V_2 \otimes V_3  }}}
    \\
    id( ( V_1 \otimes V_2 ) \otimes V_3  )
      &\underset{id(a_{V_1,V_2,V_3})}{\longrightarrow}&
    id( V_1 \otimes ( V_2 \otimes V_3 ) )
  }
  \,,
$$

and the [[unitality]] conditions

$$
  \array{
    1 \otimes id(V)
      &\overset{\epsilon \otimes id}{\longrightarrow}&
    id(1) \otimes id(V)
    \\
    {}^{\mathllap{\ell_{id(V)}}}\Big\downarrow
      &&
    \Big\downarrow{}^{\mathrlap{\mu_{1, V }}}
    \\
    id(V)
      &\overset{id(\ell_V )}{\longleftarrow}&
    id(1 \otimes V  )
  }
$$

and

$$
  \array{
    id(V) \otimes  1
      &\overset{id \otimes \epsilon }{\longrightarrow}&
    id(V) \otimes  id(1)
    \\
    {}^{\mathllap{r_{id(V)}}}\Big\downarrow
      &&
    \Big\downarrow{}^{\mathrlap{\mu_{V, 1 }}}
    \\
    id(V)
      &\overset{F(r_V )}{\longleftarrow}&
    id(V \otimes 1  )
  }
  \,,
$$

Since $\mu$ differs from the trivial monoidal isomorphism only by the sign factor, this is equivalent to the condition that the sign factors picked up in going both ways around these diagrams agree.

For [[associativity]] this is the condition

$$
  (-1)^{
    n_1 \sigma_2 + (n_1 + n_2)\sigma_3
  }
  \;=\;
  (-1)^{
    n_2 \sigma_3 + n_1 (\sigma_2 + \sigma_3)
  }
$$

for all bi-degrees $(n_i, \sigma_i) \in \mathbb{Z} \times (\mathbb{Z}/2)$,
which holds, because it already holds for the [[exponents]] themselves, as an identity in $\mathbb{Z}/2 =\mathbb{F}_2$.

For the [[unitality]] condition this is the statement that the sign given by $\mu_{1,x}$ and $\mu_{x,1}$ is the the trivial sign $(-1)^0 = 1$. This is indeed the case because the [[tensor unit]] is in degree $0 = (0,even) \in \mathbb{Z} \times (\mathbb{Z}/2)$.

Now to see that we have a [[symmetric monoidal functor]], we need to show that it intertwines the two [[symmetric monoidal category|symmetruc]] [[braiding]] [[isomorphisms]]

$$
  \array{
    id(V_1) \otimes id(V_2)
      &\stackrel{\tau_{Bernst}}{\longrightarrow}& id(V_2) \otimes id(V_1)
    \\
    {}^{\mathllap{\mu_{V_1,V_2}}}\Big\downarrow
    &&
    \Big\downarrow{}^{\mathrlap{\mu_{V_2,V_1}}}
    \\
    id(V_1\otimes V_2)
      &\overset{id(\tau_{Deligne})}{\longrightarrow}& if(W \otimes V)
  }
$$

As before, this is equivalent to a condition on the signs picked up both ways, which reads:

$$
  (-1)^{ (n_1 + \sigma_1)(n_2 + \sigma_2) + n_2 \sigma_1}
  \;=\;
  (-1)^{ n_1 \sigma_2 + n_1 n_2 + \sigma_1 \sigma_2 }
  \,.
$$

Inspection shows that this is indeed the case:

The two signs of $\tau_{Deligne}$ and $\tau_{Bernst}$ differ by the "mixed terms" that are produced in [[distributivity|multiplying out]] $(n_v + \sigma_v)(n_w + \sigma_w)$ and these two mixed terms is just what the two occurences $\mu$ provides (using that $+1 = -1 \in \mathbb{F}_2$).

=--

$\,$

### Equivalence of super Lie algebra cohomology in both conventions
 {#EquivalenceOfSuperLieAlgebraCohomology}

The different choice of signs in the de Rham complex
[above](#DifferentialFormsInSuperOddConvention) affects the signs
of any super-[[Chevalley-Eilenberg algebra]], by the identification of elements in the CE-algebra with [[left invariant differential forms]] on the supegroup.

We show now that as [[chain complexes]] the two CE-algebras are [[quasi-isomorphism|quasi-isomorphic]] and hence do have the same [[cochain cohomology]], hence that the definition of super-[[Lie algebra cohomology]] is not affected by the different perspective on signs.

(This follows abstract from Prop. \ref{EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect}, but we spell it out concretely.)

A chain-isomorphism can be obtained as follows:

Write

* $CE^\bullet$ for the [[cochain complex]] underlying the [[Chevalley-Eilenberg algebra]] with the signs given by

  $$
    ((n_1, \sigma_1), (n_2, \sigma_2)) \mapsto (-1)^{n_1 n_2 + \sigma_1 \sigma_2}
  $$

* write $CE^\bullet_{\mathbb{Z}_2}$ for the cochain complex obtained with the sign rule

  $$
    ((n_1, \sigma_1), (n_2, \sigma_2)) \mapsto (-1)^{(n_1 + \sigma_1)(n_2 + \sigma_2)}
    \,.
  $$

Say that a product of generators representing an element in $CE^k$ is in _normal order_ of all the $(\bullet,even)$-graded elements are left of the $(\bullet,odd)$-graded elements, schematically

$$
  e \wedge \cdots e \wedge \psi \wedge \cdots \psi
  \,.
$$

To define a [[linear map]] $CE^k  \longrightarrow CE^\bullet_{\mathbb{Z}_2}$ declare that elements in this normal form go to the elements given by the same expression.

This defines a linear map in each degree $k$. Notice that this is _not_ a morphism of algebras, in any sense, but it _is_ a [[chain map]]. To see this, notice that acting with the [[differential]] which is of degree $(1,even)$ on a term of the form $e \wedge \cdots e \wedge \psi \wedge \cdots \psi$ produces a sum of such terms where in each summand one generator is replaced by a term of the form $d e$ or by $d \psi$, respectively. These are in turn themselves sums of elements of the form $e \wedge \cdots e \wedge \psi \wedge \cdots \psi$. By commuting them to the middle we see that the differential of the whole term is a sum of terms of the form

$$
  e \wedge \cdots \wedge e \wedge (d e) \wedge \psi \wedge \cdots \wedge \psi
$$

and

$$
  e \wedge \cdots \wedge e \wedge (d \psi)  \wedge \psi \wedge \cdots \wedge \psi
  \,.
$$

Now observe:

1. for terms of the first type, the signs picked up by commuting $d e$ to the middle are by definition the same in $CE^\bullet$ and in $CE^\bullet_{\mathbb{Z}_2}$;

1. for terms of the second type there are a priori more signs picked up in $CE^\bullet$ than in $CE^\bullet_{\mathbb{Z}_2}$, namely one for every term $\psi$ that one commutes through. But as the differential first passes from the left into the string of $\psi$s and then $d \psi$ in turn is passed back to the left, these extra signs appear in even number and hence cancel out.

In summary this establishes a [[chain map]] $CE^\bullet \longrightarrow CE^\bullet_{\mathbb{Z}_2}$ which is an cochain [[isomorphism]] and hence in particular a [[quasi-isomorphism]], hence an [[isomorphism]] on [[cochain cohomology]].

## References

### Internalization sign rule

The [[internalization]] into the [[topos]] over [[superpoints]] is due to

* {#Schwarz84} [[Albert Schwarz]], _On the definition of superspace_, Teoret. Mat. Fiz. (1984)  Volume 60,  Number 1, Pages 37–42, ([russian original pdf](http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf))

The internalization sign rule follows for instance from applying the construction of [[internalization|internal]] [[Grassmann algebras]] in the [[category]] of [[super vector space]] as discussed in

* {#Deligne90} [[Pierre Deligne]], p. 165  _[[Catégories Tannakiennes]],  [[Grothendieck Festschrift]], vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.

The corresponding internal sign rule appears for instance in

* {#BonoraBregolaLechnerPastiTonin87} [[Loriano Bonora]], M. Bregola, [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], section 2 of _Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions_, Nuclear Physics B
Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], equations (II.2.106) and (II.2.109) in _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* {#DeligneFreed99} [[Pierre Deligne]], [[Dan Freed]], _Sign manifesto_, 1999 ([pdf](http://publications.ias.edu/sites/default/files/79_SignManifesto.pdf), [[DeligneFreedSignManifesto.pdf:file]])

  which appears in

  * [[Pierre Deligne]], [[Daniel Freed]], _Supersolutions_ ([arXiv:hep-th/9901094](http://arxiv.org/abs/hep-th/9901094)).

    which in turn appears in

    * [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], [[John Morgan]], [[David Morrison]], [[Edward Witten|E. Witten]] (eds.)  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

And with explicit acknowledgement that there is also the alternative super odd sign rule in:

* {#DeligneMorgan99} [[Pierre Deligne]], [[John Morgan]], _Notes on supersymmetry_, remark 1.2.8 in [[NotesOnNotesOnSupersymmetry.pdf:file]]

  in [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], [[John Morgan]], [[David Morrison]], [[Edward Witten|E. Witten]] (eds.)  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

* {#Freed01} [[Daniel Freed]], bottom of p. 48 in _Classical field theory and Supersymmetry_, IAS/Park City Mathematics Series Volume 11, 2001 ([pdf](https://www.ma.utexas.edu/users/dafr/pcmi.pdf))

### Super odd sign rule

* {#CarchediRoytenberg12} [[David Carchedi]], [[Dmitry Roytenberg]],  _Homological Algebra for Superalgebras of Differentiable Functions_ ([arXiv:1212.3745](https://arxiv.org/abs/1212.3745))

...


