super L-infinity algebra

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###Context###
#### Supergeometry
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[[!include supergeometry - contents]]
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#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _super $L_\infty$-algebra_ is an [[L-∞ algebra]] in the context of [[superalgebra]]: the [[higher category theory|higher category theoretical]]/[[homotopy theory|homotopy theoretical]] version of a [[super Lie algebra]]. For more background see at _[[geometry of physics -- superalgebra]]_.

In the [[supergravity]] literature the [[formal duality|formal dual]] of super $L_\infty$-algebras of [[finite type]] came to be known as "[[FDA]]"s (see remark \ref{SuperLInfintiyAsFDA} below), a decade before plain [[L-∞ algebras]] were discussed in the mathematical literature. The key example in this context are [[extended super Minkowski spacetimes]], which are [[super L-∞ algebras]] obtained by iterated [[universal higher central extension]] from the [[super Minkowski spacetime]] [[super Lie algebra]]. The super-[[L-∞ algebra cohomology]] of these (called "[[tau-cohomology]]" in the physics literature) turns out to classify [[super p-branes]] and serves as a tool for the construction of [[supergravity]] theories in the [[D'Auria-Fré formulation of supergravity]]. For more background on this see at _[[geometry of physics -- fundamental super p-branes]]_.

## Definition
 {#Definition}

Abstractly, the definition is immediate:

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

A **super $L_\infty$-algebra** is an [[L-∞ algebra]] [[internalization|internal to]] the [[symmetric monoidal category]] of [[super vector spaces]] (i.e. in [[chain complexes of super vector spaces]]).

=--

Explicitly this equivalently comes down to the following definition in components:

+-- {: .num_defn #SuperGradedSignatureOfPermutation}
###### Definition
**(super graded signature of a permutation)**

Let $V$ be a $\mathbb{Z}$-[[graded object|graded]] [[super vector space]], hence a $\mathbb{Z} \times (\mathbb{Z}/2)$-bigraded vector space.

For $n \in \mathbb{N}$ let

$$
  \mathbf{v} = (v_1, v_2, \cdots, v_n)
$$

be an [[n-tuple]] of elements of $V$ of homogeneous degree $(n_i, s_i) \in \mathbb{Z} \times \mathbb{Z}/2$, i.e. such that $v_i \in V_{(n_i,s_i)}$.

For $\sigma$ a [[permutation]] of $n$ elements, write $(-1)^{\vert \sigma \vert}$ for the [[signature of a permutation|signature of the permutation]], which is by definition equal to $(-1)^k$ if $\sigma$ is the composite of $k \in \mathbb{N}$ permutations that each exchange precisely one pair of neighboring elements.

We say that the _super $\mathbf{v}$-graded signature of $\sigma$_

$$
  \chi(\sigma, v_1, \cdots, v_n) \;\in\; \{-1,+1\}
$$

is the product of the [[signature of a permutation|signature of the permutation]] $(-1)^{\vert \sigma \vert}$ with a factor of

$$
  (-1)^{n_i n_j}(-1)^{s_i s_j}
$$

for each interchange of neighbours $(\cdots v_i,v_j, \cdots )$ to $(\cdots v_j,v_i, \cdots )$ involved in the decomposition of the permuation as a sequence of swapping neighbour pairs (see at _[[signs in supergeometry]]_ for discussion of this combination of super-grading and homological grading).

=--

Now def. \ref{SuperLInfinityAlgebra} is equivalent to the following def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity}. This is just the definiton for [L-infinity algebras](#L-infinity-algebra#DefinitionViaHigherBrackets), with the pertinent sign $\chi$ now given by def. \ref{SuperGradedSignatureOfPermutation}.

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

An _super $L_\infty$-algebra_ is

1. a $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded object|graded]] [[vector space]] $\mathfrak{g}$;

1. for each $n \in \mathbb{N}$ a [[multilinear map]], called the _$n$-ary bracket_, of the form

   $$
     l_n(\cdots)
       \;\coloneqq\;
     [-,-, \cdots, -]_n
     \;\colon\;
       \underset{n \; \text{copies}}{\underbrace{\mathfrak{g} \otimes \cdots \otimes \mathfrak{g}}}
       \longrightarrow
       \mathfrak{g}
   $$

   and of degree $n-2$

such that the following conditions hold:

1. (**super graded skew symmetry**) each $l_n$ is graded antisymmetric, in that for every [[permutation]] $\sigma$ of $n$ elements and for every [[n-tuple]] $(v_1, \cdots, v_n)$ of  homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ then

   $$
     l_n(v_{\sigma(1)}, v_{\sigma(2)},\cdots ,v_{\sigma(n)})
     =
     \chi(\sigma,v_1,\cdots, v_n) \cdot l_n(v_1, v_2, \cdots v_n)
   $$

   where $\chi(\sigma,v_1,\cdots, v_n)$ is the super $(v_1,\cdots,v_n)$-graded signature of the permuation $\sigma$, according to def. \ref{SuperGradedSignatureOfPermutation};

1. (**strong homotopy [[Jacobi identity]]**) for all $n \in \mathbb{N}$, and for all [[n-tuple|(n+1)-tuples]] $(v_1, \cdots, v_{n+1})$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ the followig [[equation]] holds

   \[
     \label{LInfinityJacobiIdentity}
     \sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}}
     \sum_{\sigma \in UnShuff(i,j)}
     \chi(\sigma,v_1, \cdots, v_{n})
     (-1)^{i(j-1)}
      l_{j} \left(
        l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right),
        v_{\sigma(i+1)} , \cdots , v_{\sigma(n)}
      \right)
     = 0
     \,,
   \]

   where the inner sum runs over all $(i,j)$-[[unshuffles]] $\sigma$ and where $\chi$ is the super graded signature sign from def. \ref{SuperGradedSignatureOfPermutation}.

A _strict [[homomorphism]]_ of super $L_\infty$-algebras

$$is
  \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2
$$

is a [[linear map]] that preserves the bidegree and all the brackets, in an evident sens.

A _[[homomorphism of L-infinity algebras|strong homotopy homomorphism]]_ ("[[sh map]]") of super $L_\infty$-algebras is something weaker than that, best defined in [[formal duality|formal duals]], below in def. \ref{SuperLInfinityCEAlgebra}.

=--

In order to define the correct homomorphisms between super $L_\infty$-algebras ("sh-maps") as well as their super-[[L-∞ algebra cohomology]], consider the following dualization of the above definition:

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

A super $L_\infty$ algebra $\mathfrak{g}$ is of _[[finite type]]_ if the underlying $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded vector space]] is degreewise of [[finite number|finite]] [[dimension]].

If $\mathfrak{g}$ is of finite type, then its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ is the [[differential graded-commutative superalgebra]] whose underlying [[graded algebra]] is the super-Grassmann algebra

$$
  \wedge^\bullet \mathfrak{g}^{\ast}
$$

of the graded degreewise [[dual vector space]] $\mathfrak{g}^\ast$, equipped with the [[differential]] which on generators is the sum of the [[dual linear maps]] of the $n$-ary brackets:

$$
  d_{\mathfrak{g}}
   \coloneqq
  [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots
  \;\colon\;
  \wedge^1 \mathfrak{g}^\ast
    \longrightarrow
  \wedge^\bullet \mathfrak{g}^\ast
$$

and extended to all of $\wedge^\bullet \mathfrak{g}^\ast$ as a super-graded [[derivation]] of degree $(1,even)$.

Notice that here the [[signs in supergeometry]] are such that for $\alpha_i \in \mathfrak{g}^\ast_{(n_i,s_i)}$ elements of homogenous bidegree, then

$$
  \alpha_1 \wedge \alpha_2
  \;=\;
  -(-1)^{n_1 n_2} (-)^{s_1 s_2}
$$

and

$$
  d_{\mathfrak{g}}
  (\alpha_1 \wedge \alpha_2)
  \;=\;
  (d_{\mathfrak{g}} \alpha_1) \wedge \alpha_2
  +
  (-1)^{n_1} \alpha_1 \wedge (d_{\mathfrak{g}} \alpha_2)
  \,.
$$

(see at _[[signs in supergeometry]]_ for more on this).

A _strong homotopy homomorphism_ ("sh-map") between super $L_\infty$-algbras of [[finite type]]

$$
 f \;\colon\; \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2
$$

is defined to be a homomorphism of [[dg-algebras]] between their [[Chevalley-Eilenberg algebras]] going the other way:

$$
  CE(\mathfrak{g}_1) \longleftarrow CE(\mathfrak{g}_2)
    \;\colon\;
  f^\ast
$$

(here $f^\ast$ is the primitive concept, and $f$ is defined as the [[formal duality|formal dual]] of $f$). Hence the [[category]] of super $L_\infty$-algebras of [[finite type]] is the [[full subcategory]]

$$
  s L_\infty Alg \hookrightarrow dgcsAlg^{op}
$$

of the [[opposite category]] of [[differential graded-commutative superalgebras]] on those that are CE-algebras as above.

Finally, the [[cochain cohomology]] of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ of a super $L_\infty$ algebra of [[finite type]] is its _[[L-∞ algebra cohomology]]_ with [[coefficients]] in $\mathbb{R}$:

$$
  H^\bullet(\mathfrak{g}, \mathbb{R})
   \;=\;
  H^\bullet(CE(\mathfrak{g}))
  \,.
$$

=--

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

Special cases of the general concept of [[super L-∞ algebras]] def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity} go by special names:

Let $\mathfrak{g}$ be a [[super L-∞ algebra]].

If $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree, it is called an _[[L-∞ algebra]]_.

If the only possibly non-vanishing brackets of $\mathfrak{g}$ are the unary one $[-]$ (which induces the structure of a [[chain complex]] on $\mathfrak{g}$) and the binary one, then $\mathfrak{g}$ is equivalently a (super-)[[dg-Lie algebra]].

If $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degrees 0 to $n-1$ then it is called a _[[super Lie n-algebra]]_.

In particular if $\mathfrak{g}$ is concentrated in degree 0, then it is equivalently a [[super Lie algebra]].

Combining this, if $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree and in $\mathbb{Z}$-degree 0 through $n-1$, then it is called a _[[Lie n-algebra]]_.

In particular if $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degree 0 and in even $\mathbb{Z}/2$-degree, then it is
equivalently a plain [[Lie algebra]].

=--

## History

+-- {: .num_remark #SuperLInfintiyAsFDA}
###### Remark
**(history of the concept of (super-)$L_\infty$ algebras)**

The identification of the concept of (super-)$L_\infty$-algebras has a non-linear history:

[[L-∞ algebras]] in the incarnation of higher brackets satisfying a higher Jacobi identity,
def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity}
and remark \ref{LInfinityTerminology}, were introduced in
[Lada-Stasheff 92](https://ncatlab.org/nlab/show/L-infinity-algebra#LadaStasheff92), based on the
example of such a structure on the [[BRST complex]] of the [[bosonic string]] that
was found in the construction of [[closed string field theory]] in [Zwiebach 92](string+field+theory#Zwiebach93).
Some of this history is recalled in [Stasheff 16](#L-infinity-algebra#Stasheff16).

The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras
(def. \ref{SuperLInfinityCEAlgebra})
is due to [Lada-Markl 94](https://ncatlab.org/nlab/show/L-infinity-algebra#LadaMarkl94).
See [Sati-Schreiber-Stasheff 08, around def. 13](#SatiSchreiberStasheff08).

But in this dual incarnation, [[L-∞ algebras]] and more generally [[super L-∞ algebras]] (of [[finite type]])
had secretly been introduced within the [[supergravity]] literature already in [D'Auria-Fré-Regge 80](#DAuriaFreRegge80) and explicitly in [van Nieuwenhuizen 82](#Nieuwenhuizen82).
The concept was picked up in the [[D'Auria-Fré formulation of supergravity]] ([D'Auria-Fré 82](#DAuriaFre82)) and eventually came to be referred to as "FDA"s (short for "free differential algebra") in the [[supergravity]] literature (but beware that these dg-algebras
are in general [[free construction|free]] only as graded-[[supercommutative superalgebras]], not as differential algebras)
The relation between super $L_\infty$-algebras and the "FDA"s of the [[supergravity]] literature is made explicit in ([FSS 13](#FSS13)).

| [[nLab:higher Lie theory]] | [[nLab:supergravity]] |
|----------------------------|-----------------------|
| $\,$ [[nLab:super Lie n-algebra]] $\mathfrak{g}$ $\,$ | $\,$ "FDA" $CE(\mathfrak{g})$ $\,$ |

The construction in [van Nieuwenhuizen 82](#Nieuwenhuizen82) in turn was motivated by the [[Sullivan algebras]]
in [[rational homotopy theory]] ([Sullivan 77](#rational+homotopy+theory#Sullivan77)). Indeed, their
dual incarnations in rational homotopy theory are [[dg-Lie algebras]] ([Quillen 69](#rational+homotopy+theory#Quillen69)),
hence a special case of $L_\infty$-algebras (remark \ref{LInfinityTerminology})

This close relation between [[rational homotopy theory]] and [[higher Lie theory]] might have remained less of a
secret had it not been for the focus of [[Sullivan minimal models]] on the non-[[simply connected topological space|simply connected]]
case, which excludes the ordinary [[Lie algebras]] from the picture. But the Quillen model of
[[rational homotopy theory]] effectively says that for $X$ a [[rational topological space]] then its [[loop space]]
[[∞-group]] $\Omega X$ is reflected, infinitesimally, by an [[L-∞ algebra]]. This perspective began to
receive more attention when the [[Sullivan construction]] in [[rational homotopy theory]] was
concretely identified as higher [[Lie integration]] in [Henriques 08](#Lie+integration#Henriques).
A modern review that makes this [[L-∞ algebra]]-theoretic nature of [[rational homotopy theory]] manifest is in
[Buijs-Félix-Murillo 12, section 2](rational+homotopy+theory#BuijsFelixMurillo12).

