L-infinity-algebra

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###Context###
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
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#### Rational homotopy theory
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[[!include differential graded objects - contents]]
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#Contents#
* table of contents
{:toc}

## Idea ##

_$L_\infty$-algebras_ (or _strong homotopy Lie algebras_) are a higher generalization (a "[[vertical categorification]]") of [[Lie algebras]]: in an $L_\infty$-algebra the [[Jacobi identity]] is allowed to hold (only) up to higher [[coherence law|coherent]] [[homotopy]].

An $L_\infty$-algebra that is concentrated in lowest degree is an ordinary [[Lie algebra]]. If it is concentrated in the lowest two degrees is is a [[Lie 2-algebra]], etc.

From another perspective: an $L_\infty$-algebra is a [[Lie ∞-algebroid]] with a single object.

$L_\infty$-algebras are [[infinitesimal space|infinitesimal]] approximations of [[smooth ∞-group]]s in analogy to how an ordinary Lie algebra is an infinitesimal approximation of a [[Lie group]]. Under [[Lie integration]] every $L_\infty$-algebra $\mathfrak{g}$ "exponentiates" to a [[smooth ∞-group]] $\Omega \exp(\mathfrak{g})$.

## History
 {#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{LInfinityDefinitionViaGeneralizedJacobiIdentity}) were introduced in [Stasheff 92](#Stasheff92), [Lada-Stasheff 92](#LadaStasheff92), based on the example of such a structure on the [[BRST complex]] of the [[bosonic string]] that  was reported in the construction of [[closed string field theory]] in [Zwiebach 92](#Zwiebach92).

[Lada-Stasheff 92](#LadaStasheff92) credit [Schlessinger-Stasheff 85](deformation+theory#SchlessingerStasheff85) with the introduction of the concept, but while that article considers many closely related structures, it does not consider $L_\infty$-algebras as such. [Lada-Markl 94](#LadaMarkl94) credit other work by Schlessinger-Stasheff as the origin, but that work appeared much later as [Schlessinger-Stasheff 12](deformation+theory#SchlessingerStasheff12).

According to [Stasheff 16, slide 25](#Stasheff16), Zwiebach had this structure already in 1989, when Stasheff recognized it in a talk by Zwiebach at a [[GUT]] conference in Chapel-Hill.   Zwiebach, in turn, was following the [[BV-formalism]] of [Batalin-Vilkovisky 83](#BatalinVilkovisky83), [Batakin-Fradkin 83](#BatakinFradkin83), whose relation to $L_\infty$-algebras was later amplified in [Stasheff 96](#Stasheff96), [Stasheff 97](#Stasheff97).

The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras
is due to [Lada-Markl 94](#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, independently of the [[BV-formalism]] of [Batalin-Vilkovisky 83](#BatalinVilkovisky83), [Batakin-Fradkin 83](#BatakinFradkin83),  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.

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](BuijsFelixMurillo12).

=--

## Definition ##

### In terms of algebras over an operad

An **$L_\infty$-algebra** is an [[algebra over an operad]] in the [[category of chain complexes]] over the [[L-∞ operad]].

In the following we spell out in detail what this means in components.

### In terms of higher brackets
 {#DefinitionViaHigherBrackets}

We now state the definition of $L_\infty$-algebras that is most directly related to the traditional definition of ordinary [[Lie algebras]], namely as $\mathbb{Z}$-[[graded vector space]] $\mathfrak{g}$ equipped with $n$-ary [[multilinear maps|multilinear]] and graded-skew symmetric maps $[-,\cdots,-]$ -- the "brackets" -- that satisfy a generalization of the [[Jacobi identity]].

To that end, we here choose grading conventions such that the following definition of $L_\infty$-algebras reduces to that of ordinary [[Lie algebras]] when $\mathfrak{g}$ is concentrated in _degree zero_. Moreover we take the [[differential]] of the underlying [[chain complex]] of the $L_\infty$-algebra to have degree $-1$ ("homological grading"). Together this means in particular that $\mathfrak{g}$ is a  [[Lie n-algebra]] for $n \in \mathbb{N}$, $n \geq 1$, if it is concentrated in degrees 0 to $n-1$.