Sigurdur Helgason

* Wikipedia entry

* [personal page](https://math.mit.edu/~helgason/)

* [institute page](https://math.mit.edu/directory/profile.html?pid=98)

## Selected writings

On [[differential geometry]], [[geometric analysis]] and [[symmetric spaces]]:

* [[Sigurdur Helgason]], *Geometric Analysis on Symmetric Spaces*,  Mathematical Surveys and Monographs **39** (1994) [[doi:10.1090/surv/039](https://doi.org/10.1090/surv/039)]

* Sigurdur Helgason, _Differential geometry, Lie groups and symmetric spaces_

* S. Helgason, _Group representations and symmetric spaces_, Proc. ICM. Nice 1970, vol. 2, 313-320, [pdf](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0313.0320.ocr.pdf), [djvu](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0313.0320.ocr.djvu)

category: people


Silvia Nagy

* [institute page](https://www.durham.ac.uk/staff/silvia-nagy/)

* [spire page](https://inspirehep.net/authors/1320707)

## Selected writings

On [[Freudenthal magic square|magic squares]] of [[magic supergravities]]  and [[U-duality]] via the [[classical double copy]]:

* {#BorstenDuffHughesNagy13} [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic square from Yang-Mills squared*, Phys. Rev. Lett. **112** (2014) 131601  [[arXiv:1301.4176](http://arxiv.org/abs/1301.4176)]

* {#ABDHN} [[Alexandros Anastasiou]], [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic pyramid of supergravities*, JHEP (2014) 178 [[arXiv:1312.6523](http://arxiv.org/abs/1312.6523)]

category: people


Silvia Vaula

* webpage?

## related $n$Lab entries

* [[M-theory super Lie algebra]]

category: people


Simen Bruinsma

* [institute page](https://www.nottingham.ac.uk/mathematics/people/simen.bruinsma)

## Selected writings

On [[operads]] in [[homotopical algebraic quantum field theory]]:

* [[Simen Bruinsma]], _Coloring Operads for Algebraic Field Theory_ in: Proceedings of _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02863](https://arxiv.org/abs/1903.02863) [doi:10.1002/prop.201910004](https://doi.org/10.1002/prop.201910004))

More on [[homotopical AQFT]]:

* [[Simen Bruinsma]], [[Christopher J. Fewster]], [[Alexander Schenkel]], *Relative Cauchy evolution for linear homotopy AQFTs* ([arXiv:2108.10592](https://arxiv.org/abs/2108.10592))

category: people


similarity transformation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

In [[linear algebra]] and [[matrix calculus]], a _similarity transformation_ between quadratic [[matrices]] $A$ and $B$ is an [[general linear group|invertible matrix]] $S$, such that

$$
  B = S^{-1} A S
  \,.
$$

This is a special case of the concept of [[conjugation action]].

## References

* wikipedia, _[Matrix similarity](https://en.wikipedia.org/wiki/Matrix_similarity)_

[[!redirects similarity transformations]]

[[!redirects matrix similarity]]
[[!redirects matrix similarities]]


Simon Burton

* [webpage](https://iris.ucl.ac.uk/iris/browse/profile?upi=SBURT21)

* [GitHub page](https://github.com/punkdit)

## Selected writings

On [[anyons]] via [[braided fusion categories]]:

* [[Simon Burton]], *A Short Guide to Anyons and Modular Functors* [[arXiv:1610.05384](https://arxiv.org/abs/1610.05384)]

On [[fractional D-branes]] charges seen under [[schreiber:Hypothesis H]] in [[equivariant stable Cohomotopy]] of point (the [[Burnside ring]]):

* [[Simon Burton]], [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Lift of fractional D-brane charge to equivariant Cohomotopy theory]]*, Journal of Geometry and Physics **161** (2021) 104034 [[doi:10.1016/j.geomphys.2020.104034](https://doi.org/10.1016/j.geomphys.2020.104034), [arXiv:1812.09679](https://arxiv.org/abs/1812.09679)]

## Related entries

* [[Burnside ring]]

* [[representation ring]]

category: people


Simon Cho

* [website](http://math.lsa.umich.edu/~simoncho/)

### Related nLab pages

* [[continuous logic]]

category: people


Simon Covez

## Selected writings

On [[Leibniz algebras]] and their [[Lie integration]] to local Lie [[racks]]:

* [[Simon Covez]], _L'intégration locale des algèbres de Leibniz_, Thesis (2010) ([pdf](http://tel.archives-ouvertes.fr/docs/00/49/54/69/PDF/THESE_Simon_Covez.pdf))

* {#Covez10}  [[Simon Covez]], _The local integration of Leibniz algebras_,  Annales de l'Institut Fourier, Volume 63 (2013) no. 1, p. 1-35 ([arXiv:1011.4112](http://arxiv.org/abs/1011.4112), [doi:10.5802/aif.2754](https://doi.org/10.5802/aif.2754))

* [[Simon Covez]], _On the conjectural cohomology for groups_, Journal of K-theory __10__:03, Dec 2012, pp 519-563  ([arXiv:1202.2269](http://arxiv.org/abs/1202.2269), [doi:10.1017/is011011011jkt195](http://dx.doi.org/10.1017/is011011011jkt195))

category: people


Simon Donaldson

Simon Donaldson is professor for pure [[mathematics]] at Imperial College London.

* [website](http://www3.imperial.ac.uk/people/s.donaldson)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Simon_Donaldson)

## Selected writings
 {#SelectedWritings}

On [[Nahm's equations]] and the [[moduli space of monopoles]]:

* [[Simon Donaldson]], _Nahm's Equations and the Classification of Monopoles_, Comm. Math. Phys., Volume 96, Number 3 (1984), 387-407, ([euclid:cmp.1103941858](https://projecteuclid.org/euclid.cmp/1103941858))

On [[Yang-Mills theory]] ([[gauge theory]]) in [[mathematical physics]]:

* {#Donaldson05} [[Simon Donaldson]], _Yang-Mills theory and geometry_ (2005) [[pdf](https://www.ma.imperial.ac.uk/~skdona/YMILLS.PDF), [[Donaldson-YangMillsGeometry.pdf:file]]]

* {#Donaldson06} [[Simon Donaldson]], _Gauge Theory: Mathematical Applications_, Encyclopedia of Mathematical Physics, Academic Press (2006) 468-481 [[doi:10.1016/B0-12-512666-2/00075-4](https://doi.org/10.1016/B0-12-512666-2/00075-4), [author pdf](http://wwwf.imperial.ac.uk/~skdona/EMP.PDF), [[DonaldsonGaugeTheory.pdf:file]]]

## Related $n$Lab entries

* [[Donaldson-Thomas invariant]]

* [[Donaldson theory]]

category: people

[[!redirects S. Donaldson]]


Simon Forest

* [webpage](https://pageperso.lis-lab.fr/simon.forest/)

## Selected writings

* [[Simon Forest]], _Unifying notions of pasting diagrams_, Higher Structures, **6**, 2022 ([arxiv:1903.00282](https://arxiv.org/abs/1903.00282), [doi:10.21136/HS.2022.01](https://doi.org/10.21136/HS.2022.01))

category: people


Simon Friedrich

* [webpage](http://simonfriederich.eu/)

## related $n$Lab entries

* [[fine-tuning]]

category: people


Simon Henry

* [webpage](http://www.normalesup.org/~henry/)

## Selected writings

On [[measure theory]] via [[Boolean toposes]]:

* {#Henry14} [[Simon Henry]], _Measure theory over boolean toposes_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 163 Issue 1, 2016 ([arXiv:1411.1605](https://arxiv.org/abs/1411.1605), [doi:10.1017/S0305004116000700](https://doi.org/10.1017/S0305004116000700))

On [[formal category theory]]:

* [[Ivan Di Liberti]], [[Simon Henry]], [[Mike Lieberman]], [[Fosco Loregian]], _Formal category theory_, course notes (2017) [[pdf](https://tetrapharmakon.github.io/stuff/course_muni-formal.pdf), [[DLHLL-FormalCategoryTheory.pdf:file]]]

On [[weak model category|weak model categories]];

* [[Simon Henry]], _Weak model categories in classical and constructive mathematics_ ([arXiv:1807.02650](https://arxiv.org/abs/1807.02650))

* [[Simon Henry]], _Combinatorial and accessible weak model categories_ ([arXiv:2005.02360](https://arxiv.org/abs/2005.02360))

On a [[constructive model structure on simplicial sets]]:

* {#Henry19} [[Simon Henry]], _A constructive account of the Kan-Quillen model structure and of Kan's $Ex^\infty$ functor_ ([arXiv:1905.06160](https://arxiv.org/abs/1905.06160))

On the [[locale of real numbers]]:

* [[Madeleine Birchfield]], [[Simon Henry]], *The field structure on the locale of real numbers* (2022) [[MO:q/434706](https://mathoverflow.net/q/434706/381)]

On the [[homotopy hypothesis]] for [[Grothendieck infinity-groupoid|Grothendieck 3-groupoids]]:

* {#HenryLanari} [[Simon Henry]], [[Edoardo Lanari]], *On the homotopy hypothesis in dimension 3*, Theory and Applications of Categories **39** 26 (2023) 735-768 [[arxiv/1905.05625](https://arxiv.org/abs/1905.05625), [tac:39-26](http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/39/26/39-26.pdf)].