=--

## Properties

* Every super $L_\infty$-Lie algebra has a [[Lie integration]] to a [[super ∞-groupoid]] and a [[smooth super ∞-groupoid]]. See at _[[Lie integration]]_ for more on this.

## Examples

In the context of [[supergravity]]/[[string theory]] the

* [[super Poincaré Lie algebra]]

and its super-$L_\infty$-extensions to the

* [[supergravity Lie 3-algebra]] ([[m2brane]])

* [[supergravity Lie 6-algebra]]

* [[type II supergravity Lie 2-algebra]]

play a central role. Their exceptional [[infinity-Lie algebra cohomology]] governs the consistent [[Green-Schwarz action functionals]] for super-$p$-[[brane|branes]]. (See the discusson of the _[brane scan](Green-Schwarz+action+functional#BraneScan)_) there.

See at _[[geometry of physics -- fundamental super p-branes]]_ for more on this.

$\,$

The [[BRST complex]] of the [[superstring]] might form a super $L_\infty$-algebra whose brackets give the [[n-point function]] of the string, in analogy to what happens for the bosonic string in Zwiebach's [[string field theory]]. (...)

## Related concepts

* [[model structure on chain complexes of super vector spaces]]

* [[model structure on differential graded-commutative superalgebras]]

## References
 {#References}

In their [[formal duality|formal dual]] incarnations as super-graded commutative [[dg-algebras]] (super [[Chevalley-Eilenberg algebras|Chevalley-Eilenberg algebras]]), super $L_\infty$-algebras of [[finite type]] had secretly been introduced in

* {#DAuriaFreRegge80} [[Riccardo D'Auria]], [[Pietro Fré]] [[Tullio Regge]], _Graded Lie algebra, cohomology and supergravity_, Riv. Nuov. Cim. 3, fasc. 12 (1980) ([spire](http://inspirehep.net/record/156191))

* {#Nieuwenhuizen82} [[Peter van Nieuwenhuizen]], _Free Graded Differential Superalgebras_, in *Istanbul 1982, Proceedings, Group Theoretical Methods In Physics*, 228-247 and CERN Geneva - TH. 3499 ([spire](http://inspirehep.net/record/182644/))

and hence a whole decade before the explicit appearance of plain (non-super) [[L-∞ algebras]] in [Lada-Stasheff 92](L-infinity-algebra#LadaStasheff92).

The concept was picked up in the [[D'Auria-Fré formulation of supergravity]]

* {#DAuriaFre82}  [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140 ([[GeometricSupergravityErrata.pdf:file]])

and eventually came to be referred to as "FDA"s (short for "free differential algebra") in the [[supergravity]] literature (where in [[rational homotopy theory]] one says "[[semifree dga]]", since these dg-algebras are crucially not required to be free as _differential_ algebras).

The relation between super $L_\infty$-algebras and the "FDA"s of the [[supergravity]] literature is made explicit in:

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §2 in: *[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]*, International Journal of Geometric Methods in Modern Physics **12** 02 (2015) [[arXiv:1308.5264](https://arxiv.org/abs/1308.5264), [doi:10.1142/S0219887815500188](http://www.worldscientific.com/doi/abs/10.1142/S0219887815500188)]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §2 in: _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_, Adv. Theor. Math. Phys. **22** 5 (2018) [[arXiv:1611.06536](https://arxiv.org/abs/1611.06536), [doi:10.4310/ATMP.2018.v22.n5.a3](https://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3)]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], (21) in: _[[schreiber:The rational higher structure of M-theory]]_, in: _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/)_: _[[Higher Structures in M-Theory 2018]]_, August 2018, Fortschritte der Physik **67** 8-9 (2019) [[arXiv:1903.02834](https://arxiv.org/abs/1903.02834), [doi:10.1002/prop.201910017](https://doi.org/10.1002/prop.201910017)]

also:

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], §3.2 in: _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Comm. Math. Phys. **371** (2019) 425-524 [[arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](https://doi.org/10.1007/s00220-019-03442-3)]

* [[Urs Schreiber]], p. 33, pp. 48 in: *[[schreiber:Higher Prequantum Geometry]]*, in *[[New Spaces for Mathematics and Physics]]*, Cambridge University Press (2021) [[arXiv:1601.05956](https://arxiv.org/abs/1601.05956), [doi:10.1017/9781108854399.008](https://doi.org/10.1017/9781108854399.008)]

Lecture notes:

* [[Urs Schreiber]], *[Super L ∞-cohomology and FDAs](geometry+of+physics+--+fundamental+super+p-branes#SuperLInfinityCohomologyAndFDAs)*, Section 2 in: *[[geometry of physics -- fundamental super p-branes]]*, originating in [[schreiber:Super Lie n-algebra of Super p-branes|talk notes (2016/17)]]

* [[Urs Schreiber]], *[[schreiber:Introduction to Higher Supergeometry]]*, lecture at _[[nLab:Higher Structures in M-Theory 2018]]_, [Durham Symposium](http://www.maths.dur.ac.uk/lms/) (2019)

Further discussion:

* [[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)]

[[!redirects super L-∞ algebra]]
[[!redirects super L-∞ algebras]]

[[!redirects super L-infinity algebras]]

[[!redirects super Lie n-algebra]]
[[!redirects super Lie n-algebras]]

[[!redirects super Lie-infinity algebra]]

[[!redirects super Lie-∞ algebra]]
[[!redirects super Lie-∞ algebras]]

[[!redirects FDA]]
[[!redirects FDAs]]


super Lie algebra

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

#Contents#
* table of contents
{:toc}

## Idea

A _super Lie algebra_ is the analog of a [[Lie algebra]] in [[superalgebra]]/[[supergeometry]].

See also at *[[supersymmetry]]*.

## Definition

There are various equivalent ways to state the definition of super Lie algebras. Here are a few (for more discussion see at _[[geometry of physics -- superalgebra]]_):

### As Lie algebras internal to super vector spaces
 {#AsLieAlgebrasInternalToSuperVectorSpaces}

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

A _[[super Lie algebra]]_ is a [[Lie algebra object]] [[internalization|internal]] to the [[symmetric monoidal category]] $sVect = (Vect^{\mathbb{Z}/2}, \otimes_k, \tau^{super} )$ of [[super vector spaces]] (a _[[Lie algebra object]] in [[super vector spaces]]_).
Hence this is

1. a [[super vector space]] $\mathfrak{g}$;

1. a homomorphism

   $$
     [-,-] \;\colon\; \mathfrak{g} \otimes_k \mathfrak{g} \longrightarrow \mathfrak{g}
   $$

   of super vector spaces (the _super Lie bracket_)

such that

1. the bracket is skew-symmetric in that the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       \mathfrak{g} \otimes_k \mathfrak{g}
         &
           \overset{\tau^{super}_{\mathfrak{g},\mathfrak{g}}}{\longrightarrow}
         &
       \mathfrak{g} \otimes_k \mathfrak{g}
       \\
       {}^{\mathllap{[-,-]}}\downarrow && \downarrow^{\mathrlap{[-,-]}}
       \\
       \mathfrak{g} &\underset{-1}{\longrightarrow}& \mathfrak{g}
     }
   $$

   (here $\tau^{super}$ is the [[braiding]] [[natural isomorphism]] in the category of [[super vector spaces]])

1. the [[Jacobi identity]] holds in that the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g}
       &&
         \overset{\tau^{super}_{\mathfrak{g}, \mathfrak{g}} \otimes_k id }{\longrightarrow}
       &&
       \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g}
       \\
       & {}_{\mathllap{\left[-,\left[-,-\right]\right]} - \left[\left[-,-\right],-\right] }\searrow && \swarrow_{\mathrlap{\left[-,\left[-,-\right]\right]}}
       \\
       && \mathfrak{g}
     }
     \,.
   $$

=--

### As super-graded Lie algebras

Externally this means the following:

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

A [[super Lie algebra]] according to def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces} is equivalently

1. a $\mathbb{Z}/2$-[[graded vector space]] $\mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}$;

1. equipped with a [[bilinear map]] (the _super Lie bracket_)

   $$
     [-,-] : \mathfrak{g}\otimes_k \mathfrak{g} \to \mathfrak{g}
   $$

   which is _graded_ skew-symmetric: for $x,y \in \mathfrak{g}$ two elements of homogeneous degree $\sigma_x$, $\sigma_y$, respectively, then

   $$
     [x,y] = -(-1)^{\sigma_x \sigma_y} [y,x]
     \,,
   $$

1. that satisfies the $\mathbb{Z}/2$-graded [[Jacobi identity]] in that
for any three elements $x,y,z \in \mathfrak{g}$ of homogeneous super-degree $\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2$ then

   \[
     \label{GradedJacobiIdentity}
     [x, [y, z] ] = [ [x,y],z] + (-1)^{\sigma_x \cdot \sigma_y} [y, [x,z] ]
     \,.
   \]

A [[homomorphism]] of super Lie algebras is a homomorphisms of the underlying [[super vector spaces]]
which preserves the Lie bracket. We write

$$
  sLieAlg
$$

for the resulting [[category]] of super Lie algebras.

=--

### As formal duals of a Chevalley-Eilenberg super-algebras

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

For $\mathfrak{g}$ a [[super Lie algebra]] of [[finite number|finite]] [[dimension]], then its _[[Chevalley-Eilenberg algebra]]_ $CE(\mathfrak{g})$ is the super-[[Grassmann algebra]] on the [[dual vector space|dual]] super vector space

$$
  \wedge^\bullet  \mathfrak{g}^\ast
$$

equipped with a [[differential]] $d_{\mathfrak{g}}$ that on generators is the linear dual of the super Lie bracket

$$
  d_{\mathfrak{g}} \;\coloneqq\; [-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast
$$

and which is extended to $\wedge^\bullet \mathfrak{g}^\ast$ by the graded Leibniz rule (i.e. as a graded [[derivation]]).

$\,$

Here all elements are $(\mathbb{Z} \times \mathbb{Z}/2)$-bigraded, the first being the _cohomological grading_ $n$ in $\wedge^\n \mathfrak{g}^\ast$, the second being the _super-grading_ $\sigma$ (even/odd).

For $\alpha_i \in CE(\mathfrak{g})$ two elements of homogeneous bi-degree $(n_i, \sigma_i)$, respectively,
the [[signs in supergeometry|sign rule]] is

$$
  \alpha_1 \wedge \alpha_2 = (-1)^{n_1 n_2} (-1)^{\sigma_1 \sigma_2}\; \alpha_2 \wedge \alpha_1
  \,.
$$

(See at _[[signs in supergeometry]]_ for discussion of this sign rule and of an alternative sign rule that is also in use. )

=--

We may think of $CE(\mathfrak{g})$ equivalently as the [[dg-algebra]] of [[invariant differential form|left-invariant]]
[[super differential forms]] on the [[Lie theory|corresponding]] simply connected [[super Lie group]] .

The concept of [[Chevalley-Eilenberg algebras]] is traditionally introduced as a means to define
[[Lie algebra cohomology]]:

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

Given a [[super Lie algebra]] $\mathfrak{g}$, then

1. an _$n$-cocycle_ on $\mathfrak{g}$ (with [[coefficients]] in $\mathbb{R}$) is an element of degree $(n,even)$ in its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ (def. \ref{CEAlgebraofSuperLieAlgebra}) which is $d_{\mathbb{g}}$ closed.

1. the cocycle is non-trivial if it is not $d_{\mathfrak{g}}$-exact

1. hene the _super-[[Lie algebra cohomology]]_ of $\mathfrak{g}$ (with [[coefficients]] in $\mathbb{R}$) is the [[cochain cohomology]] of its [[Chevalley-Eilenberg algebra]]

   $$
     H^\bullet(\mathfrak{g}, \mathbb{R}) = H^\bullet(CE(\mathfrak{g}))
     \,.
   $$

=--

The following says that the [[Chevalley-Eilenberg algebra]] is an equivalent incarnation of the [[super Lie algebra]]:

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

The functor

$$
  CE \;\colon\; sLieAlg^{fin}  \hookrightarrow dgAlg^{op}
$$

that sends a finite dimensional [[super Lie algebra]] $\mathfrak{g}$ to its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$
(def. \ref{CEAlgebraofSuperLieAlgebra}) is a [[fully faithful functor]] which hence exibits [[super Lie algebras]]
as a [[full subcategory]] of the [[opposite category]] of [[differential-graded algebras]].

=--

### As super-representable Lie algebras in the topos over superpoints
 {#AsSuperRepresentableLieAlgebrasOverSuperpoints}

Equivalently, a super Lie algebra is a "super-representable" [[Lie algebra object]] [[internalization|internal]] to the [[cohesive (∞,1)-topos]] [[Super∞Grpd]] over the site of [[super points]] ([Sachse 08, Section 3.2, towards cor. 3.3](#Sachse08)).

See the discussion at [[superalgebra]] for details on this.

## Properties

### Classification
 {#Classification}

([Kac 77a](Kac77a), [Kac 77b](#Kac77b)) states a classification of super Lie algebras which are

1. finite dimensional

1. simple

1. over a [[field]] of [[characteristic zero]].

Such an algebra is called of _classical type_ if the [[action]] of its even-degree part on the odd-degree part is completely reducible. Those simple finite dimensional algebras not of classical type are of _Cartan type_.

1. classical type

   1. four infinite series

      1. $A(m,n)$

      1. $B(m,n) = $ [[osp]]$(2m+1,2n)$ $m\geq 0$, $n \gt 0$

      1. $C(n)$

      1. $D(m,n) = $[[osp]]$(2m,2n)$ $m \geq 2$, $n \gt 0$

   1. two exceptional ones

      1. $F(4)$

      1. [[G(3)]]

   1. a family $D(2,1;\alpha)$ of deformations of $D(2,1)$

   1. two "strange" series

      1. $P(n)$

      1. $Q(n)$

1. Cartan type

   (...)

The underlying even-graded [[Lie algebra]] for type 2 is as follows

| $\mathfrak{g}$ |  $\mathfrak{g}_{even}$  |  $\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{odd}$ |
|----------------|-------------------------|------------------|
| $B(m,n)$  | $B_m \oplus C_n$  | vector $\otimes$ vector |
| $D(m,n)$  | $D_m \oplus C_n$  | vector $\otimes$ vector |
| $D(2,1,\alpha)$ |  $A_1 \oplus A_1 \oplus A_1$ | vector $\otimes$ vector $\otimes$ vector |
| $F(4)$ | $B_3\otimes A_1$ | spinor $\otimes$ vector |
| $G(3)$ | $G_2\oplus A_1$ | spinor $\otimes$ vector |
| $Q(n)$ | $A_n$ | adjoint |

For type 1 the $\mathbb{Z}/2\mathbb{Z}$-grading lifts to an $\mathbb{Z}$-grading with $\mathfrak{g} = \mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$.

| $\mathfrak{g}$ |  $\mathfrak{g}_{even}$  |  $\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{{-1}}$ |
|----------------|-------------------------|------------------|
| $A(m,n)$ |  $A_m \oplus A_n \oplus \mathbb{C}$ | vector $\otimes$ vector $\otimes$ $\mathbb{C}$ |
| $A(m,m)$ | $A_m \oplus A_m$ | vector $\otimes$ vector |
| $C(n)$ | $C_{n-1} \oplus \mathbb{C}$ | vector $\otimes$ $\mathbb{C}$ |

reviewed e.g. in ([Farmer 84, p. 25,26](#Farmer84), [Minwalla 98, section 4.1](#Minwalla98)).

### Relation to dg-Lie algebras
 {#RelationToDGLieAlgebras}

A [[dg-Lie algebra]] $(\mathfrak{g}, \partial, [-,-])$ may be understood equivalently as a [[super Lie algebra]]

$$
  (\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}, [-,-,])
$$

equipped with

1. a [[lift]] of the $\mathbb{Z}_2$-[[graded vector space|grading]] of the underlying [[vector space]] to a $\mathbb{Z}$[-graded vector space](graded+vector+space#SpecialCaseOfZGradedVectorSpaces) through the projection $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} = \mathbb{Z}_2$, hence

   $$
     \mathfrak{g}_{even}
     \;\simeq\;
     \underset{n }{\oplus} \mathfrak{g}_{2n}
     \,,
     \phantom{AAA}
     \mathfrak{g}_{odd}
     \;\simeq\;
     \underset{n }{\oplus} \mathfrak{g}_{2n+1}
   $$

   such that

   $$
     [-,-] \;\colon\; \mathfrak{g}_{n_1} \otimes \mathfrak{g}_{n_2} \longrightarrow \mathfrak{g}_{n_1 + n_2}
   $$

1. an element $Q \in \mathfrak{g}_{-1}$

   such that

   \[
     \label{NilpotencyOfQ}
      [Q,Q] = 0
   \]

(See also at "[[NQ-supermanifold]]".)

Given this, define the [[differential]] to be the [[adjoint action]] by $Q$:

$$
  \partial \;\coloneqq\; [Q,-]
  \,.
$$

That this differential squares to 0 follows by the super-Jacobi identity (eq:GradedJacobiIdentity) and by the nilpotency (eq:NilpotencyOfQ):

$$
  [Q,[Q,-] ]
  \;=\;
  [ \underset{= 0}{\underbrace{[Q,Q]}}, -  ] - [  Q, [Q, -]  ]
  \phantom{AA}
  \Rightarrow
  \phantom{AA}
  [ Q,[Q,-] ] = 0
$$

and the [[derivation]]-property of the differential over the bracket follows again with the super Jacobi identity (eq:GradedJacobiIdentity):

$$
  [Q,[x,y] ]
  \;=\;
  [ [Q,x],y]
  + (-1)^{deg(x)} [x, [Q,y] ]
  \,.
$$

## Examples
 {#Examples}

### Basic examples

Some obvious but important classes of examples are the following:

+-- {: .num_example #SuperVectorSpaceAsAbelianSuperLieAlgebra}
###### Example

every $\mathbb{Z}/2$-[[graded vector space]] $V$ becomes a
[[super Lie algebra]]
(def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}, prop. \ref{SuperLieAlgebraTraditional})
by taking the super Lie bracket to be the [[zero morphism|zero map]]

$$
  [-,-] = 0
  \,.
$$

These may be called the "abelian" super Lie algebras.

=--

+-- {: .num_example #OrdinaryLieAlgebraAsSuperLieAlgebra}
###### Example

Every ordinary [[Lie algebras]] becomes a [[super Lie algebra]]
(def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}, prop. \ref{SuperLieAlgebraTraditional})
concentrated in even degrees. This constitutes a [[fully faithful functor]]

$$
  LieAlg \hookrightarrow sLieAlg
  \,.
$$

which is a  [[coreflective subcategory]] inclusion in that it has a [[left adjoint]]

$$
  LieAlg
    \underoverset
     {\underset{ \overset{ \rightsquigarrow}{(-)} }{\longleftarrow}}
     {\hookrightarrow}
     {\bot}
  sLieAlg
$$

given on the underlying super vector spaces by restriction to the even graded part

$$
  \overset{\rightsquigarrow}{\mathfrak{s}} = \mathfrak{s}_{even}
  \,.
$$

=--

### Super-Poincaré super Lie algebras (supersymmetry)

* The [[super Poincaré Lie algebra]] and various of its polyvector extension are super-extension of the ordinary [[Poincare Lie algebra]]. These are the [[supersymmetry algebras]] in the strict original sense of the word.  For more on this see at _[[geometry of physics -- supersymmetry]]_.

* [[super q-Schur algebra]]

* For every [[connected topological space]], the [[Whitehead product]] makes its [[homotopy groups]] into a super Lie algebra over the [[ring]] of [[integers]] -- the [Whitehead super Lie algebra](Whitehead+product#SuperLieAlgebraStructure).

* higher super Lie algebras

  Just as [[Lie algebras]] are [[vertical categorification|categorified]] to [[L-infinity algebra]]s and [[L-infinity algebroid]]s, so super Lie algebras categorifie to [[super L-infinity algebra]]s.  A secretly famous example is the

  * [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]

### Embedding tensors and tensor hierarchy
 {#SuperLieAlgebraInducedFromVectorSpace}

The following example is highlighted in [Palmkvist 13, 3.1](tensor+hierarchy#Palmkvist13) (reviewed more clearly in [Lavau-Palmkvist 19, 2.4](tensor+hierarchy#LavauPalmkvist19)) where it is attributed to I. L. Kantor (1970).

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

Let $V$ be a [[finite-dimensional vector space]] over some [[ground field]] $k$.

Define a $\mathbb{Z}$-[[graded vector space]]

$$
  \widehat V \;\in \; Vect_k^{\mathbb{Z}}
  \,,
$$

concentrated in degrees $\leq 1$, [[recursion|recursively]] as follows:

For $n =1$ we set

$$
  \widehat V_{1}
  \;\coloneqq\;
  V
  \,.
$$

For $n \leq 0 \in \mathbb{Z}$, the component space in degree $n-1$ is taken to be the vector space of [[linear maps]] from $V$ to the component space in degree $n$:

$$
  \widehat V_{n-1}
  \;\coloneqq\;
  Hom_k( V, \widehat V_n )
  \,.
$$

Hence:

\[
  \label{ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace}
  \begin{aligned}
    \widehat V_1 & = V
    \\
    \widehat V_0 & = Hom_k(V,V) = \mathfrak{gl}(V)
    \\
    \widehat V_{-1} & =  Hom_k(V, Hom_k(V,V)) \simeq  Hom_k(V \otimes V, V)
    \\
    \widehat V_{-2} & = Hom_k(V, Hom_k(V, Hom_k(V,V))) \simeq  Hom_k(V \otimes V \otimes V, V)
    \\
    \vdots
  \end{aligned}
\]

Consider then the [[direct sum]] of these component spaces as a [[super vector space]] with the [[even number]]/[[odd number]]-degrees being in super-even/super-odd degree, respectively.

On this [[super vector  space]] consider a [[super Lie bracket]] defined [[recursion|recusively]] as follows:

For all $v_1, v_2 \in \widehat V_1 = V$ we set

$$
  [v_1, v_2] \;=\; 0
  \,.
$$

For $f \in \widehat V_{n \leq 0}$ and $v \in \widehat V_1 = V$ we set

\[
  \label{SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace}
  [f, v]
  \;\coloneqq\;
  f(v)
\]

Finally, for $f\in \widehat V_{ deg(f) \leq 0 }$ and $g\in \widehat V_{deg(g) \leq 0}$ we set

\[
  \label{SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace}
  \begin{aligned}
  [f, g]
  &
  \colon\;
  v
  \;\mapsto\;
  [f, g(v)]
  -
  (-1)^{
    deg(f) deg(g)
  }
  [ g, f(v) ]
  \\
  \end{aligned}
\]

=--

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

By (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace) the definition (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) is equivalent to

$$
  [ [f,g],v ]
  \;=\;
  [f, [g,v] ]
  -
  (-1)^{
    deg(f) deg(g)
  }
  [ g, [f,v] ]
$$

Hence (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) is already implied by (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace) if the bracket is to satisfy the super Jacobi identity (eq:GradedJacobiIdentity).

=--

It remains to show that:

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

Def. \ref{TheSuperLieAlgebraInducedFromVectorSpace} indeed gives a [[super Lie algebra]] in that the bracket (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) satisfies the super Jacobi identity (eq:GradedJacobiIdentity).

=--

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

We proceed by [[induction]]:

By Remark \ref{ConstraintsOnBracketInSuperLieAlgebraInducedByVectorSpace} we have that the super Jacobi identity holds for for all [[triples]] $f_1, f_2, f_3 \in \widehat{V}$ with $deg(f_3) \geq 0$.