## Related entries

* [[Boolean topos]]

* [[measurable space]]

* [[modular theory]]

* [[weak model structure on semi-simplicial sets]]

category: people


Simon Huber

* [webpage](http://www.cse.chalmers.se/~simonhu/)

## related $n$Lab entries

* [[cubical set]]

* [[cubical type theory]]

category: people


Simon J. Devitt

## Selected writings

On [[quantum error correction]]:

* {#DevittNemotoMunro09} [[Simon J. Devitt]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Error Correction for Beginners*, Rep. Prog. Phys. **76** (2013) 076001 [[arXiv:0905.2794](https://arxiv.org/abs/0905.2794)]

On [[classically controlled quantum computation]]:

* {#Devitt14} [[Simon J. Devitt]], *Classical Control of Large-Scale Quantum Computers*, In: S. Yamashita, S. Minato  (eds.) *Reversible Computation* Lecture Notes in Computer Science, vol 8507. Springer 2014 ([arXiv:1405.4943](https://arxiv.org/abs/1405.4943), [doi:10.1007/978-3-319-08494-7_3](https://doi.org/10.1007/978-3-319-08494-7_3))

On [[classically controlled quantum computation]] for [[quantum error correction]]:

* {#PalerHerrDevitt19} Alexandru Paler, Daniel Herr, [[Simon J. Devitt]], *Really Small Shoe Boxes - On Realistic Quantum Resource Estimation*, Computer **52** 6 (2019) [[arXiv:1902.08104](https://arxiv.org/abs/1902.08104), [doi:10.1109/MC.2019.2908621](https://doi.org/10.1109/MC.2019.2908621)]

category: people

[[!redirects Simon Devitt]]


Simon John Ambler

* [institute page](https://www.cs.le.ac.uk/people/sambler/)

## Selected writings

On [[linear logic]]/[[linear type theory]] with [[categorical semantics]] in [[symmetric monoidal category|symmetric]] [[closed monoidal categories]]:

* [[Simon John Ambler]], *First Order Linear Logic in Symmetric Monoidal Closed Categories*, PhD thesis, Edinburgh (1991) [[ECS-LFCS-92-194](https://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194/ECS-LFCS-92-194.pdf), [[Ambler-FOLL.pdf:file]]]

category: people

[[!redirects Simon J. Ambler]]
[[!redirects Simon Ambler]]


Simon Kochen

Simon Kochen is Professor and Associate Chair in the Department of Mathematics at Princeton University.

* [Wikipedia entry](https://en.wikipedia.org/wiki/Simon_B._Kochen)

* [Faculty page](https://dof.princeton.edu/about/clerk-faculty/emeritus/simon-bernard-kochen)

## Selected writings

On [[model theory]] and introducing the notation for "$\vDash$-[[judgements]]" (cf. [there](judgment#History)):

* {#Kochen61} [[Simon Kochen]], p. 223 in: *Ultraproducts in the Theory of Models*, Ann. Math.  **74** 2 (1961) 221-261 [[doi:10.2307/1970235](https://doi.org/10.2307/1970235)]

Introducing the [[Kochen-Specker theorem]]:

* {#KochenSpecker67} [[Simon Kochen]], [[Ernst Specker]], _The problem of hidden variables in quantum mechanics_, Journal of Mathematics and Mechanics **17** (1967) 59-87 [[pdf](http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1968/17/17004/pdf), [jstor:24902153](https://www.jstor.org/stable/24902153), [doi:10.1007/978-94-010-1795-4_17](https://doi.org/10.1007/978-94-010-1795-4_17)]

Strengthening of the KS theorem and relating it to the "[[free will theorem]]":

* [[Simon Kochen]], *On the Free Will Theorem* [[arXiv:2207.06295](https://arxiv.org/abs/2207.06295)]

## Related entries

* [[Kochen-Specker theorem]]

* [[Timeline of category theory and related mathematics]]

category: people

[[!redirects Simon B. Kochen]]


Simon L. Peyton Jones

* [personal page](https://simon.peytonjones.org/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Simon_Peyton_Jones)

## Selected writings

Introducing the [[IO-monad]] (then adopted in [[Haskell]]):

* [[Simon L. Peyton Jones]], [[Philip Wadler]], *Imperative functional programming*, Principles of programming languages (1993) 71–84 [[doi:10.1145/158511.158524](https://doi.org/10.1145/158511.158524), [pdf](https://www.microsoft.com/en-us/research/wp-content/uploads/1993/01/imperative.pdf)]

On the history of the development of the [[Haskell]] [[programming language]]:

* [[Paul Hudak]], [[John Hughes]], [[Simon Peyton Jones]], [[Philip Wadler]], *A history of Haskell: being lazy with class*, History of Programming Languages (2007) 12-1 - 12-55 [[doi:10.1145/1238844.1238856](https://doi.org/10.1145/1238844.1238856)]

category: people

[[!redirects Simon Peyton Jones]]


Simon Lentner

* [personal page](http://simon.lentner.net/)

* [institute page](https://grk1670.math.uni-hamburg.de/researchers/lentner/)

## Selected writings

On [[braided monoidal category|braided]] [[tensor categories]] arising as [[representation categories]] of [[logarithmic CFTs|logarithmic]] [[vertex operator algebras]]:

* [[Thomas Creutzig]], [[Simon Lentner]], [[Matthew Rupert]], *Characterizing braided tensor categories associated to logarithmic vertex operator algebras* ([arXiv:2104.13262](https://arxiv.org/abs/2104.13262))

category: people


Simon McNamara

* [arXiv page](https://arxiv.org/search/hep-th?searchtype=author&query=McNamara%2C+S)

## Selected writings

On relation of [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[knot invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative Field Theory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

category: people


Simon Perdrix

* [personal page](https://members.loria.fr/SPerdrix/)

* [GoogleScholar page](https://scholar.google.com/citations?user=HDjzFscAAAAJ&hl=en)

## Selected writings

Inroducing the [[ZX-calculus]] as a tool for organizing [[measurement-based quantum computation]]:

* [[Ross Duncan]], [[Simon Perdrix]], *Rewriting Measurement-Based Quantum Computations with Generalised Flow*, in: *Automata, Languages and Programming. ICALP 2010*, Lecture Notes in Computer Science **6199**, Springer (2010) [[doi:10.1007/978-3-642-14162-1_24](https://doi.org/10.1007/978-3-642-14162-1_24)]

category: people


Simon Plouffe

* [Wikipedia entry](https://en.wikipedia.org/wiki/Simon_Plouffe)

## Selected writings

On efficient representations in [[exact real computer arithmetic]] of [[pi]] and related [[irrational numbers]] ([[Bailey-Borwein-Plouffe algorithm]]):

* [[David Bailey]], [[Peter Borwein]], [[Simon Plouffe]]: *On the Rapid Computation of Various Polylogarithmic Constants*, Math. Comp. **66** (1997) 903-913 [[doi:1997-66-218/S0025-5718-97-00856-9](https://www.ams.org/journals/mcom/1997-66-218/S0025-5718-97-00856-9)]

* [[Simon Plouffe]], *On the computation of the $n^{th}$ decimal digit of various transcendental numbers* [[arXiv:0912.0303](https://arxiv.org/abs/0912.0303)]

category: people


Simon Ross

## Selected writings

On non-uniqueness of [[black rings]]:

* [[Iosif Bena]], [[Simon Ross]], [[Nicholas Warner]], _Coiffured Black Rings_, Class.Quant.Grav. 31 (2014) 165015 ([arXiv:1405.5217](https://arxiv.org/abs/1405.5217))

category: people


Simon Salamon

* [webpage](http://calvino.polito.it/~salamon/)

## Selected writings

On [[twistor fibrations]]:

* [[Simon Salamon]], _Harmonic and holomorphic maps_,  In: E. Vesentini  (eds.) _Geometry Seminar "Luigi Bianchi" II - 1984_, Lecture Notes in Mathematics, vol 1164. Springer 1985 ([doi:10.1007/BFb0081912](https://doi.org/10.1007/BFb0081912))

* [[James Eells]], [[Simon Salamon]], Section 8 of: _Twistorial construction of harmonic maps of surfaces into four-manifolds_,  Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 4, p. 589-640 ([numdam:ASNSP_1985_4_12_4_589_0](http://www.numdam.org/item/ASNSP_1985_4_12_4_589_0))

* John Armstrong, [[Simon Salamon]], _Twistor Topology of the Fermat Cubic_, SIGMA 10 (2014), 061, 12 pages ([arXiv:1310.7150](https://arxiv.org/abs/1310.7150))

On the [[metric cone over complex projective 3-space]] and the [[twistor fibration]]:

* [[Bobby Acharya]], [[Robert Bryant]], [[Simon Salamon]], _A circle quotient of a $G_2$ cone_, Differential Geometry and its Applications Volume 73, December 2020, 101681  ([arXiv:1910.09518](https://arxiv.org/abs/1910.09518), [doi:10.1016/j.difgeo.2020.101681](https://doi.org/10.1016/j.difgeo.2020.101681))

On [[quaternion-Kähler manifolds]]:

* {#Salamon82} [[Simon Salamon]], _Quaternionic Kähler manifolds_, Invent Math (1982) 67: 143. ([doi:10.1007/BF01393378](https://doi.org/10.1007/BF01393378))

* {#Salamon86} [[Simon Salamon]], _Differential Geometry of Quaternionic Manifolds-, Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55.