Now assume that the super Jacobi identity has been shown for triples $(f_1, f_2, f_3(v))$ and $(f_1, f_3, f_2(v))$, for any $v \in V$. The following computation shows that then it holds for $(f_1, f_2, f_3)$:

$$
  \begin{aligned}
    [f_1, [f_2, f_3] ] (v)
    &
    =
    [ f_1,  [f_2, f_3](v) ]
    -
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3))}
    [  [ f_2, f_3 ], f_1(v)  ]
    \\
    & =
    [ f_1,  [  f_2, f_3(v) ] ]
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_2)deg(f_3)}
    [ f_1,  [  f_3, f_2(v) ] ]
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3))}
    [  f_2,  [ f_3,  f_1(v) ] ]
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3)) + deg(f_2)deg(f_3)}
    [  f_3,  [ f_2,  f_1(v) ] ]
    \\
    & =
    [ f_1,  [  f_2, f_3(v) ] ]
    -
    (-1)^{deg(f_1) deg(f_2)}
    {
    \color{green}
    [ f_2,  [  f_1, f_3(v) ] ]
    }
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_2) deg(f_3)}
    \big(
    [ f_1,  [  f_3, f_2(v) ] ]
    -
    (-1)^{deg(f_1) deg(f_3)}
    {
    \color{orange}
    [ f_3,  [  f_1, f_2(v) ] ]
    }
    \big)
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3))}
    \big(
    [  f_2,  [ f_3,  f_1(v) ] ]
    -
    (-1)^{deg(f_1)deg(f_3)}
    {
    \color{blue}
    [  f_2,  [ f_1,  f_3(v) ] ]
    }
    \big)
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1) deg(f_2 ) + deg(f_1) deg(f_3) + deg(f_2) deg(f_3)}
    \big(
    [  f_3,  [ f_2,  f_1(v) ] ]
    -
    (-1)^{deg(f_1) deg(f_2)}
    {
    \color{cyan}
    [  f_3,  [ f_1,  f_2(c) ] ]
    }
    \big)
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1) deg(f_2)}
    \big(
    \underset{
      = 0
    }{
    \underbrace{
    +
    {
    \color{green}
    [ f_2,  [  f_1, f_3(v) ] ]
    }
    -
    {
    \color{blue}
    [  f_2,  [ f_1,  f_3(v) ] ]
    }
    }
    }
    \big)
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1) deg(f_3) + deg(f_2) deg(f_3)}
    \big(
    \underset{
      = 0
    }{
    \underbrace{
    {
    \color{orange}
    [  f_3,  [ f_1,  f_2(v) ] ]
    }
    -
    {
    \color{cyan}
    [ f_3,  [  f_1, f_2(v) ] ]
    }
    }
    }
    \big)
    \\
    & =
    \big[
      [f_1, f_2], f_3(v)
    \big]
    \\
    & \phantom{=}
    -
    (-1)^{ deg(f_2) deg(f_3) }
    \big[
      [f_1, f_3], f_2(c)
    \big]
    \\
    & \phantom{=}
    +
    (-1)^{ deg(f_1) deg(f_2)  }
    \big[
      f_2, [f_1, f_3](v)
    \big]
    \\
    & \phantom{=}
    -
    (-1)^{ deg(f_3)( deg(f_1) + deg(f_2) ) }
    \big[
      f_3, [f_1, f_2](v)
    \big]
    \\
    & =
    \big[
     [f_1, f_2], f_3
    \big](v)
    +
    (-1)^{deg(f_1)deg(f_2))}
    \big[
      f_2, [f_1, f_3]
    \big](v)
  \end{aligned}
$$

> (Fine, but is this sufficient to induct over the full range of all three degrees?)

=--

+-- {: .num_example #LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace}
###### Example

For $f,g \in \widehat V_0 = Hom_k(V,V)$ (eq:ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace) we have that the bracket on $\widehat V$ in Def. \ref{SuperLieAlgebraInducedFromVectorSpace} restricts to

$$
  [f,g](v)
  \;=\;
  [f,g(v)] - [g,f(v)]
  \;=\;
  f(g(v)) - g(f(v))
$$

(by combining (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) with (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace)).

This is the [[Lie bracket]] of the [[general linear Lie algebra]] $\mathfrak{gl}(V)$, as indicated on the right in (eq:ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace).

=--

+-- {: .num_prop #EmbeddingTensorViaSuperLieAlgebraInducedFromVectorSpace}
###### Proposition
**([[embedding tensors]] are square-0 elements in $\widehat{V}$)**

Let $k$ be a [[ground field]] of [[characteristic zero]].

An element in degree -1 of the [[super Lie algebra]] $\widehat V$ from Def. \ref{SuperLieAlgebraInducedFromVectorSpace},

$$
  \Theta \in \widehat V_{-1} \simeq Hom_{k}(V, \mathfrak{gl}(V))
  \,,
$$

which by Example \ref{LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace} is identified with a [[linear map]]

$$
  \Theta \;\colon\; V \longrightarrow \mathfrak{g} \coloneqq \mathfrak{gl}(V)
$$

from $V$ to the [[general linear Lie algebra]] on $V$, is square-0 (eq:NilpotencyOfQ) precisely if it is an _[[embedding tensor]]_, in that:

$$
  [\Theta, \Theta]
  \;=\;
  0
  \phantom{AAA}
  \Leftrightarrow
  \phantom{AAA}
  [\Theta(v_1), \Theta(v_2) ]
  \;=\;
  \Theta( \rho_{\Theta(v)1)}(v_2) )
  \,.
$$

Here on the right, $[-,-]$ denotes the [[Lie bracket]] in $\mathfrak{gl}(V)$, while $\rho$ denotes the canonical [[Lie algebra action]] of $\mathfrak{gl}(V)$ on $V$.

=--

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

By unwinding of the definition (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace) and (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) and using again Example \ref{LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace} we compute as follows:

$$
  \begin{aligned}
    \big( \tfrac{1}{2} [Q,Q](v_1) \big)(v_2)
    & =
    [Q, Q(v_1)](v_2)
    \\
    & =
    [Q,
      \underset{
        \mathclap{
          = \rho_{\Theta(v_1)}(v_2)
        }
      }
      {
        \underbrace{
          (Q(v_1))(v_2)
        }
      }
    ]
    -
    [Q(v_1), Q(v_2)]
    \\
    & =
    \Theta( \rho_{\Theta(v_1)}(v_2) )
    -
    [ \Theta(v_1), \Theta(v) ]
  \end{aligned}
$$

=--

+-- {: .num_prop #EmbeddingTensorsInduceTensorHierarchies}
###### Remark
**([[embedding tensors]] induce [[tensor hierarchies]])**

In view of the relation between [[super Lie algebras]] and [[dg-Lie algebras]] ([above](#RelationToDGLieAlgebras)), Prop. \ref{EmbeddingTensorViaSuperLieAlgebraInducedFromVectorSpace} says that every choice of an [[embedding tensor]] for a [[faithful representation]] on a [[vector space]] $V$ induces a [[dg-Lie algebra]]
$(\widehat V, [-,-], \partial \coloneqq [\Theta, -])$.

According to [Palmkvist 13, 3.1](tensor+hierarchy#Palmkvist13), [Lavau-Palmkvist 19, 2.4](tensor+hierarchy#LavauPalmkvist19) this [[dg-Lie algebra]] (or some extension of some sub-algebra of it) is the _[[tensor hierarchy]]_ associated with the embedding tensor.

=--

## Related entries

* [[Haag–Łopuszański–Sohnius theorem]]

* [[geometry of physics -- supersymmetry]]

## References

According to [Kac77b](#Kac77b) the definition of super Lie algebra is originally due to

* [[Felix Berezin]], G. I. Kac, Math. Sbornik 82, 343—351 (1970) (Russian)

See also

* {#Kantor70} [[Isaiah Kantor]], _Graded Lie algebras_, Trudy Sem. Vektor. Tenzor. Anal 15 (1970): 227-266.

The original references on the classification of super Lie algebras are

* {#Kac77a} [[Victor Kac]], _Lie superalgebras_, Advances in Math. 26 (1977), no. 1, 8--96 (<a href="https://doi.org/10.1016/0001-8708(77)90017-2">doi:10.1016/0001-8708(77)90017-2</a>)

* {#Kac77b} [[Victor Kac]], _A sketch of Lie superalgebra theory_, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. ([euclid:cmp/1103900590](https://projecteuclid.org/euclid.cmp/1103900590))

See also

* [[Werner Nahm]], V. Rittenberg, [[Manfred Scheunert]], _The classification of graded Lie algebras_ , Physics Letters B Volume 61, Issue 4, 12 April 1976, Pages 383–384 ([publisher](http://www.sciencedirect.com/science/article/pii/0370269376905943))

* M. Parker, _Classification Of Real Simple Lie Superalgebras Of Classical Type_, J.Math.Phys. 21 (1980) 689-697 ([spire](http://inspirehep.net/record/157627?ln=en))

Further discussion of classification related specifically to [classification of supersymmetry](#supersymmetry#Classification) is due to

* {#Nahm78} [[Werner Nahm]], _[[Supersymmetries and their Representations]]_, Nucl.Phys. B135 (1978) 149 ([spire](https://inspirehep.net/record/120988/), [pdf](http://cds.cern.ch/record/132743/files/197709213.pdf))

* {#Shnider88} [[Steven Shnider]], _The superconformal algebra in higher dimensions_, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383

* [[Victor Kac]], _Classification of supersymmetries_, Proceedings of the ICM, Beijing 2002, vol. 1, 319--344 ([arXiv:math-ph/0302016](http://arxiv.org/abs/math-ph/0302016))

Introductions and surveys include

* {#Farmer84} Richard Joseph Farmer, _Orthosymplectic superalgebras in mathematics and science_, PhD Thesis (1984) ([web](http://eprints.utas.edu.au/19542/), [pdf](http://eprints.utas.edu.au/19542/1/whole_FarmerRichardJoseph1985_thesis.pdf))

* L. Frappat, A. Sciarrino, P. Sorba, _Dictionary on Lie Superalgebras_ ([arXiv:hep-th/9607161](http://arxiv.org/abs/hep-th/9607161))

* Groeger, _Super Lie groups and super Lie algebras_, lecture notes 2011 ([pdf](http://www.mathematik.hu-berlin.de/~groegerj/teaching_files/lecture12.pdf))

* L. Frappat, A. Sciarrino, P. Sorba, _Dictionary on Lie Superalgebras_ ([arXiv:hep-th/9607161](http://arxiv.org/abs/hep-th/9607161))

* D. Leites, _Lie superalgebras_, J. Soviet Math. 30 (1985), 2481--2512 ([web](http://dx.doi.org/10.1007/BF02249121))

* [[Manfred Scheunert]], _The theory of Lie superalgebras. An introduction_, Lect. Notes Math. 716 (1979)

* D. Westra, _Superrings and supergroups_ ([pdf](http://www.mat.univie.ac.at/~michor/westra_diss.pdf))

* {#Minwalla98} [[Shiraz Minwalla]], _Restrictions imposed by superconformal invariance on quan tum field theories_ Adv. Theor. Math. Phys. 2, 781 (1998)
([arXiv:hep-th/9712074](http://arxiv.org/abs/hep-th/9712074)).

Discussion in the topos over superpoints is in

* {#Sachse08} [[Christoph Sachse]], _A Categorical Formulation of Superalgebra and Supergeometry_ ([arXiv:0802.4067](http://arxiv.org/abs/0802.4067))

Discussion of  [[Lie algebra extensions]] for super Lie algebras includes

* [[Dmitri Alekseevsky]], [[Peter Michor]], Wolfgang Ruppert, _Extensions of super Lie algebras_, J. Lie Theory 15 (2005) No. 1, 125--134 ([arXiv:math/0101190](http://arxiv.org/abs/math/0101190))

On [[Lie algebra weight systems]] arising from [[super Lie algebras]]:

* {#FFKV97} [[José Figueroa-O’Farrill]], [[Takashi Kimura]], [[Arkady Vaintrob]], _The universal Vassiliev invariant for the Lie superalgebra $\mathfrak{gl}(1\vert1)$_, Commun. Math. Phys. 185 (1997) 93-127 ([arXiv:q-alg/9602014](https://arxiv.org/abs/q-alg/9602014))

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))

[[!redirects super Lie algebras]]

[[!redirects Lie superalgebra]]
[[!redirects Lie superalgebras]]

[[!redirects super Lie bracket]]
[[!redirects super Lie brackets]]


super Lie algebroid

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

#Contents#
* table of contents
{:toc}

## Idea

The evident [[Lie algebroid]]-version of what is discussed at _[[super L-∞ algebra]]_.

## Examples

* The super-dg-algebra which si the de Rham complex of [[superdifferential forms]] on a [[supermanifold]] is the [[Chevalley-Eilenberg algebra]] of the super-[[tangent Lie algebroid]] of that manifold. See the discussion at _[[signs in supergeometry]]_.

The [[BRST complex]] of a supersymmetric field theory (e.g [FFK 89](#FFK89) for the [[superstring]]) is the [[Chevalley-Eilenberg algebra]] of the corresponding super-[[action Lie algebroid]].

## References

* {#FFK89} [[José Figueroa-O'Farrill]], Takashi Kimura, _The BRST cohomology of the NSR string: vanishing and ``no-ghost'' theorems_, Comm. Math. Phys.
Volume 124, Number 1 (1989), 105-132. ([Euclid](http://projecteuclid.org/euclid.cmp/1104179078))

* [[Alexander Belopolsky]], _De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology_, Phys.Lett. B403 (1997) 47-50 ([arXiv:hep-th/9609220](http://arxiv.org/abs/hep-th/9609220))

[[!redirects super Lie algebroids]]


super line 2-bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _super line 2-bundle_ is a [[line 2-bundle]] in ([[higher geometry|higher]]) [[supergeometry]].

We discuss line 2-bundles in [[supergeometry]] and their relation to [[twisted K-theory]]. This follows the discussion in chapter 1 of ([Freed](#Freed12)), which in turn follows the classical text ([Donovan-Karoubi](#DonovanKaroubi)) on [[twisted K-theory]] and ([Wall](#Wall)) on _[Picard 2-groupoids of superalgebras](super+algebra#Picard2Groupoid)_. What we add to this here, following ([Fiorenza-Sati-Schreiber 12](#FSS)) is that we make explit the incarnation of these constructions as the [[∞-stack|higher stack]] on [[supermanifolds]] $2\mathbf{sLine}$ of super line 2-bundles. This is a supergeometric refinement of the [[moduli infinity-stack|moduli 2-stack]] $\mathbf{B}^2\mathbb{C}^\times$ for bare complex line 2-bundles, $\mathbb{C}^\times$-[[principal 2-bundles]].

## Definition

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

Let $\mathbf{H} \coloneqq $ [[SmoothSuper∞Grpd]] be the [[cohesive (∞,1)-topos]] of [[SmoothSuper∞Grpd|smooth super-∞-groupoids]]. With [[CartSp]]${}_{th}$ the [[site]] given by the [[full subcategory]] of the category of [[supermanifolds]] on those of the form $\mathbb{R}^{p|q}$ for $p,q \in \mathbb{N}$ this is the corresponding [[(∞,1)-category of (∞,1)-sheaves]]

$$
  SmoothSuper\infty Grpd \simeq Sh_\infty(CartSp_{th})
$$

This is cohesive over [[Super∞Grpd]] $ \simeq Sh_\infty(SuperPoints)$

$$
  \Gamma \colon SmoothSuper\infty Grpd  \to Super \infty Grpd
  \,.
$$

=--

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

Let $\mathbb{K} \in \mathbf{H}$ be the canonical [[affine line|affine]] [[line object]], whose underlying sheaf of sets assigns

$$
  \mathbb{K} \colon \mathbb{R}^{p|q} \mapsto C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even}
 \,.
$$

=--

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

By the discussion at _[[superalgebra]]_ we have that $\Gamma(\mathbb{K})$-algebras in [[Super∞Grpd]] are, externally, superalgebras over the [[complex numbers]].

=--

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

Write

$$
  2\mathbf{sVect} \in SmoothSuper \infty Grpd
$$

for the object which over $\mathbb{R}^{p|q}$ is the [[2-groupoid]] whose

* objects are semisimple $\mathbb{K}(\mathbb{R}^{p|q})$-algebras;

* [[1-morphisms]] are invertible [[bimodules]];

* [[2-morphisms]] are invertible bimodule homomorphisms.

This is naturally a [[braided monoidal 2-category]] object. Write

$$
 2 \mathbf{sLine} \in SmoothSuper \infty Grpd
$$

for the maximal [[braided 3-group]] inside this on the invertible objects.

=--

## Properties

We now want to analyse the super 2-stack $2 \mathbf{sLine}$. In order to do so, first notice the following classical results about the [[Picard 3-group]] of superalgebras.

### The Brauer 3-group of superalgebras
 {#TheBrauer3Group}

+-- {: .num_theorem #AzumayaAreCentralSimple}
###### Theorem

A superalgebra is invertible/Azumaya (see [here](super+algebra#Azumaya)) precisely if it is finite dimensional and central simple (see [here](super+algebra#CentralSimple)).

=--

This is due to ([Wall](#Wall)).

+-- {: .num_theorem }
###### Theorem

The [[Brauer group]] of superalgebras over the [[complex numbers]] is the [[cyclic group of order 2]]. That over the [[real numbers]] is cyclic of order 8:

$$
  sBr(\mathbb{C}) \simeq \mathbb{Z}_2
$$
$$
  sBr(\mathbb{R}) \simeq \mathbb{Z}_8
  \,.
$$

The non-trivial element in $sBr(\mathbb{R})$ is that presented by the superalgebra $\mathbb{C} \oplus \mathbb{C} u$ of the example [here](super+algebra#ComplexCl1), with $u \cdot u = 1$.

=--

This is due to ([Wall](#Wall)).

The following generalizes this to the higher [[homotopy groups]].

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

The [[homotopy groups]] of the [[braided 3-group]] $sAlg^\times$ of Azumaya superalgebra are

| | $sAlg^\times_{\mathbb{C}}$ | $sAlg^\times_{\mathbb{R}}$ |
|--|--|--|
| $\pi_2$ | $\mathbb{C}^\times$ | $\mathbb{R}^\times$
| $\pi_1$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$
| $\pi_0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_8$

where the [[groups of units]] $\mathbb{C}^\times$ and $\mathbb{R}^\times$ are regarded as [[discrete groups]].

=--

This is recalled for instance in ([Freed 12, (1.38)](#Freed12)).

### The homotopy type of the 2-stack of super 2-lines
 {#HomtopyTypeOfSuper2Lines}

Now we can analyse the super 2-stack $2\mathbf{sLine}$ of super 2-line 2-bundles.

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

The object $2\mathbf{sLine} \in SmoothSuper\infty Grpd$ is equivalent to that which to $\mathbb{R}^{p|q}$ assigns the 2-groupoid whose

* objects are the algebra $C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even}$ and that algebra tensored with $Cl_1(\mathbb{C})$;

* morphisms are $C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even}$ regarded as a bimodule over itself and that bimodule tensored with $Cl_1(\mathbb{C})$;

* 2-morphisms form the group $C^\infty(\mathbb{R}^p, \mathbb{C}^\times)$.

=--

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

First, the $\mathbb{K}$-algebras in the topos of supergeometry are externally the ordinary [[superalgebras]], by the discussion at _[superalgebra -- As algebras in the topos over superpoints](super+algebra#AlgebraOverSuperpoints)_.

With this the statement is a straightforward generalization of the discussion at  _[superalgebra -- Picard 2-groupoid](super+algebra#Picard2Groupoid)_ from superalgebras over $\mathbb{C}$ to those over $C^\infty(\mathbb{R}^p, \mathbb{C})$.

While the invertible ordinary $\mathbb{C}^\infty(\mathbb{R})$-algebras are equivalent to that algebra itself (hence there is only one, up to equivalence); the invertible superalgebras are equivalent either to the ground field or to the complex [[Clifford algebra]] $Cl_1(\mathbb{C})$ (hence there are two, up to equivalence, the two elements in the [[Brauer group]] $\mathbb{Z}_2 = \pi_0(2\mathbf{sLine})$ ). Similarly for the invertible bimodules. Finally the invertible intertwiners are pointwise $\mathbb{C}^\times$.

=--

It follows that

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

The [[homotopy groups]] of the [[geometric realization]]
${\vert 2\mathbf{sLine} \vert}$ of $2\mathbf{sLine}$ are

| | ${\vert 2\mathbf{sLine}_{\mathbb{C}} \vert}$ | ${\vert 2\mathbf{sLine}_{\mathbb{R}} \vert}$ |
|--|--|--|
| $\pi_3$ | $\mathbb{Z}$ | $\mathbb{Z}$ |
| $\pi_2$ | 0 | 0 |
| $\pi_1$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |
| $\pi_0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_8$ |

=--

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

Therefore we have a canonical morphism

$$
  \mathbf{B}^2 \mathbb{C}^\times \simeq 2\mathbf{Line} \to 2\mathbf{sLine}
$$

in [[SmoothSuper∞Grpd]] (a [[n-monomorphism|2-monomorphism]]) from the [[moduli ∞-stack|moduli 2-stack]] of $\mathbb{C}^\times$-[[principal 2-bundles]]/[[bundle gerbes]], which picks the "ordinary" super 2-line bundle (as opposed to its "superpartner"), ignores the odd auto-[[gauge transformations]] of that and keeps all the [[higher gauge transformations]].

=--

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

In [[bosonic string theory]] over a [[spacetime]] $X$ the [[background gauge field]] called the [[B-field]] is a line 2-bundle [[circle n-bundle with connection|with connection]] given by a morphism

$$
  X \to 2\mathbf{Line} \simeq \mathbf{B}^2 \mathbb{C}^\times
  \,.
$$

However in [[type II superstring theory]] the [[B-field]] is actually a super line 2-bundle, hence given by a morphism

$$
  X \to 2\mathbf{sLine}
$$

in [[SmoothSuper∞Grpd]]. This observation (formulated in less stacky language) is due to the analysis of [[orientifold]] background fields in ([Precis](#Precis)).

=--

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

The first [[k-invariant]] of $\vert 2\mathbf{sLine}\vert$ is the essentially unique nontrivial

$$
  \mathbb{Z}_2 \to \mathbf{B}^2 \mathbb{Z}_2
$$

given by the [[Steenrod square]]. This is represented by the [[braided monoidal category|braiding]] equivalence on the [[tensor product]] of $Cl_1^{\mathbb{C}} \simeq \langle 1, e\rangle_{[e^2 = 1]}$

$$
  Cl_1^{\mathbb{C}}
  \otimes_{\mathbb{C}}
  Cl_1^{\mathbb{C}}
  \stackrel{\simeq}{\to}
  Cl_1^{\mathbb{C}}
  \otimes_{\mathbb{C}}
  Cl_1^{\mathbb{C}}
$$

given by the algebra homomorphism

$$
  \begin{aligned}
    1 \otimes 1 & \mapsto 1 \otimes 1
    \\
    e \otimes e & \mapsto - e \otimes e
    \\
    1 \otimes e & \mapsto e \otimes 1
    \\
    e \otimes 1 & \mapsto 1 \otimes e
  \end{aligned}
$$

(exchange the tensor factors and introduce a sign when exchanging two odd graded ones).

=--

For instance ([Freed 12, 1.42](#Freed12)).

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

The second [[k-invariant]]

$$
  \mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^4 \mathbb{Z}
$$

is the delooping of that of super lines $\mathbf{sLine}$, being the image under the [[Bockstein homomorphism]] of

$$
  \mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^3 U(1)
$$

which sends $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \to \mathbb{Z}_2 \hookrightarrow U(1)$.

(...)

=--

For instance ([Freed 12, 1.44](#Freed12)).

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

Comparison with the notation and terminology of
([Freed-Hopkins-Teleman 07](#FreedHopkinsTeleman)):

the "$\mathcal{B}\mathbb{T}^{\pm}$" on the top of p. 14 there is
$\Omega (2\mathbf{sLine})$ here; a graded line bundle over some $X$ there is a map of stacks $X \to \Omega (2\mathbf{sLine})$ here.

For $X$ 1-truncated, hence a groupoid, a graded central extension in the sense there is a map $X \to 2\mathbf{sLine}$ which factors as
$X \to \mathbf{B}\Omega (2\mathbf{sLine}) \to 2\mathbf{sLine}$.

=--

### Relation to the unconnected delooping of the $\infty$-group of units of $KU$
 {#RelationToUnconnectedDeloopingOfUnitsOfKU}

In ([Sagave 11](#Sagave11)) is introduced a "non-connected delooping" $bgl_1^\ast(E)$ of the [[∞-group of units]] $gl_1(E)$ of an [[E-∞ ring]] $E$, fitting into a [[homotopy cofiber sequence]]

$$
  gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E)
  \,.
$$

See at _[∞-Group of units -- Augmented definition](infinity-group+of+units#AugmentedDefinition)_.

By ([Sagave 11, theorem 12 and example 4.10](#Sagave11)) and comparing to the above discussion we have an [[equivalence]]

$$
  {\vert 2\mathbf{sLine}\vert}
  \simeq
  bgl_1^\ast(KU) \langle0,..,4\rangle
$$

of the [[geometric realization]] of the super-2-stack of super line 2-bundles with the [[n-truncated object in an (infinity,1)-category|4-truncation]] of the connected delooping of the [[infinity-group of units]] of $KU$.

hitting the Donovan-Karoubi twists of K-theory. This is what in ([Freed-Distler-Moore](#Precis), [Freed](#Freed12)) is written $R^{-1}$.