* {#PoonSalamon91} Y. S. Poon, [[Simon Salamon]], _Quaternionic Kähler 8-manifolds with positive scalar curvature_, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 ([euclid:1214446322](https://projecteuclid.org/euclid.jdg/1214446322))

* {#LeBrunSalamon94} Claude LeBrun, [[Simon Salamon]], _Strong rigidity of positive quaternion Kähler manifolds_, Inventiones Mathematicae 118, 1994, 109–132 ([dml:144231](https://eudml.org/doc/144231), [doi:10.1007/BF01231528](https://doi.org/10.1007/BF01231528))

On [[SU(2)-structure]] on [[5-manifolds]]:

* {#ContiSalamon05} Diego Conti, [[Simon Salamon]], _Generalized Killing spinors in dimension 5_, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5319-5343 ([arXiv:math/0508375](https://arxiv.org/abs/math/0508375))

* Oscar Garcia-Prada, Jean Pierre Bourguignon, [[Simon Salamon]] (eds.) _The Many Facets of Geometry: A Tribute to Nigel Hitchin_, Oxford University Press 2010 ([doi:10.1093/acprof:oso/9780199534920.001.0001](http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199534920.001.0001/acprof-9780199534920))

On the construction of [[G2-manifolds]]:

* [[Robert Bryant]], [[Simon Salamon]],  _On the construction of some complete metrics with exceptional holonomy_, Duke Mathematical Journal 58: 829–850 (1989) ([euclid:dmj/1077307681](https://projecteuclid.org/euclid.dmj/1077307681))

## Related $n$Lab pages

* [[SU(2)-structure]] on [[5-manifolds]]

* [[quaternion-Kähler manifold]]

* [[Wolf space]]

* [[exceptional geometry]]

* [[G2]]

* [[bundle gerbe]]

* [[motives in physics]], [[motivic quantization]]

* [[sigma-model]]

* [[super line 2-bundle]]

* [[Chern-Simons theory]]

category: people


Simon Saunders

* [personal page](https://users.ox.ac.uk/~lina0174/Saunders.html)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Simon_Saunders)

* [institute page](https://www.ox.ac.uk/news-and-events/find-an-expert/professor-simon-saunders)

## Selected writings

On the [[many-worlds interpretation of quantum mechanics]]:

* [[Simon Saunders]], *Chance in the Everett Interpretation*, in: *Many Worlds?*, Oxford University Press (2010) 181–205 [[doi:10.1093/acprof:oso/9780199560561.003.0008](https://doi.org/10.1093/acprof:oso/9780199560561.003.0008)]

category: people


Simon Thompson

* [personal page](https://www.kent.ac.uk/computing/people/3164/thompson-simon)

* [GoogleScholar page](https://scholar.google.com/citations?user=twtKi84AAAAJ&hl=en)

## Selected writings

On [[type theory]] and [[functional programming]]:

* [[Simon Thompson]], *[[Type Theory and Functional Programming]]*, Addison-Wesley (1991) [ISBN:0-201-41667-0, [webpage](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP), [pdf](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf)]

Introduction to [[functional programming]] with examples in [[Haskell]]:

* [[Simon Thompson]], *Haskell: the Craft of Functional Programming*, Addison-Wesley (1996) [[webpage](https://www.haskellcraft.com/craft3e/Home.html), [pdf](https://simonjohnthompson.github.io/craft3e/craft3e.pdf)]

category: people


Simon Wassermann

* [institute page](https://www.gla.ac.uk/schools/mathematicsstatistics/staff/awassermann/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=71380)

## Selected writings

On [[continuous bundles of C-star algebras]]:

* [[Eberhard Kirchberg]], [[Simon Wassermann]], *Operations on continuous bundles of $C^\ast$-algebras*, Mathematische Annalen **303** 4 (1995) 677-698 [[eudml:165387](https://eudml.org/doc/165387), [[KirchbergWassermann-ContBndlsOfCStarAlgs.pdf:file]]]

category: people


Simon Willerton

* [webpage](http://www.simonwillerton.staff.shef.ac.uk/)

## Selected writings

On [[Vassiliev invariants]] of [[knots]] and [[braids]]:

* [[Simon Willerton]], _On the Vassiliev invariants for knots and for pure braids_, 1997 ([hdl:1842/11581](http://hdl.handle.net/1842/11581), [ethos.663801](https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.663801), [[WillertonVasdilievInvariants1997.pdf:file]])

On [[knot invariants]] via [[Feynman amplitudes]] in [[Chern-Simons theory]] ([[Vassiliev invariants]] on [[Jacobi diagrams]]):

* {#Willerton99} [[Simon Willerton]], _The Kontsevich integral and algebraic structures on the space of diagrams_ in: Knots in Hellas ’98. Series on Knots and Everything, vol. 24, World Scientific, 2000, 530–546 ([arXiv:math/9909151](https://arxiv.org/abs/math/9909151))

On [[Rozansky-Witten weight systems]] as [[Lie algebra weight systems]] for [[Lie algebra objects]] in the [[derived category of quasi-coherent sheaves]], and unified [[Wheels theorem]]:

* {#RobertsWillerton} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

category: people


Simon Wood

* [personal page](https://www.simonwood.eu/)

* [Institute page](https://www.cardiff.ac.uk/people/view/466630-Dr-Simon-Wood)

## Selected writings

On the [[Verlinde formula]] for [[logarithmic CFTs]]:

* [[David Ridout]], [[Simon Wood]], *The Verlinde formula in logarithmic CFT*,  J. Phys.: Conf. Ser. **597** 012065 $[$[arXiv:1409.0670](https://arxiv.org/abs/1409.0670), [doi:10.1088/1742-6596/597/1/012065](https://doi.org/10.1088/1742-6596/597/1/012065)$]$

Survey of [[algebra|algebraic]] [[structures]] in [[2d conformal field theory]] ([[vertex operator algebras]], [[conformal blocks]], [[modular tensor category|modular]] [[fusion categories]], [[modular functors]], [[FRS theorem]], [[string-net models]]):

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

category: people


Simona Paoli

Simona Paoli is a Senior Lecturer in Mathematics Institute, University of Aberdeen, Aberdeen U.K.

* [Aberdeen profile](https://www.abdn.ac.uk/people/Simona.Paoli).

* [Personal webpage](https://sites.google.com/view/simona-paoli-webpage)

## Selected writings

On [[higher category theory]] in terms of [[n-fold categories]] and comparison with the Tamsamani Simpson model:

* [[T. M.  Fiore]], S. Paoli, [[D. A.  Pronk]],  _Model structures on the category of small double categories_, Algebraic and Geometric Topology, 8 (2008) 1855-1959.

* S. Paoli,  _Weakly globular $cat^n$-groups and Tamsamani’s model_, Advances in Mathematics, 222, (2009) 621-727.

* [[D. Blanc]], S.Paoli, _Comonad cohomology of track categories_, Journal of Homotopy and Related Structures, published online 14 May 2019,  arXiv.1801.05065

Specifically on a [[Thomason model structure]] on [[n-categories]] (cf. [[model structure on DblCat]]):

* {#FPPModel} [[Thomas M. Fiore]], [[Simona Paoli]], [[Dorette A. Pronk]], *Model structures on the category of small double categories*, _Algebraic & Geometric Topology_ **8** (2008) 1855–1959 [[doi:10.2140/agt.2008.8.1855](https://doi.org/10.2140/agt.2008.8.1855)]

* [[Thomas M. Fiore]], [[Simona Paoli]], *A Thomason model structure on the category of small $n$-fold categories*, Algebr. Geom. Topol. **10** 4 (2010) 1933-2008 [[arXiv:0808.4108](http://arxiv.org/abs/0808.4108), [doi:10.2140/agt.2010.10.1933](https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-10/issue-4/A-Thomason-model-structure-on-the-category-of-small-nfold/10.2140/agt.2010.10.1933.full)]

Introducing [[weakly globular n-fold categories]]:

* [[Simona Paoli]], *Weakly globular $N$-fold categories as a model of weak $N$-categories*, in: *Simplicial Methods for Higher Categories*, Algebra and Applications **26** Springer (2019)  [[arXiv:1609.04072](https://arxiv.org/abs/1609.04072), [doi:10.1007/978-3-030-05674-2_12](https://doi.org/10.1007/978-3-030-05674-2_12)]

* [[Simona Paoli]], _Simplicial Methods for Higher Categories -- Segal-type Models of Weak $n$-Categories_, Springer (2019)  [[doi:10.1007/978-3-030-05674-2](https://doi.org/10.1007/978-3-030-05674-2), [toc pdf](https://link.springer.com/content/pdf/bfm%3A978-3-030-05674-2%2F1.pdf)]

category: people


Simone Gutt

* [webpage](http://homepages.ulb.ac.be/~sgutt/)

## Selected writings

On the [[formal deformation quantization]] of the [[cotangent bundle]] of a [[Lie group]], [[fiber]]-wise via [[universal enveloping algebras]]L

* {#Gutt83} [[Simone Gutt]], *An explicit $\ast$-product on the cotangent bundle of a Lie group*, Letters in Mathematical Physics **7** (1983) 249–258 [[doi:10.1007/BF00400441](https://doi.org/10.1007/BF00400441)]

On [[Poisson manifolds]] and their [[formal deformation quantization]]:

* {#Gutt11} [[Simone Gutt]], _Deformation quantization of Poisson manifolds_, Geometry and Topology Monographs **17** (2011) 171-220 [[gtm:17](https://msp.org/gtm/2011/17/p003.xhtml), [pdf](http://msp.org/gtm/2011/17/gtm-2011-17-003p.pdf), [arXiv:math/0003107](https://arxiv.org/abs/math/0003107)]

## Related entries

* [[metaplectic group]]

category: people


Simone Noja

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=231366)

## Selected writings

On [[superstring field theory]], [[supergeometry]], [[integration over supermanifolds]] of [[differential forms on supermanifolds]] and on [[picture changing operators]]:

* {#CatenacciGrassiNoja18} [[Roberto Catenacci]], [[Pietro Grassi]], [[Simone Noja]], _Superstring Field Theory, Superforms and Supergeometry_, Journal of Geometry and Physics Volume 148, February 2020, 103559 ([arXiv:1807.09563](https://arxiv.org/abs/1807.09563))

On [[Lie algebra cohomology]] of [[super Lie algebras]] (see also the [[Green-Schwarz action functional|brane scan]]) in relation to [[integration over supermanifolds|integrable forms]] of coset [[supermanifolds]]:

* [[Roberto Catenacci]], [[C. A. Cremonini]], [[Pietro Antonio Grassi]], [[Simone Noja]], _Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces_ ([arXiv:2012.05246](https://arxiv.org/abs/2012.05246))

On [[differential forms on supermanifolds]]:

* [[Simone Noja]], *On the Geometry of Forms on Supermanifolds* ([arXiv:2111.12841](https://arxiv.org/abs/2111.12841))

On [[D=3 supergravity]] via [[super Cartan geometry]] for [[super L-infinity algebras]]:

* [[Konstantin Eder]], [[John Huerta]], [[Simone Noja]], §16 in: *Poincaré Duality for Supermanifolds, Higher Cartan Geometry and Geometric Supergravity* [[arXiv:2312.05224](https://arxiv.org/abs/2312.05224)]

category: people


Simone Speziale

* [institute page](https://www.cpt.univ-mrs.fr/~speziale/index.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=U0gwxYMAAAAJ&hl=en)

## Selected writings

On [[discrete light cone quantization]]:

* [[Glenn Barnich]], Sucheta Majumdar, [[Simone Speziale]], Wen-Di Tan, *Lessons from discrete light-cone quantization for physics at null infinity: Bosons in two dimensions* [[arXiv:2401.14873](https://arxiv.org/abs/2401.14873)]

category: people


Simone Virili

[[!redirects Simone Vrili]]
[[!redirects Simone Vrili]]

* [GoogleScholar page](https://scholar.google.com/citations?user=TSbjyfUAAAAJ)

## Selected writings

On [[t-structures]] and [[reflective factorization systems]]:

* [[Fosco Loregian]], [[Simone Virili]] *Triangulated factorization systems and t-structures*, Journal of Algebra **550** (2020) 219-241 [[doi:10.1016/j.jalgebra.2019.12.021](https://doi.org/10.1016/j.jalgebra.2019.12.021)]

category: people


Simons Foundation

* [webpage](https://www.simonsfoundation.org/)

## Selected activities

On [[holography]] relating [[quantum information]] and [[quantum gravity]] (via [[holographic entanglement entropy]], holographic [[quantum error correcting codes]], ...):

* {#SimonsFoundationItFromQBit} *[It from Qubit: Simons Collaboration on Quantum Fields, Gravity and Information](https://www.simonsfoundation.org/mathematics-physical-sciences/it-from-qubit/)*

category: people


Simons-Sullivan structured bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What [Simons-Sullivan 08](#SimonsSullivan08) call a _structured bundle_ is a certain [[equivalence class]] of [[vector bundles]] with [[connection on a bundle|connection]], such that the corresponding [[Grothendieck group]] is a model for [[differential K-theory]].

The [[equivalence relation]] divided out is relation by a [[concordance]] such that the corresponding [[Chern-Simons form]] is exact.

## Details

Let $V \to X$ be a [[complex vector bundle]] with [[connection on a bundle|connection]] $\nabla$ and [[curvature]] 2-form

$$
  F = F_\nabla \in \Omega^2(X,End(V))
  \,.
$$

**Definition**

The **[[Chern character]]** of $\nabla$ is the inhomogenous [[curvature characteristic form]]

$$
 ch(\nabla) := \sum_{j \in \mathbb{N}}
   k_j
   tr( F_\nabla \wedge \cdots \wedge F_\nabla)
  \;\;
  \in \Omega^{2 \bullet}(X)
  \,,
$$

where on the right we have $j$ wedge factors of the [[curvature]] .

**Definition**

Let $(V,\nabla)$ and $(V',\nabla')$ be two complex
vector bundles with connection.

A [[Chern-Simons form]] for this pair is a differential form

$$
  CS(\nabla,\nabla') + d \omega \in
  \Omega^{2 \bullet + 1}(X)
$$

obtained from the [[concordance]] bundle $\bar V \to X \times [0,1]$ given by [[pullback]] along $X \times [0,1] \to X$ equipped with a connection $\bar \nabla$ such that ..., by

$$
  CS(\nabla,\nabla') =
  \int_0^1 \psi_t^* (\iota_{\partial/\partial t} ch(\bar \nabla))
  + d (...)
  \,.
$$

**Proposition** This is indeed well defined in that it is independent of the chosen [[concordance]], up to an exact term.

**Definition**

A **structured bundle** in the sense of the Simons-Sullivan model is a complex vector bundle $V$ equipped with the equivalence class $[\nabla]$ of a connection under the equivalence relation that identifies two connections $\nabla$ and $\nabla'$ if their Chern-Simons form $CS(\nabla,\nabla')$ is exact.

Two structured bundles are isomorphic if there is a vector bundle isomorphism under which the two equivalence classes of connections are identified.

**Definition**

Let $Struc(X)$ be the set of [[isomorphism]] classes of structured bundles on $X$.

Under [[direct sum]] and [[tensor product]] of vector bundles, this becomes a commutatve [[rig]].

Let
$$
  \hat K(X) := K(Struct(X))
$$

be the additive [[group completion]] of this rig as usual in [[K-theory]].

So as an additive group $\hat K(X)$ is the quotient of the [[monoid]] induced by [[direct sum]] on pairs $(V,W)$ of isomorphism classes in $Struc(X)$, modulo the sub-monoid consisting of pairs $(V,V)$.

Hence the pair $(V,0)$ is the additive inverse to $(0,V)$ and $(V,W)$ may be written as $V - W$.

**Theorem**

$\hat K(X)$ is indeed a [[differential cohomology]] refinement of ordinary K-theory $K(X)$ of $X$ (i.e. of the 0th [[cohomology group]] of [[K-theory spectrum|K-cohomology]]).

Moreover...

## Refereces

* {#SimonsSullivan08} [[James Simons]], [[Dennis Sullivan]], _Structured vector bundles define differential K-theory_ ([arXiv:0810.4935](http://arxiv.org/abs/0810.4935))

[[!redirects Simons-Sullivan structured bundles]]


SimpAb

**$\Simp\Ab$** is the category of simplicial abelian groups, i.e., of [[simplicial object]]s in the category [[Ab]] of [[abelian group]]s, i.e., it is the [[functor category]] of functors from the [[opposite category]] $\Delta^{op}$ of the [[simplex category]] $\Delta$ to $Ab$:

$$
  \Simp\Ab = [\Delta^{op}, Ab]
  \,.
$$

category: category

[[!redirects SAb]]


simple algebra

A [[unital algebra|unital]] [[associative algebra]] $A$ over a commutative ring $k$ is **simple** if it is it is a [[simple object]] in the category of $A$-$A$-[[bimodules]].

This can be stated in more elementary terms in any of the following equivalent ways:

*  $A$ is nontrivial and has no nontrivial two-sided [[ideals]].
*  $A$ has exactly two two-sided ideals (which must be $A$ itself and its [[zero ideal]]).

In [[constructive mathematics|constructive]] algebra, this is too strong; we must say:

*  For each two-sided ideal $I$, $I$ is the zero ideal if and only if $I$ is proper (not equal to $A$).

 By the Artin--Wedderburn theorem, any finite-dimensional simple algebra over $k$ is a matrix algebra with entries lying in some [[division algebra]] whose center is $k$.


simple foliation

A regular [[foliation]] $\mathcal{P} \hookrightarrow T X$ is called **simple** if the [[leaf space]] $X/\mathcal{P}$ is a [[smooth manifold]] and the [[quotient]] [[projection]] $X \to X/\mathcal{P}$ is a [[surjective submersion]].

Conversely, a simple foliation is a foliation by [[leaves]] of a [[surjective submersion]].

[[!redirects simple foliations]]


simple function

# Simple functions
* table of contents
{: toc}

## Idea

Simple functions are (almost) the most basic notion of [[measurable function]] in [[measure theory]].  Given a [[measure]], it\'s easy to define the [[integral]] of a simple function, and we extend this to more general functions by continuity.

## Definitions

Let $X$ be a [[measurable space]].  We may want $X$ to be equipped with some more data; if $X$ is a [[measure space]], then this is plenty of data.  However, for the most basic definitions, it\'s enough if $X$ is simply a measurable space.  This is the [[domain]] of our simple functions.

Another necessary datum is the simple functions\' [[codomain]] $K$, which we will eventually want to be at least a [[Banach space]] over the [[real numbers]].  (In the simplest example, $K$ is $\mathbb{R}$ itself, or perhaps the space $\mathbb{C}$ of [[complex numbers]].)  We take $K$ to be a measurable space using its [[Borel sets]].

+-- {: .num_defn #trad}
###### Definition

A [[measurable function]] from $X$ to $K$ is __simple__ if its [[range]] is [[finite set|finite]].
=--

Since a simple function $f$ is measurable and a [[singleton subset|singleton]] is Borel, each [[fibre]] of $f$ is a [[measurable set]] in $X$; the function $f$ is given by the (finitely many) nonempty fibres and their (singleton) images.  This suggests another way to look at simple functions:

+-- {: .num_defn #lincomb}
###### Definition

A __simple function__ from $X$ to $K$ is a formal $K$-[[linear combination]] of [[measurable subsets]] of $X$.
=--

Here we identify a measurable set $A$ with its [[characteristic function]] $\chi_A$, so the formal linear combination $\sum_i c_i A_i$ is identified with the function $\sum_i c_i \chi_{A_i}$, which is measurable and whose range is contained in the finite set of sums of the $c_i$.  (If there are $n$ terms in the linear combination, then there are at most $2^n$ such sums.)