(...)

## References

Line 2-bundles in [[supergeometry]] as a model for the [[B-field]] and [[orientifolds]] are discussed (even if not quite explicitly in the language of higher bundles) in

* {#Freed12} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_, lectures at ESI Vienna, 2012 ([[FreedESI2012.pdf:file]])

based on the old results about the [[Picard 2-groupoid]] of complex [[super algebras]]

* {#Wall} [[C. T. C. Wall]], _Graded Brauer groups_, J. Reine Angew. Math. 213 (1963/1964), 187-199.

and based on the discussion of [[twisted K-theory]] in

* {#DonovanKaroubi} [[Peter Donovan]], [[Max Karoubi]], _Graded Brauer groups and $K$-theory with local coefficients_, Publications Mathématiques de l'IHÉS, 38 (1970), p. 5-25  ([numdam](http://www.numdam.org/item?id=PMIHES_1970__38__5_0))

as refined in

* {#FreedHopkinsTeleman} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Loop groups and twisted K-theory I_,  Journal of Topology (2011) 4 (4): 737-798 ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906))

and developing constructions in

* {#Precis} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

See also

* [[Tadeusz Józefiak]], _Semisimple Superalgebras_,  Volume 1352 of the series Lecture Notes in Mathematics pp 96-113.

Similar comments appear earlier on p. 8 and following of

* {#FreedHopkinsTeleman07} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Consistent Orientation of Moduli Spaces_ ([arXiv:0711.1909](http://arxiv.org/abs/0711.1909)), chapter XIX, pages 395-419 in: 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))

The above higher supergeometric story is made explicit in

* {#FSS} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_

The "unconnected delooping" of the [[infinity-group of units]] of an $E_\infty$-ring $E$ is introduced in

* {#Sagave11} [[Steffen Sagave]], _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))

and the specific example for the case of  $E = KU$ is in example 4.10 there.

[[!redirects super line 2-bundles]]

[[!redirects super 2-line bundle]]
[[!redirects super 2-line bundles]]


super Minkowski spacetime

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{: toc}

## Idea

Super-Minkowski spacetime is a [[super spacetime]] which is an analog in [[supergeometry]] of ordinary [[Minkowski spacetime]]. It is a [[super Cartesian space]] whose odd coordinates form a [[real spin representation]].

## Definition

Ordinary $(d+1)$-dimensional [[Minkowski space]] may be understood as the  quotient $Iso(\mathbb{R}^{d-1,1})/(Spin(d-1,1))$ of the [[Poincare group]] by the [[spin group]] cover of  [[Lorentz group]] -- the [[translation group]].

Analogously, the for each real irreducible [[spin representation]] $N$ the
$dim(N)$-extended [[supermanifold]] **Minkowski superspace** or **super Minkowski space** is the quotient of [[supergroups]] of the [[super Poincaré group]] by the corresponding [[spin group]] (a [[super Klein geometry]]).

The _[[super-translation group]]_. See there for more details.

Alternatively, regarded as a [[super Lie algebra]] this is the quotient of the [[super Poincaré Lie algebra]] by the relevant [[Lorentz Lie algebra]].

## Properties

### Canonical coordinates
 {#CanonicalCoordinates}

We briefly review some basics of the canonical [[coordinates]] and the super [[Lie algebra cohomology]] of the [[super Poincaré Lie algebra]] and [[super Minkowski space]] (see also at _[[super Cartesian space]]_ and at _[[signs in supergeometry]]_).

By the general discussion at [[Chevalley-Eilenberg algebra]], we may characterize the [[super Poincaré Lie algebra]] $\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$ by its CE-algebra $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ "of [[left-invariant 1-forms]]" on its group manifold.

Let $d \in \mathbb{N}$ and let $N$ be a real [[spin representation]] of $Spin(d-1,1)$. See at _[[Majorana representation]]_ for details.

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

The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ is generated on

* elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$

* and elements $\{\psi^\alpha\}$ of degree $(1,odd)$

where $a \in \{0,1, \cdots, d-1\}$ is a spacetime index, and where $\alpha$ is an index ranging over a basis of the chosen [[Majorana spinor]] representation $N$.

The CE-differential defined as follows

$$
  d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
$$

and

$$
  d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi
  \,.
$$

(which is the differential for the [[semidirect product]] of the [[Poincaré Lie algebra]] [[action|acting]] on the given [[Majorana spinor]] representation)

and

$$
  d_{CE} \, e^{a }
    =
   \omega^a{}_b \wedge e^b + \overline{\psi} \wedge \Gamma^a \psi
$$

where on the right we have the spinor-to-vector pairing in $N$ ([def.](Majorana+spinor#SpinorToVectorBilinearPairing)).

This defines the [[super Poincaré super Lie algebra]]. After discarding the terms involving $\omega$ this becomes the CE algebra of the [[super translation algebra]] underlying super Minkowski spacetime.

=--

In this way the [[super-Poincaré Lie algebra]] and its extensions is usefully discussed for instance in ([D'Auria-Fré 82](#DAuriaFre82)) and in ([Azcárraga-Townsend 89](AzcarragaTownsend89), [CAIB 99](#CAIB99)). In much of the literature instead the following equivalent notation is popular, which more explicitly involves the [[coordinates]] on [[super Minkowski space]].

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

The abstract generators in def. \ref{CEAlgebraOfSuperPoincare} are identified with [[left invariant 1-forms]] on the [[super-translation group]] (= super Minkowski spacetime) as follows.

Let $N$ be a [[real spin representation]] and let $(x^a, \theta^\alpha)$ be the canonical [[coordinates]] on the [[supermanifold]] $\mathbb{R}^{d-1,1\vert N}$ underlying the super-Minkowski super translation group. Then the canonical [[super vielbein]] is the $\mathbb{R}^{d-1,1\vert N}$-valued [[super differential form]] with components

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

* $e^a \coloneqq \mathbf{d} x^a + \overline{\theta} \Gamma^a \mathbf{d} \theta$.

Notice that this then gives the above formula for the differential of the [[super-vielbein]] in def. \ref{CEAlgebraOfSuperPoincare} as

$$
  \begin{aligned}
    d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta)
    \\
    & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta
    \\
    & = \frac{i}{2} \overline{\psi}\Gamma^a \psi
  \end{aligned}
  \,.
$$

=--

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

The term $\frac{i}{2}\bar \psi \Gamma^a \psi$ is sometimes called the *[[supertorsion]]* of the [[super-vielbein]] $e$, because the defining equation

$$
  d_{CE} e^{a } -\omega^a{}_b \wedge e^b = \frac{i}{2}\bar \psi \Gamma^a \psi
$$

may be read as saying that $e$ is [[torsion]]-free except for that term. Notice that this term is the only one that appears when the differential is applied to "Lorentz scalars", hence to object in $CE(\mathfrak{siso})$ which have "all indices contracted". See also at _[[torsion constraints in supergravity]]_.

Notably we have

$$
  d \left(
    \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}
  \right)
  \propto
  \left(
    \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}}
  \right)
  \wedge
  \left(
    \overline{\Psi} \wedge \Gamma_{a_p} \Psi
  \right)
  \,.
$$

This remaining operation "$e \mapsto \Psi^2$" of the differential acting on Loretz scalars is sometimes denoted "$t_0$", e.g. in ([Bossard-Howe-Stelle 09, equation (8)](#BossardHoweStelle09)).

This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of $(D,p)$ a [[Fierz identity]] implies that the term

$$
  \left(
    \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}}
  \right)
  \wedge
  \left(
    \overline{\Psi} \wedge \Gamma_{a_p} \Psi
  \right)
$$

vanishes identically, and hence in these dimensions the term

$$
    \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}
$$

is a [[cocycle]]. See also the [brane scan](#BraneScan) table below.

=--

### Cohomology and super $p$-branes

As opposed to ordinary [[Minkowski space]], the [[de Rham cohomology]] of [[left invariant forms]] of super-Minkowski space contains nontrivial exceptional [[cocycles]] (the _[[brane scan]]_). These serve as the [[Wess-Zumino-Witten model|WZW terms]] for the [[Green-Schwarz action functional]] (see there for more) of super-$p$-[[branes]] propagating on super-Minkowski space ([FSS 13](#FSS13)).

The corresponding $L_\infty$-extensions are _[[extended superspacetime]]_.

### As a central extension of the superpoint
 {#AsCentralExtensionOfTheSuperpoint}

Regarded as a [[super Lie algebra]], super Minkowski spacetime $\mathbb{R}^{d-1,1|N}$ has the single nontrivial super-Lie bracket given by the spinor bilinear pairing

$$
  \overline{(-)}\Gamma (-)
  \colon
  S \otimes S \longrightarrow V
$$

discussed in detail at _[[spin representation]]_.

Notice that this means that if one regards the [[superpoint]] $\mathbb{R}^{0|dim(N)}$ as an [[abelian super Lie algebra]], then super Minkowski spacetime is the [[Lie algebra extension]] of that by this bilinear pairing regarded as a super-[[Lie algebra cohomology|Lie algebra cocycle]] with [[coefficients]] in $\mathbb{R}^{d}$.

$$
  \array{
    \mathbb{R}^{d} &\longrightarrow& \mathbb{R}^{d-1,1|N}
    \\
    && \downarrow
    \\
    && \mathbb{R}^{0|dim(N)}
  }
$$

## Related concepts

* [[super anti de Sitter spacetime]]

[[!include local and global geometry - table]]

## References

The $d = 4$, $N =2$ super Minkowski spacetime was originally introduced in

* [[Abdus Salam]] J.A. Strathdee, _Supergauge Transformations_, Nucl.Phys. B76 (1974) 477-482 ([spire](http://inspirehep.net/record/89208))

* [[Abdus Salam]] J.A. Strathdee, Physical Review D11, 1521-1535 (1975)

see at "[[superspace]] in physics".

Further discussion includes:

* _Super spacetimes and super Poincaré-group_ ([pdf](http://www.math.ucla.edu/~vsv/papers/ch6.pdf))

* {#Freed01} [[Daniel Freed]], lecture 6 of _Classical field theory and Supersymmetry_, IAS/Park City Mathematics Series Volume 11 (2001) ([pdf](https://www.ma.utexas.edu/users/dafr/pcmi.pdf))

* [[Daniel Freed]], _Lecture 4 of [[Five lectures on supersymmetry]]_

* [[Veeravalli Varadarajan]], section 7 of _Supersymmetry for mathematicians: An introduction_

* [[Leonardo Castellani]], [[Riccardo D'Auria]], _[[Pietro Fre]], page 370, part II section II.3.3 of _[[Supergravity and Superstrings - A Geometric Perspective]]_

Discussion of how [[super L-infinity algebra]] [[L-infinity cocycle|extensions]] of super Minkowski spacetime yield all the [[brane scan]] of [[string theory]]/[[M-theory]] is in

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet]]_ (2013)

[[!redirects super Minkowski spacetime]]
[[!redirects super-Minkowski spacetime]]

[[!redirects super Minkowski spacetimes]]
[[!redirects super-Minkowski spacetimes]]

[[!redirects super Minkowski space]]
[[!redirects super Minkowski space]]
[[!redirects super Minkowski space]]

[[!redirects super Minkowski super Lie algebra]]
[[!redirects super Minkowski super Lie algebras]]


super module

[[!redirects super modules]]
[[!redirects super prevector space]]
[[!redirects super prevector spaces]]

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#Contents#
* table of contents
{:toc}

## Idea ##

A more general version of [[super vector space]].

## Definition ##

### $\mathbb{Z}/2\mathbb{Z}$-graded $R$-modules ###

Given a [[commutative ring]] $R$, an **$\mathbb{Z}/2\mathbb{Z}$-graded $R$-module** is a $R$-[[module]] $V$ with decomposition functions $\mathcal{D}_0:V \to V$ and $\mathcal{D}_1:V \to V$, such that

* for all $v:V$, $v = \mathcal{D}_0(v) + \mathcal{D}_1(v)$
* for all $a:R$, $b:R$, $v:V$, and $w:V$, $\mathcal{D}_0(a v + b w) = a \mathcal{D}_0(v) + b \mathcal{D}_0(w)$
* for all $a:R$, $b:R$, $v:V$, and $w:V$, $\mathcal{D}_1(a v + b w) = a \mathcal{D}_1(v) + b \mathcal{D}_1(w)$
* for all $v:V$, $\mathcal{D}_0(\mathcal{D}_0(v)) = \mathcal{D}_0(v)$
* for all $v:V$, $\mathcal{D}_0(\mathcal{D}_1(v)) = 0$
* for all $v:V$, $\mathcal{D}_1(\mathcal{D}_0(v)) = 0$
* for all $v:V$, $\mathcal{D}_1(\mathcal{D}_1(v)) = \mathcal{D}_1(v)$

As a result, the [[image]] of the two decomposition functions $\im(\mathcal{D}_0)$ and $\im(\mathcal{D}_1)$ are $R$-[[modules]]  and there exists a [[linear isomorphism]] $i:V \cong \im(\mathcal{D}_0) \otimes \im(\mathcal{D}_1)$, where $A \otimes B$ is the [[tensor product of modules]].