However, the naïve notion of equality of linear combinations is finer than equality of the corresponding functions, so we must combine Definition \ref{lincomb} with a definition of equality:

+-- {: .num_defn #equal}
###### Definition

Two simple functions from $X$ to $K$, in the sense of Definition \ref{lincomb}, are __equal__ if their corresponding [[functions]] from $X$ to $K$ are equal as functions.
=--

Then we have a [[canonical transformation|canonical]] [[bijection]] between the set of simple functions as in Definition \ref{trad} and the set of [[equivalence classes]] of simple functions as in Definition \ref{lincomb}.

Arguably, even this is not really the correct notion of equality, since functions may be equal for the purpose of integration without being literally equal.  If $X$ is equipped with a $\sigma$-[[sigma-ideal|ideal]] of [[null sets]] (or a $\delta$-[[delta-filter|filter]] of [[full sets]]), then we may consider a yet coarser notion of equality:

+-- {: .num_defn #almostequal}
###### Definition

Two simple functions, in the sense of either Definition \ref{trad} or Definition \ref{lincomb}, are __almost equal__ if they (or their corresponding functions) are [[almost equality|equal almost everywhere]].
=--

Sometimes, we wish to restrict attention to those simple functions which we expect to have a finite integral.  If $X$ is equipped with an [[ideal]] of [[bounded sets]] (which in a measure space are sets with finite measure), then we may do this:

+-- {: .num_defn #bounded}
###### Definition

A __simple function of bounded support__ is a simple function in the sense of Definition \ref{trad} such that the fibre over every non-zero number is bounded, or equivalently (in the sense of Definition \ref{lincomb}) a formal linear combination of bounded measurable sets.
=--

In some approaches to [[measure theory]], one *starts* with a $\delta$-[[delta-ring|ring]] of measurable sets, which may be reinterpreted as the bounded sets in the generated $\sigma$-[[sigma-algebra|algebra]] of [[relatively measurable sets]], and then the simple functions will automatically have bounded support.

Finally, there is one more useful restriction (and slight generalisation) of simple functions, applicable when $K$ is ordered:

+-- {: .num_defn #positive}
###### Definition

A __positive simple function__ is a simple function in the sense of Definition \ref{trad} whose range is contained in the [[positive cone]] $K^+$ of $K$, or equivalently (in the sense of Definition \ref{lincomb}) a formal $K^+$-linear combination of measurable sets.  An __extended positive simple function__ (note the [[red herring]]) takes values in the [[extended positive cone]] $\bar{K}^+$, or equivalently is a $\bar{K}^+$-linear combination.
=--

## Integration

Let $X$ be equipped with a [[measure]] $\mu$, so $(X,\mu)$ is a [[measure space]].  (In particular, $X$ has the structure necessary for all of the definitions above, including both Definitions \ref{almostequal} and \ref{bounded}.)

If $f$ is a simple function from $X$ to $K$, then we wish to define the [[integral]] of $f$.  In general, this is a little tricky, but it\'s easy if $f$ either is [positive](#positive) or has [bounded support](#bounded).  It is easiest to write down the definition if we think of simple functions using Definition \ref{lincomb}.  Then we have:

+-- {: .num_defn #integral}
###### Definition

The __integral__ of the simple function $f$, represented by the linear combination $\sum_i c_i A_i$, is $\sum_i c_i \mu(A_i)$.
=--

+-- {: .num_prop #intpositive}
###### Proposition

The integral of a positive simple function always exists (but may be infinite).  It is finite if $\mu$ is a [[finite measure]], and it is positive (possibly $0$ or $\infty$) if $\mu$ is a [[positive measure]].  Also, if $\mu$ is positive, then the integral of an extended positive simple function always exists.
=--

(However, the integral of an extended positive simple function with respect to a finite positive measure need not be finite.)

+-- {: .num_prop #intbounded}
###### Proposition

The integral of a simple function with bounded support always exists and is finite (being a finite linear combination of finite numbers).
=--

+-- {: .num_prop #intequal}
###### Proposition

Two (positive or with bounded support) simple functions $f$ and $g$ are almost equal (with respect to $\mu$) if and only if the integral of $f - g$ is zero.
=--

+-- {: .num_defn #norm}
###### Definition

The __$L^1$-norm__ of a simple function is the integral of its pointwise norm (which is a positive simple function to $\mathbb{R}$) with respect to the [[absolute value]] of the measure $\mu$ (which is a positive measure):
$$ {\|f\|}_1 \coloneqq \int {\|f(x)\|} {|\mu(\mathrm{d}x)|} .$$
=--

In this context, we usually start with a positive measure $\mu$; in that case, of course, there is no need to bother taking the absolute value of $\mu$.

+-- {: .num_prop #NVS}
###### Proposition

The simple functions of bounded support form a [[normed vector space]] $Simp_c$ under the $L^1$-norm, if we consider them up to [almost equality](#almostequal).
=--

If we don\'t use almost equality, then we get in general only a [[seminorm]], but if we pass to a [[quotient space]] with a norm, then Proposition \ref{intequal} tells us that we are now using almost equality (and shows that Definition \ref{integral} is well defined when applied to Definition \ref{trad}).

+-- {: .num_defn #L1}
###### Definition

The [[complete space|completion]] of the normed vector space $Simp_c$ (under the $L^1$-norm) is the [[Banach space]] $L^1$ of __[[absolutely integrable functions]]__ (an example of a [[Lebesgue space]]).
=--

+-- {: .num_prop #integration}
###### Proposition

Taking the integral of a simple function of bounded support is a [[continuous linear functional]] on $Simp_c$, so it extends to all of $L^1$.
=--

In this way, we may define the integral of any absolutely integrable function.

+-- {: .query}
There might be some technical requirements for this to be true.  I\'ll try to check on that.
=--

[[!redirects simple function]]
[[!redirects simple functions]]


simple group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

### Standard definition

A **simple group** is a [[group]] $G$ with exactly two [[quotient group]]s: the [[trivial quotient group]] $\{1\} \cong G/G$ and the group $G \cong G/\{1\}$ itself.

Equivalently, a simple group is a group possessing exactly two [[normal subgroup|normal subgroups]]: the [[trivial subgroup]] $\{1\}$ and the group $G$ itself. One can also say that a normal subgroup is trivial iff it is not $G$, or trivial iff proper (compare the definition in constructive mathematics below).

Note that the [[trivial group]] does not itself count as simple, on the grounds that it has only *one* quotient group (or only one normal subgroup).  It may be possible to find authors that use "at most" in place of "exactly", thereby allowing the trivial group to be simple.  (Compare [[too simple to be simple]].)

### In constructive algebra

In [[constructive mathematics]], we consider a group $G$ equipped with a [[tight apartness]] $\ne$ such that the group operations are [[strongly extensional function|strongly extensional]] and use the theory of [[antisubgroups]] (which classically are the complements of subgroups).  Then $G$ is __simple__ if, given any [[normal antisubgroup]] $N$ of $G$, $N$ is trivial iff it is proper.  Explicitly, this says $N$ owns every nonidentity element (every $x$ such that $x \ne 1$) iff $N$ is [[inhabited subset|inhabited]] (with some $x$ such that necessarily $x \ne 1$).  Replacing 'iff' with 'if' here would allow the trival group to be simple.

## Examples

### Finite simple groups

Simple groups are most commonly encountered in the theory of [[finite group]]s. Every finite group $G$ admits a [[composition series]], i.e., a finite filtration of subgroups

$$1 = G_0 \subseteq G_1 \subseteq \ldots \subseteq G_n = G$$

where each inclusion $G_i \subseteq G_{i+1}$ is a [[normal subgroup]] and the quotient $G_{i+1}/G_i$ (called a **composition factor**) is simple. The condition of simplicity means that that the filtration cannot be further refined by addition of strict inclusions of normal subgroups. Furthermore, the [[Jordan-Hölder theorem]] ensures that any two composition series have the same length and the same composition factors (up to permutation).

Thus finite simple groups are in some sense the primitive building blocks of finite groups generally. The massive program of classifying all finite simple groups was announced as completed by Daniel Gorenstein in 1983, although some doubts remained because there were some gaps in proofs. Most if not all the gaps are considered by experts in the area to have been filled, but there remain some notable skeptics, including for example [[Jean-Pierre Serre]], who said in an interview

> Whenever I asked the specialists, they replied something like: "Oh no,
  it is not a gap; it is just something which has not been written, but
  there is an incomplete unpublished 800-page manuscript on it."  For me,
  it was just the same as a "gap," and I could not understand why it was
  not acknowledged as such. ([Source](https://cs.nyu.edu/pipermail/fom/2008-November/013181.html))

and possibly also [[John H. Conway]], although according to [Joe Shipman](https://cs.nyu.edu/pipermail/fom/2008-November/013178.html),

> A few months ago I was discussing the COFSG with John Conway and he was still pessimistic. (Meaning of course that he was confident the classification was correct, "optimistic" in his case means there are previously undiscovered finite simple groups, because they're beautiful and interesting objects and it would be disappointing to have no more.)

See [[classification of finite simple groups]].

### Directed colimits

+-- {: .num_prop #directed}
###### Proposition
Let $S_\alpha$ be a directed system of simple groups and monomorphisms between them. Then $colim_\alpha S_\alpha$ is also simple.
=--

+-- {: .proof}
###### Proof
Suppose $N$ is a normal subgroup of $colim_\alpha S_\alpha$ with a non-identity element. Then $N_\alpha = N \cap S_\alpha$ is normal in $S_\alpha$ (if $x \in S_\alpha$ and $y \in N \cap S_\alpha$, then clearly $x y x^{-1}$ belongs to both $N$ and $S_\alpha$). As soon as $\alpha$ is large enough that $S_\alpha$ contains a non-identity element of $N$, it follows from simplicity of $S_\alpha$ that $N_\alpha = S_\alpha$. By directedness, this shows $N_\alpha = S_\alpha$ for all $\alpha$, and since for any $s \in S$ there is $\alpha$ such that $s \in S_\alpha$, we conclude $s \in S_\alpha = N_\alpha \subseteq N$, i.e., $N$ contains any $s \in S$.
=--

### Infinite simple groups

There are simple groups of any infinite [[cardinality]] $\kappa$; take for example the smallest normal subgroup of the [[automorphism group]] $Aut(\kappa)$ containing all 3-cycles (this is the infinite version of the [[alternating group]]).