The elements of $\im(\mathcal{D}_0)$ are called **even elements** or **bosonic elements**, and the elements of $\im(\mathcal{D}_1)$ are called **odd elements** or **fermionic elements**.

### Super modules ###

The tensor product of $\mathbb{Z}/2\mathbb{Z}$-graded $R$-modules $A \otimes B$ for $a:A$, $b:B$ is defined as the following:

$$\mathcal{D}_0(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)$$

$$\mathcal{D}_1(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)$$

This plus the linearity of the $\mathcal{D}_0$ and $\mathcal{D}_1$ functions result in the category of $\mathbb{Z}/2\mathbb{Z}$-graded $R$-modules to be a [[monoidal category]].

A **super module** is an object of the category of $\mathbb{Z}/2\mathbb{Z}$-graded $R$-modules with the braiding for the tensor product $A \otimes B$:

$$t_{A, B}:(A \otimes B) \to (B \otimes A)$$

such that

$$\mathcal{D}_0(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) - \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)$$

$$\mathcal{D}_1(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)$$

## See also ##

* [[module]]

* [[graded module]]

* [[super vector space]]

* [[commutative ring]]

* [[super abelian group]]

* [[super algebra]]


super multiplet

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#Contents#
* table of contents
{:toc}

## Idea

A _supermultiplet_ is an [[irreducible representation]] of a [[supersymmetry]] [[supergroup]] or [[super Lie algebra]], decomposed as a [[direct sum]] ([[branching rule|branching]]) of [[irreps]] of the underlying ordinary group.

For the case of [[unitary representations of the super Poincaré group]], then under the canonical inclusion of the ordinary [[Poincaré group]] into the [[super Poincaré group]], the supermultiplets for super-isometries of [[super Minkowski spacetime]] decompose into a [[direct sum]] of irreducible [[unitary representations of the Poincaré group]]. Via the [[Wigner classification]] of fundamental [[particles]] as irreps of the [[Poincaré group]], this gives a decomposition of supermultiplets into [[particle]] species. The particles in this direct sum related by odd supersymmetry transformations are called _superpartners_.

Similar statements hold for the [[super anti de Sitter group]] or [[superconformal group]] etc.

By [[Deligne's theorem on tensor categories]] the most general regular [[tensor categories]] are those spanned by supermultiplets for some [[supergroup]].

## Related concepts

* [[superconformal multiplet]]

* [[adinkra]]

## References

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter II.4 in _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Veeravalli Varadarajan]], section 5.3 of _Unitary representations of super Lie groups_, 2006 ([pdf](http://www.math.ucla.edu/~vsv/ur.pdf))

See also:

* Wikipedia, _[Supermultiplet](https://en.wikipedia.org/wiki/Supermultiplet)_

Relation to [[L-infinity algebras|$L_\infty$-algebras]] via [[pure spinor]]-formalism:

* [[Martin Cederwall]], Simon Jonsson, [[Jakob Palmkvist]], [[Ingmar Saberi]], *Canonical supermultiplets and their Koszul duals* [[arXiv:2304.01258](https://arxiv.org/abs/2304.01258)]

[[!redirects super multiplets]]

[[!redirects supermultiplet]]
[[!redirects supermultiplets]]

[[!redirects super-multiplet]]
[[!redirects super-multiplets]]

[[!redirects superpartner]]
[[!redirects superpartners]]

[[!redirects super partner]]
[[!redirects super partners]]

[[!redirects vector multiplet]]
[[!redirects vector multiplets]]

[[!redirects tensor multiplet]]
[[!redirects tensor multiplets]]

[[!redirects hypermultiplet]]
[[!redirects hypermultiplets]]

[[!redirects hyper multiplet]]
[[!redirects hyper multiplets]]


super parallel transport

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* table of contents
{:toc}

## Idea

The notion of [[parallel transport]] in [[supergeometry]].

## Related concepts

* [[superconnection]]

## References

The basic ideas are discussed in

* [[Florin Dumitrescu]], _Superconnections and Parallel Transport_ ([arXiv:0711.2766](http://arxiv.org/abs/0711.2766))

The relation to the [[Chern character]] is established in

* [[Florin Dumitrescu]], _A geometric view of the Chern character_ ([arXiv:1202.2719](http://de.arxiv.org/abs/1202.2719))


super Poincaré group

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#Contents#
* table of contents
{:toc}

## Idea

The _super Poincaré group_ is the [[Lie integration]] in [[supergeometry]] of the [[super Poincaré Lie algebra]]. This is a [[super Lie group]]-extension of the ordinary [[Poincaré group]].

In [[physics]] an [[action]]/[[symmetry]] of the super Poincaré group is also called a _[[supersymmetry]]_.

The [[coset]] ([[super Klein geometry]]) of a super Poincaré group by the [[Spin group]]/[[Pin group]] inside it is a [[super translation group]], whose underlying [[supermanifold]] is a [[super Minkowski spacetime]].

## Related concepts

* [[Poincaré group]], [[Poincaré spinor group]]

* [[R-symmetry group]]

* [[super Poincaré Lie algebra]]

* [[super translation algebra]]

* [[unitary representation of the super Poincaré group]], [[spin representation]], [[BPS state]]

[[!include table of orthogonal groups and related]]

## References

* [[Pierre Deligne]], [[Daniel Freed]], section 1.1 of _Supersolutions_ ([arXiv:hep-th/9901094](http://arxiv.org/abs/hep-th/9901094))

  in [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], (eds.), _[[Quantum Fields and Strings]]_

* {#Freed01} [[Daniel Freed]], lecture 6 of _Classical field theory and Supersymmetry_, IAS/Park City Mathematics Series Volume 11 (2001) ([pdf](https://www.ma.utexas.edu/users/dafr/pcmi.pdf))

* [[Daniel Freed]], _Lecture 3 of [[Five lectures on supersymmetry]]_

* [[Veeravalli Varadarajan]], section 6,5 of  _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I 2004 ([pdf](http://www.math.ucla.edu/~vsv/papers/ch6.pdf))

* [[Antoine Van Proeyen]], _Tools for supersymmetry_ ([arXiv:hep-th/9910030](http://arxiv.org/abs/hep-th/9910030))

[[!redirects super Poincaré groups]]

[[!redirects super Poincare group]]
[[!redirects super Poincare groups]]

[[!redirects super-Poincare group]]
[[!redirects super-Poincare groups]]

[[!redirects super-Poincaré group]]
[[!redirects super-Poincaré groups]]


super Poincaré Lie algebra

[[!redirects super Poincare Lie algebra]]
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#Contents#
* table of contents
{:toc}

## Idea

The _super Poincaré Lie algebra_ is a [[super Lie algebra]] extension of a [[Poincaré Lie algebra]].

The corresponding [[super Lie group]] is the [[super Euclidean group]] (except for the signature of the metric).

## Definition
 {#Definition}

Let $d \in \mathbb{N}$ and consider [[Minkowski spacetime]] $\mathbb{R} ^{d-1,1}$ of [[dimension]] $d$. For $Spin(d-1,1)$ the corresponding [[spin group]], let

$$
  S \in Rep(Spin(d-1,1))
$$

be a real [[spin representation]] ([[Majorana spinor]]), which has the property ([def.](Majorana+spinor#SpinorToVectorBilinearPairing), [prop.](Majorana+spinor#SpinorToVectorPairing)) that there exists a [[linear map]]

$$
  \Gamma \;\colon\; S \otimes S \longrightarrow \mathbb{R}^{d-1,1}
$$

which is

1. symmetric:

1. $Spin(V)$ [[equivariance|equivariant]] (a [[homomorphism]] of [[spin representations]]).

For a classification of spin representations with this property see at _[[spin representations]]_ the sections _[real irreducible spin representations in Lorentz signature](spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature)_ and _[super Poincaré brackets](spin+representation#SuperPoincareBrackets)_. For explicit construction in components see at _[[Majorana spinor]]_ the section _[The spinor pairing to vectors](Majorana+spinor#TheSpinorPairingToVectors)_.

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

The **super Poincaré Lie algebra** $\mathfrak{siso}_S(d-1,1)$ of $d$-dimensional [[Minkowski spacetime]] with respect to the [[spin representation]] $S$ with symmetric and $Spin(V)$-equivariant pairing $\Gamma \colon S \otimes S \to \mathbb{R}^{d-1,1}$ is the [[super Lie algebra|super]] [[Lie algebra extension]] of the [[Poincaré Lie algebra]] by $\Pi S$ (the [[vector space]] underlying $S$ taken in odd degree)

$$
  \Pi S \longrightarrow \mathfrak{siso}_S(d-1,1)
  \longrightarrow \mathfrak{iso}(d-1,1) \simeq \mathbb{R}^{d-1,1} \ltimes \mathfrak{so}(d-1,1)
  \,,
$$

where the [[Lie bracket]] of elements in $\mathfrak{so}(d-1,1)$ with those in $S$ is the given [[action]], the Lie bracket of elements of $\mathbb{R}^{d-1,1}$ with those on $S$ is trivial, and the Lie bracket of two elements $s_1, s_2 \in S$ is given by $\Gamma$:

$$
  [s_1,s_2] \coloneqq \Gamma(s_1,s_2)
  \,.
$$

=--

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

It is precisely the symmetry and $Spin(V)$-equivariant assumption on $\Gamma$ that makes this a well defined [[super Lie algebra]]: the symmetry corresponds to the graded skew-symmetry of the [[Lie bracket]] on elements in $S$, which are regarded as odd, and the $Spin(V)$-equivariance yields the nontrivial [[Jacobi identity]] for $o \in \mathfrak{so}(d-1,1)$ and $s_1, s_2 \in S$:

$$
  \Gamma([o,s_1], s_2)
  +
  \Gamma(s_1, [o,s_2])
  =
  [o, \Gamma(s_1,s_2)]
  \,.
$$

=--

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

By the general discussion at [[Chevalley-Eilenberg algebra]], we may characterize the [[super Poincaré Lie algebra]] $\mathfrak{siso}_S(D-1,1)$ by its CE super-[[dg-algebra]] $CE(\mathfrak{siso}_S(D-1,1))$ "of [[left-invariant 1-forms]]" on its group manifold.

Write $\{\omega_a{}^b\}_{a,b}$ for the canonical basis of the [[special orthogonal Lie algebra|special orthogonal]] [[matrix Lie algebra]] $\mathfrak{so}(D-1,1)$ and write $\{\psi_\alpha\}_\alpha$ for a corresponding [[basis]] of the [[spin representation]] $S$.

The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{siso}_N(d-1,1))$ is generated on

* elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$

* and elements $\{\psi^\alpha\}$ of degree $(1,odd)$

with the differential defined by

$$
  d_{CE} \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
$$

$$
  d_{CE} e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
$$

$$
  d_{CE} \psi = \frac{1}{4} \omega^{ a b} \wedge \Gamma_{a b} \psi
  \,.
$$

=--

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

Removing all terms involving $\omega$ here yields the [[Chevalley-Eilenberg algebra]] of the [[super translation algebra]] $\mathbb{R}^{D;N}$.

=--

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

The abstract generators in def. \ref{CEAlgebraOfSuperPoincare} are identified with [[left invariant 1-forms]] on the [[super-translation group]] as follows.

Let $(x^a, \theta^\alpha)$ be the canonical [[coordinates]] on the [[supermanifold]] $\mathbb{R}^{d|N}$ underlying the super translation group. Then the identification is

* $\psi^\alpha = d \theta^\alpha$.

* $e^a = d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta$.

This then gives the formula for the differential of the super-[[vielbein]] in def. \ref{CEAlgebraOfSuperPoincare} as

$$
  \begin{aligned}
    d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta)
    \\
    & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta
    \\
    & = \frac{i}{2} \overline{\psi}\Gamma^a \psi
  \end{aligned}
  \,.
$$

=--

## Properties

### Lie algebra cohomology
 {#LieAlgebraCohomology}

The super Poincaré Lie algebra has, on top of the [[Lie algebra cohomology|Lie algebra cocycles]] that it inherits from $\mathfrak{so}(n)$, a discrete number of exceptiona cocycles bilinear in the spinors, on the [[super translation algebra]], that exist only in very special dimensions.