To see that $Alt(\kappa)$ is simple, it is enough to observe that it is a directed colimit of $Alt(X)$ where $X$ ranges over finite subsets of $\kappa$ of cardinality at least $5$; then simplicity of $A_n$ for integers $n \geq 0$, coupled with Proposition \ref{directed}, yields the desired result.

A rather deeper set of examples is afforded by the following result, which follows from a theorem named after [Baer, Ulam, and Schreier](https://groupprops.subwiki.org/wiki/Baer-Schreier-Ulam_theorem).

+-- {: .num_theorem}
###### Theorem
For an infinite set $X$, every proper normal subgroup of the permutation group $Sym(X)$ is contained in a maximal such normal subgroup $N_X$, consisting of all permutations that move fewer than ${|X|}$ many elements of $X$, where ${|X|}$ is the cardinality of $X$.
=--

It follows that the quotient group $Q_X = Sym(X)/N_X$ is simple, and we have the following corollary.

+-- {: .num_cor}
###### Corollary
Every group embeds into a simple group.
=--

+-- {: .proof}
###### Proof
For finite groups $G$ (WLOG, of cardinality at least $3$), we have the Cayley embedding $G \hookrightarrow Sym(G)$ into the permutation group of the underlying set, and there is an embedding $Sym(G) \hookrightarrow Alt(G + 2)$ which carries an even permutation on $G$ to the obvious even permutation on $G + 2$ that fixes the elements $a, b$ of $2$, and an odd permutation $\pi$ on $G$ to the even permutation $\pi (a\; b)$. Hence $G$ embeds in a simple group $Alt(G + 2)$.

An infinite group $G$ embeds in $Q_G$ via the Cayley embedding

$$G \hookrightarrow Sym(G) \to Sym(G)/N_G = Q_G$$

noting that for any non-identity $g \in G$, the permutation $Cayley(g) = (h \mapsto g h)$ has no fixed points, hence does not belong to $N_G$, so that $G \to Q_G$ is indeed monic.
=--

#### Tarski monsters

Among the infinite simple groups, there are curious examples called "Tarski monsters" for primes $p$, infinite groups with the property that every subgroup is either trivial, of prime order $p$, or the entire group. Their existence (for all primes $p \gt 10^{75}$, according to [Wikipedia](https://en.wikipedia.org/wiki/Tarski_monster_group)) was established only in 1979, by Olshanskii. It may be shown that all such groups must be simple, and generated by two elements, and also that for each prime $p \gt 10^{75}$ there are continuum many monsters for $p$ that are distinct up to isomorphism.

[[!redirects simple group]]
[[!redirects simple groups]]


simple homotopy theory

# Simple homotopy theory
* table of contents
{: toc}

## Idea and a bit of history

The aim of much of [[homotopy theory]] in the 1930s and 1940s was to try to lay _combinatorial_ foundations for determining whether two [[topological spaces]] were of the same [[homotopy type]] and, if so, was it possible to build up the homotopy equivalences in some simple way.  The model for this idea came, it seems, from [[Tietze transformation|Tietze's theorem]] in the theory of group presentations, where any presentation of a group could be deformed into any other by a series of 'moves'.  The same process was apparent in the development of the classification theorem for surfaces.  The question thus was: could one find some simple set of 'moves' that would generate all the homotopy equivalences with a given domain, $X$?

The theory was initially developed by Reidemeister, and then [[Henry Whitehead|Whitehead]], culminating in his 1950 paper _Simple homotopy types_.  The theory received a further stimulus with [[John Milnor|Milnor]]'s classic paper in 1966 in which emphasis was put on [[elementary expansion]]s.

Further interesting approaches were developed by [[Eckmann]] and [[Siebenmann]] in 1970 (see references).

##Elementary expansions and contractions

We will work here with finite CW-complexes. These are built up by induction by gluing on $n$-cells, that is copies of $D^n=\{x\in \mathbb{R}^n\mid \sum x_i^2\leq 1\}$, at each stage.  Each $D^n$ has a boundary an $(n-1)$-sphere, $S^{n-1}=\{x\in \mathbb{R}^n\mid \sum x_i^2= 1\}$.  The construction of objects in the category of finite CW-complexes is by attaching cells by means of maps defined on part of all of the boundary of a cell.  This will usually change the homotopy type of the space, creating or filling in a 'hole'. The homotopy type will not be changed if the attaching map has domain a hemisphere.  We write $S^{n-1} = D^{n-1}_-\cup  D^{n-1}_+$, with each hemisphere homeomorphic to a $(n-1)$-cell, and their intersection being the equatorial $(n-2)$-sphere, $S^{n-2}$, of $S^{n-1}$.

Given, now, a finite CW-complex, $X$, we can build a new complex $Y$, consisting of $X$ and two new cells, $e^n$ and $e^{n-1}$ together with a continuous map, $\varphi : D^n\to Y$ satisfying

(i) $\varphi(D^{n-1}_+)\subseteq X_{n-1}$;

(ii) $\varphi(S^{n-2}) \subseteq X_{n-2}$;

(iii) the restriction of $\varphi$ to the interior of $D^n$ is a homeomorphism onto $e^n$;

and

(iv) the restriction of $\varphi$ to the interior of $D^{n-1}_-$ is a homeomorphism onto $e^{n-1}$.

There is an obvious inclusion map, $i: X\to Y$, which is called an _elementary expansion_.  There is also a retraction map $r : Y\to X$, homotopy inverse to $i$, and which is called an _elementary contraction_.  Both are [[homotopy equivalences]].

* Can all homotopy equivalences between finite CW-complexes be built by composing such elementary ones?

More precisely, if we have a homotopy equivalence $f: X\to X^\prime$, is $f$ homotopic to a composite of a finite sequence of elementary expansions and contractions?  Such a homotopy equivalence would be called _simple_.  Whitehead showed that not all homotopy equivalences are simple and constructed a group of obstructions for the problem with given space $X$, each non-identity element of the group corresponding to a distinct homotopy class of non-simple homotopy equivalence.

## References

Some original sources:

*  [[J. H. C. Whitehead]], _Simple homotopy type_, Amer. J. Math., 72,  (1950), 1 - 57.

* [[J. Milnor]], _Whitehead torsion_, Bull. Amer. math. Soc., 72 (1966), 358 - 426.

* [[B. Eckmann]], _Simple homotopy type and categories of fractions_, Symp. Math. V (1970), 285 - 299.

* [[B. Eckmann]] and S. Maumary, _Le groupe des types simples d'homotopie sur un polyèdre_, Essays on Topology and related topics, Mémoires dédiés à Georges de Rham, Springer (1970).

* [[L. C. Siebenmann]], _Infinite simple homotopy types_, Indag. math. 32, (1970), 479 - 495.

A very useful textbook is:

* M. M. Cohen, _A course in Simple Homotopy Theory_, Grad. Texts in Math, 10, Springer, 1973.

A more abstract, but at the same time geometric, approach to simple homotopy theory was explored in Cohen's book as well as in the papers by Eckmann, Eckmann and Maumary, and Siebenmann, listed above. Some of this is treated in

* K. H. Kamps, [[Tim Porter]], _Abstract homotopy and simple homotopy theory_, World Scientific 1997.
[[!redirects simple homotopy theory]]


simple Lie algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

# Simple Lie algebras
* table of contents
{: toc}

## Definition

A __simple Lie algebra__ is a [[Lie algebra]] $\mathfrak{g}$ such that:

*  $\mathfrak{g}$ is not [[abelian Lie algebra|abelian]], and
*  the only [[proper ideal]] of $\mathfrak{g}$ is the zero ideal.

Equivalently, a simple Lie algebra is a [[simple object]] of [[LieAlg]] that *also* is nonabelian.  Note that there are only two abelian Lie algebras whose only proper ideal is the zero ideal: the [[trivial Lie algebra]] (which is not a simple object in $Lie Alg$ either, since the zero ideal is not proper either) and the [[line]] (which is a simple object in $Lie Alg$ but is still not considered a simple Lie algebra).

## Classification

Simple Lie algebras over an [[algebraically closed field]] of [[characteristic]] [[zero]], like many other things in [[mathematics]], may be classified by [[Dynkin diagram]]s.  We have:

* $\mathfrak{a}_n = \mathfrak{sl}_{n+1}$, the [[special linear Lie algebra]] of rank $n$.  We count this only for $n \geq 1$, since $\mathfrak{a}_0$ is the [[trivial Lie algebra]] (which is not simple but is still [[semisimple Lie algebra|semisimple]]).

* $\mathfrak{b}_n = \mathfrak{so}_{2n+1}$, the odd-dimensional [[special orthogonal Lie algebra]] of rank $n$.  We count this only for $n \geq 2$, since $\mathfrak{b}_n = \mathfrak{a}_n$ for $n \lt 2$.

* $\mathfrak{c}_n = \mathfrak{sp}_n$, the [[symplectic Lie algebra]] of rank $n$.  We count this only for $n \geq 3$, since $\mathfrak{c}_n = \mathfrak{b}_n$ for $n \lt 3$.