The following theorem has been stated at various placed in the physics literature (known there as the _[[brane scan]]_ for $\kappa$-symmetry in _[[Green-Schwarz action functionals]]_ for super-$p$-[[branes]] on [[super-Minkowski spacetime]]). A full proof is in [Brandt 12-13](#Brandt12-13). The following uses the notation in terms of [[division algebras]] ([Baez-Huerta 10](#BaezHuerta10)).

**Theorem**

* In dimensional $d =  3,4,6, 10$, $\mathfrak{siso}(d-1,1)$ has a nontrivial 3-cocycle given by

  $$
    (\psi, \phi, A) \mapsto g(\psi \cdot \phi, A)
  $$

  for spinors $\psi, \phi \in \mathcal{S}$ and vectors $A \in \mathcal{T}$, and 0 otherwise.

* In dimensional $d =  4,5,7, 11$, $\mathfrak{siso}(d-1,1)$ has a nontrivial 4-cocycle given by

  $$
    (\Psi, \Phi, \mathcal{A}, \mathcal{B}) \mapsto
    \langle \Psi , (\mathcal{A}\mathcal{B}- \mathcal{B} \mathcal{A})\Phi \rangle
  $$

  for spinors $\Psi, \Phi \in \mathcal{S}$ and vectors $\mathcal{A}, \mathcal{B} \in \mathcal{V}$, with the commutator taken in the Clifford algebra.

The 4-cocycle in $d = 11$ is the one that induces the [[supergravity Lie 3-algebra]].

All these cocycles are controled by the relevant [[Fierz identities]].

### Extensions

#### Super $L_\infty$-algebra extensions

The [[super L-infinity algebra]] [[infinity-Lie algebra cohomology]] of the super Poincaré Lie algebra corresponding to the [above](#LieAlgebraCohomology) cocycles involves

[[supergravity Lie 6-algebra]] $\to$ [[supergravity Lie 3-algebra]] $\to$ **super-Poincaré Lie algebra**

#### Extended super Poincaré Lie algebra -- Polyvector extensions
 {#PolyvectorExtensions}

The super-Poincaré Lie algebra has a class of super [[Lie algebra extensions]] called _[[extended supersymmetry]]_ algebras or _polyvector extensions_ , because they involve additional generators that transforn as skew-symmetric [[tensors]]. A complete classification is in ([ACDP](#ACDP)).

For instance the "[[M-theory Lie algebra]]" is a polyvector extension of the super Poincaré Lie algebra $\mathfrak{siso}_{N=1}(10,1)$ by polyvectors of rank $p = 2$ and $p=5$ (the [[M2-brane]] and the [[M5-brane]] in the [[brane scan]]), see below [Polyvector extensions as automorphism Lie algebras](#PolyvectorExtensionsAsAutomorphismLieAlgebras).

##### As current algebras of the GS super $p$-branes

The polyvector extensions arise as the super Lie algebras of [[conserved currents]] of the [[Green-Schwarz super p-brane sigma-models]] ([AGIT 89](#AGIT89)).

##### As automorphism Lie algebras of Lie $n$-superalgebras
 {#PolyvectorExtensionsAsAutomorphismLieAlgebras}

At least some of the [polyvector extensions](#PolyvectorExtensions) of the super Poincaré Lie algebra arise as the [[automorphism]] super Lie algebras of the [[Lie n-algebra]]
[[infinity-Lie algebra cohomology|extensions]] classified by the cocycles discussed above.

For instance the automorphisms of the [[supergravity Lie 3-algebra]] gives the "[[M-theory Lie algebra]]"-extension of super-Poincaré in 11-dimensions ([FSS 13](#FSS13)). This is also discussed at _[supergravity Lie 3-algebra -- Polyvector extensions](supergravity+Lie+3-algebra#Polyvector)_.

## Related concepts

* [[super Minkowski spacetime]], [[signs in supergeometry]]

* [[super Poincaré group]], [[supersymmetry]]

* [[supermultiplet]], [[BPS state]]

* [[division algebra and supersymmetry]]

* [[R-symmetry]]

* [[super translation algebra]]

* [[Green-Schwarz action functional]], [[brane scan]]

* [[4d supergravity Lie 2-algebra]], [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]

## References

Introducing the [[super Poincaré Lie algebra]] ("[[supersymmetry]]"):

* {#GolfandLikhtman72} [[Yuri Golfand]], [[Evgeny Likhtman]],_On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators_, in: [[Victor Ginzburg]] et al. (eds.) _I. E. Tamm Memorial Volume Problems of Theoretical Physics_, (Nauka, Moscow 1972), page 37,

  translated and reprinted in: [[Mikhail Shifman]] (ed.) _[[The Many Faces of the Superworld]]_ pp. 44-53,  World Scientific (2000) ([doi:10.1142/9789812793850_0006](https://doi.org/10.1142/9789812793850_0006))

### General

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.2.1 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

The seminal classification result of simple supersymmetry algebras is due to

* [[Werner Nahm]], _Supersymmetries and their Representations_, Nucl.Phys. B135 (1978) 149 ([spire](http://inspirehep.net/record/120988?ln=en))

Lecture notes include

* _Super spacetimes and super Poincaré-group_ ([pdf](http://www.math.ucla.edu/~vsv/papers/ch6.pdf))

* {#Freed01} [[Daniel Freed]], lecture 6 of _Classical field theory and Supersymmetry_, IAS/Park City Mathematics Series Volume 11 (2001) ([pdf](https://www.ma.utexas.edu/users/dafr/pcmi.pdf))

* [[Daniel Freed]], _Lecture 4 of [[Five lectures on supersymmetry]]_

* {#Varadarajan04} [[Veeravalli Varadarajan]], section 7 of _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)

See also

* {#CAIB99} C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

for discussion in the view of the [[brane scan]] and [[schreiber:The brane bouquet]] of super-$p$-[[brane]] [[Green-Schwarz sigma-models]].

### Polyvector extensions
 {#PolyvectorExtensionRefs}

The Polyvector extensions of $\mathfrak{Iso}(\mathbb{R}^{10,1|32})$ (the "[[M-theory super Lie algebra]]") were first considered in

* {#DAuriaFre82}  [[Riccardo D'Auria]], [[Pietro Fré]] _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140

 * [[Jan-Willem van Holten]], [[Antoine Van Proeyen]], _$N=1$ supersymmetry algebras in $d=2,3,4 \,mod\, 8$_ J.Phys. A15, 3763 (1982).

Polyvector extensions were found as the algebra of [[conserved currents]] of the [[Green-Schwarz super p-branes]] in

* {#AGIT89} [[José de Azcárraga]], [[Jerome Gauntlett]], J.M. Izquierdo, [[Paul Townsend]], _Topological Extensions of the Supersymmetry Algebra for Extended Objects_, Phys.Rev.Lett. 63 (1989) 2443 ([spire](https://inspirehep.net/record/26393?ln=en))

reviewed in section 8.8. of

* [[José de Azcárraga]], José M. Izquierdo, _Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics_ , Cambridge monographs of mathematical physics, (1995)

and specifically for super-[[D-branes]] this discussion is in

* Hanno Hammer, _Topological Extensions of Noether Charge Algebras carried by D-p-branes_, Nucl.Phys. B521 (1998) 503-546 ([arXiv:hep-th/9711009](http://arxiv.org/abs/hep-th/9711009))

The role of polyvector extended supersymmetry algebras in [[supergravity]] and [[string theory]] is further highlighted in

* [[Paul Townsend]], _$p$-Brane Democracy_ ([arXiv:hep-th/9507048](http://arxiv.org/abs/hep-th/9507048))

A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in

* {#ACDP} [[Dmiti Alekseevsky]], [[Vicente Cortés]], C. Devchand, [[Antoine Van Proeyen]], _Polyvector Super-Poincaré Algebras_ Commun.Math.Phys. 253 (2004) 385-422 ([arXiv:hep-th/0311107](http://arxiv.org/abs/hep-th/0311107))

### Super Lie algebra cohomology
 {#ReferencesLieAlgebraCohomology}

Discussion of the super-[[Lie algebra cohomology]] of the [[super Poincare Lie algebra]] goes back to work on [[Green-Schwarz sigma models]] in

* {#AzcarragaTownsend89} [[José de Azcárraga]], [[Paul Townsend]], _Superspace geometry and classification of supersymmetric extended objects_, Phys. Rev. Lett. 62, 2579--2582 (1989)

A rigorous classification of these cocycles was later given in

* {#Brandt12-13} [[Friedemann Brandt]], _Supersymmetry algebra cohomology_

  _I: Definition and general structure_  J. Math. Phys.51:122302, 2010, ([arXiv:0911.2118](http://arxiv.org/abs/0911.2118))

  _II: Primitive elements in 2 and 3 dimensions_, J. Math. Phys. 51 (2010) 112303 ([arXiv:1004.2978](http://arxiv.org/abs/1004.2978))

  _III: Primitive elements in four and five dimensions_, J. Math. Phys. 52:052301, 2011 ([arXiv:1005.2102](http://arxiv.org/abs/1005.2102))

  _IV: Primitive elements in all dimensions from $D=4$ to $D=11$_, J. Math. Phys. 54, 052302 (2013) ([arXiv:1303.6211](http://arxiv.org/abs/1303.6211))

A classification of some special cases of signature/supersymmetry of this is also in the following (using a computer algebra system):

* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries_ ([arXiv:1011.4731](http://arxiv.org/abs/1011.4731))

* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra_, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483–503 ([arXiv:1106.0335](http://arxiv.org/abs/1106.0335))

For applications of this classification see also at _[[Green-Schwarz action functional]]_ and at _[[brane scan]]_.

An introduction to the exceptional fermionic cocycles on the super Poincaré Lie algebra, and their description using
[[normed division algebras]], are discussed here:

* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry I_ ([arXiv:0909.0551](http://arxiv.org/abs/0909.0551))

* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry II_ ([arXiv:1003.34360](http://arxiv.org/abs/1003.3436))
 {#BaezHuerta10}

* [[John Huerta]], _Division Algebras, Supersymmetry and Higher Gauge Theory_, ([arXiv:1106.3385](http://arxiv.org/abs/1106.3385))

This subsumes some of the results in ([Azcárraga-Townend](#AzcarragaTownsend89))

Discussion of the corresponding [[super L-∞ algebra]] [[L-∞ extensions]] in the context of [[Green-Schwarz action functionals]] and [[schreiber:∞-Wess-Zumino-Witten theory]] is in

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_

A direct constructions of ordinary (Lie algebraic) extensions of the super Poincaré Lie algebra by means of [[division algebras]] is in

* Jerzy Lukierski, Francesco Toppan, _Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory_ ([pdf](http://cbpfindex.cbpf.br/publication_pdfs/NF00102.2010_08_03_10_47_48.pdf))

For more on this see at _[[division algebra and supersymmetry]]_.

[[!redirects super Poincare Lie algebras]]

[[!redirects super Poincaré Lie algebra]]
[[!redirects super Poincaré Lie algebras]]

[[!redirects super Poincaré super Lie algebra]]
[[!redirects super Poincaré super Lie algebras]]

[[!redirects super-Poincaré Lie algebra]]


super q-Schur algebra

The super $q$-Schur algebra $S_{m|n}^d$ is the analogue over the [[Lie superalgebra]] $\mathfrak{sl}(m|n)$ of the usual $q$-[[q-Schur algebra|Schur algebra]].  That is, if we let $V$ denote the deformation of the defining representation of $\mathfrak{sl}(m|n)$, then
$$S_{m|n}^d=End_{U_q(\mathfrak{sl}(m|n))}(V^{\otimes d}).$$

When $q$ is generic, this is a quotient of the Hecke algebra by the ideal generated by all representations whose corresponding representations are not hooks.

In fact, when $m=n=1$ the weight spaces of $V^{\otimes d}$ are irreducible representations, with arm length of the hook  corresponding to weight.
+--{: .query}
[[Ben Webster]]: Does $S_{m|n}^d$ inherit a basis from the Hecke algebra in the usual way?  It certainly seems that the representation $V^{\otimes d}$ has a basis on which positive integral combination of the Kazhdan-Lusztig basis vectors act with positive integral coefficients.

Also, does this picture have a categorification?  One natural candidate to use in the $m=n=1$ case is the [categorifications of the cell modules](http://front.math.ucdavis.edu/0702.5811) due to Stroppel and Mazorchuk.  Also possibly of interest is their [categorification of Wedderburn basis](http://front.math.ucdavis.edu/0708.3949) of $S_n$
=--

* See also MathOverflow: [Does the super Temperley-Lieb algebra have a Z-form?](http://mathoverflow.net/questions/3299/does-the-super-temperley-lieb-algebra-have-a-z-form/3442#3442)