* $\mathfrak{d}_n = \mathfrak{so}_{2n}$, the even-dimensional [[special orthogonal Lie algebra]] of rank $n$.  We count this only for $n \geq 4$, since $\mathfrak{d}_n = \mathfrak{a}_n$ for $n \lt 2$ and $n = 3$, while $\mathfrak{d}_2 = \mathfrak{a}_2 \oplus \mathfrak{a}_2$ (which is not simple but is still [[semisimple Lie algebra|semisimple]]).

* $\mathfrak{e}_n$, an [[exceptional Lie algebra]] that only exists for rank $n \lt 9$.  We count this only for $n \geq 6$ (thus for $n = 6, 7, 8$ in all), since $\mathfrak{e}_5 = \mathfrak{d}_5$, $\mathfrak{e}_4 = \mathfrak{a}_4$, $\mathfrak{e}_n = \mathfrak{a}_{n-1} \oplus \mathfrak{a}_1$ (which is not simple but is still [[semisimple Lie algebra|semisimple]]) for $2 \leq n \lt 4$, and $\mathfrak{e}_n = \mathfrak{a}_n$ for $n \lt 2$.

* the [[exceptional Lie algebra]]s $\mathfrak{f}_4$ and $\mathfrak{g}_2$, which exist only for those ranks.

If you want to classify simple objects in $Lie Alg$, then there is one other possibility: the [[line]] (which has no corresponding Dynkin diagram).

It is much more difficult to classify simple Lie algebras over non-closed fields, over fields with positive characteristic, and especially over non-fields.

## Semisimple Lie algebras

A __[[semisimple Lie algebra]]__ is a [[direct sum]] of simple Lie algebras.  In particular, every simple Lie algebra is semisimple, but there are many more.

## Simple Lie groups

A [[Lie group]] is a [[simple Lie group]] if the Lie algebra corresponding to it under [[Lie integration]] is simple.

## Related concepts

* [[exceptional Lie algebra]]

[[!redirects simple Lie algebra]]
[[!redirects simple Lie algebras]]


simple Lie group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

# Simple Lie groups
* table of contents
{: toc}

## Definition

A __simple Lie group__ is a non-[[abelian group|abelian]] [[connected space|connected]] [[Lie group]] $G$ with no nontrivial [[connected space|connected]] [[normal subgroups]].

Via [[Lie's theorems]] this is equivalent to the [[Lie algebra]] $\mathfrak{g}$ of $G$ being a [[simple Lie algebra]] (over the [[real numbers]]).

+-- {: .num_remark}
###### Remark

This is *not* the same thing as a [[simple object]] in [[LieGrp]] (or even in $Conn Lie Grp$).  However, the [[Lie algebra]] associated to a simple Lie group is always a [[simple Lie algebra]] (although that definition also requires a non-abelian clause).

=--

## Classification

The classification of simple Lie groups consists of four infinite series -- the [[classical Lie group]]s -- and five separate cases -- the [[exceptional Lie group]]s.

See Wikipedia\'s [list of simple Lie groups](http://en.wikipedia.org/wiki/List_of_simple_Lie_groups).

> graphics grabbed from [Schwichtenberg](http://jakobschwichtenberg.com/classification-of-simple-lie-groups/)

See also at _[[ADE classification]]_

## Examples

* The [[special unitary group]] $SU(N+1)$ is simple for all $N \geq 1$.

* The [[special orthogonal group]] $SO(n)$ is simple at least for $n \geq 8$ (corresponding to [[SO(8)]]), hence so are the corresponding [[spin groups]].

## References

* Notes on simple Lie algebras and Lie groups ([pdf](https://math.berkeley.edu/~ianagol/261A.F09/Simplegroups.pdf
))

See also

* Wikipedia, _[Simple Lie group](https://en.wikipedia.org/wiki/Simple_Lie_group)_

[[!redirects simple Lie group]]
[[!redirects simple Lie groups]]

[[!redirects classification of simple Lie groups]]


simple object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Definition

+-- {: .num_defn}
###### Definition

An [[object]] $X$ in a [[category]] $C$ with a [[terminal object]] $1$ is __simple__ if there are precisely two [[quotient objects]] of $X$, namely $1$ and $X$.

=--

+-- {: .num_remark}
###### Remark

If $C$ is [[abelian category|abelian]], then the terminal object is a [[zero object]] and we may use [[subobjects]] in place of quotient objects in the definition, and this is more common; the result is the same.

=--

+-- {: .num_remark}
###### Remark

The [[terminal object ]] itself is *not* simple, as it has only *one* quotient object.  It is [[too simple to be simple]].

=--

+-- {: .num_remark}
###### Remark

In [[constructive mathematics]], we want to phrase the definition as follow: A quotient object of $X$ is $X$ if and only if it is not $1$.

=--

+-- {: .num_defn}
###### Definition

An [[object]] which is a [[direct sum]] of simple objects is called a **[[semisimple object]]**.

=--

## Properties

### In an abelian category

\begin{proposition}
**([[Schur's lemma]])**

In an [[abelian category]] $C$, every [[morphism]] between simple objects is either a [[zero morphism]] or an [[isomorphism]], and the [[endomorphism]] [[algebra]] of any simple object is a [[division ring]].

If $C$ is also [[enriched category|enriched]] in [[finite-dimensional vector spaces]] over an [[algebraically closed field]] $k$, then $hom(X, Y)$ has [[dimension]] $0$ or $1$ for any pair of simple objects $X$ and $Y$.  In this case the endomorphism algebra of any simple object is $k$.
\end{proposition}

\begin{proof}
Suppose $X$ and $Y$ are simple objects in an abelian category $C$. If $f \colon X \to Y$ is any morphism then the kernel of $f$ must be either $0$ or $X$, while its cokernel must be $0$ or $Y$. If the kernel and cokernel are both $0$, $f$ is an isomorphism; otherwise $f = 0$. It follows that every element of the endomorphism ring $End(X)$ of a simple object $X$ is zero or invertible, so $End(X)$ is a division ring.

Next suppose $C$ is enriched over finite-dimensional vector spaces over an algebraically closed field $k$.   In this case $End(X)$ is a finite-dimensional division algebra over $k$, but any such algebra is isomorphic to $k$.   Post-composing with an isomorphism $f: X \to Y$ gives a vector space isomorphism $End(X) \to hom(X,Y)$, so if such an isomorphism $f$ exists then $hom(X,Y)$ is one-dimensional.  If no such isomorphism exists all the morphisms from $X$ to $Y$ are zero.
\end{proof}

+-- {: .num_remark}
###### Remark

If an abelian category is enriched over finite-dimensional vector spaces over a field $k$ that is not algebraically closed, the endomorphism algebra of a simple object can be a division algebra other than $k$.  For example consider $k = \mathbb{R}$.  In the category of real representations of the [[Lie group]] [[SO(2)]], the usual action of rotations on the plane gives a simple object (that is, [[irreducible representation]]) $X$ with $End(X) \cong \mathbb{C}$. In the category of real representations of [[Sp(1))]], the action of this group by right multiplication on the [[quaternions]] gives a simple object $X$ with $End(X) \cong \mathbb{H}$.

=--

## Examples

* In a [[top]] bounded  [[partial order]] its simple objects are its [[atom|coatoms]].

*  In the category [[Vect]] of [[vector spaces]] over some field $k$, the simple objects are precisely the [[line]]s: the $1$-dimensional vector spaces, i.e. $k$ itself, up to [[isomorphism]].

*  A [[simple group]] is a simple object in [[Grp]].  (Here it is important to use quotient objects instead of subobjects.)

*  For $G$ a [[group]] and $Rep(G)$ its [[category of representations]], the simple objects are the [[irreducible representations]].

*  A [[simple ring]] is a simple object in [[Ring]]. Equivalently, it is a ring $R$ that is simple in its category of [[bimodules]].

*  A [[simple Lie algebra]] is a simple object in [[LieAlg]] that (by conventional fiat) is not [[abelian Lie algebra|abelian]].  As an abelian Lie algebra is simply a [[vector space]], the only simple object of $Lie Alg$ that is not accepted as a simple Lie algebra is the $1$-dimensional Lie algebra.

## Related entries

* [[length of an object]]

* [[semisimple category]]

  * [[fusion category]]

  * [[modular tensor category]]

[[!redirects simple object]]
[[!redirects simple objects]]
[[!redirects simple module]]
[[!redirects simple modules]]

[[!redirects irreducible object]]
[[!redirects irreducible objects]]


simple ring

A [[ring]] $R$ is **simple** if it is it is a [[simple object]] in the category of $R$-$R$-[[bimodules]].

This can be stated in more elementary terms in any of the following equivalent ways:

*  $R$ is nontrivial and has no nontrivial two-sided [[ideal]]s.
*  $R$ has exactly two two-sided ideals (which must be $R$ itself and its [[zero ideal]]).

In [[constructive mathematics|constructive]] algebra, this is too strong; we must say:

*  For each two-sided ideal $I$, $I$ is the zero ideal if and only if $I$ is proper (not equal to $R$).

## Examples

* A [[field]], or a [[division ring]], is simple.

* If $D$ is a division ring, then the ring $M_n(D)$ of $n \times n$ [[matrices]] with entries in $D$ is a simple ring.

* The [[Weyl algebra]] $k\langle x, y\rangle/(x y - y x - 1)$ over a field $k$ is a simple ring. (In different language: this is the ring of differential operators with polynomial coefficients in one variable $t$, obtained as the image of the ring homomorphism from the noncommutative [[polynomial ring]] $k \langle x, y \rangle$ to the ring of $k$-linear endomorphisms $Vect(k[t], k[t])$ that sends $x$ to the derivative operator $\frac{d}{d t}$ and $y$ to the multiplication operator $t \cdot -$.) An explanation of why this is simple may be found [here](https://qchu.wordpress.com/2012/05/30/the-jacobson-radical/) at [[Qiaochu Yuan]]'s blog.