Beware that there are also other conventions possible, and there are other conventions in use, for both these choices, leading to different signs in the following formulas.

+-- {: .num_defn #GradedSignatureOfPermutation}
###### Definition
**(graded signature of a permuation)**

Let $V$ be a $\mathbb{Z}$-[[graded vector space]], and 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 $\vert v_i \vert  \in \mathbb{Z}$, i.e. such that $v_i \in V_{\vert v_i\vert}$.

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 _$\mathbf{v}$-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 $(-1)^{\vert v_i \vert \vert v_j \vert}$ 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.

=--

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

An $L_\infty$-algebra is

1. a $\mathbb{Z}$-[[graded vector space]] $\mathfrak{g}$;

1. for each $n \in \mathbb{N}$, $n \geq 1$ 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 $n-2$

   (if one includes here $n = 0$ then one speaks of a _[[curved L-infinity algebra]]_)

such that the following conditions hold:

1. (**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 $(v_1,\cdots,v_n)$-graded signature of the permuation $\sigma$, according to def. \ref{GradedSignatureOfPermutation};

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

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

   where the inner sum runs over all $(i,j-1)$-[[unshuffles]] $\sigma$ and where $\chi$ is the graded signature sign from def. \ref{GradedSignatureOfPermutation}.

=--

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

In lowest degrees the generalized Jacobi identity says that

1. for $n = 1$: the unary map $\partial \coloneqq l_1$ squares to 0:

   $$
     \partial (\partial(v_1)) = 0
   $$

1. for $n = 2$: the unary map $\partial$ is a graded [[derivation]] of the binary map

   $$
     - [\partial v_1, v_2]
     - (-1)^{\vert v_1 \vert \vert v_2 \vert}
     [\partial v_2, v_1]
     +
     \partial [v_1, v_2]
     =
     0
   $$

   hence

   $$
     \partial [v_1, v_2] =  [\partial v_1, v_2] + (-1)^{\vert v_1 \vert} [v_1, \partial v_2]
     \,.
   $$

=--

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

When all higher brackets vanish, $l_{k \gt 2}= 0$ then for $n = 3$:

$$
  [[v_1,v_2],v_3]
  +
  (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)}
  [[v_2,v_3],v_1]
  +
  (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)}
  [[v_1,v_3],v_2]
  =
  0
$$

this is the graded [[Jacobi identity]]. So in this case the $L_\infty$-algebra
is equivalently a [[dg-Lie algebra]].

=--

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

When $l_3$ is possibly non-vanishing, then on elements $x_i$ on which $\partial = l_1$ vanishes, the generalized Jacobi identity for $n = 3$ gives

$$
  [[v_1,v_2],v_3]
  +
  (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)}
  [[v_2,v_3],v_1]
  +
  (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)}
  [[v_1,v_3],v_2]
  =
  - \partial [v_1, v_2, v_3]
  \,.
$$

This shows that the Jacobi identity holds up to an "exact" term, hence up to homotopy.

=--

### In terms of semifree differential coalgebra
 {#ReformulationInSemifreeDgCoalgebra}

In ([Lada-Stasheff 92](#LadaStasheff92)) it was pointed out that the higher brackets of an $L_\infty$-algebra (def. \ref{LInfinityDefinitionViaGeneralizedJacobiIdentity}) induce on the graded-co-commutative [[cofree coalgebra]] $\vee^\bullet \mathfrak{g}$ over the underlying graded vector space $\mathfrak{g}$ the structure of a [[differential graded coalgebra]], with differential $D = [-] + [-,-] + [-,-,-] + \cdots$ the sum of the higher brackets, extended as graded [[coderivations]]. The higher Jacobi identity is equivalently the condition that $D^2 = 0$. In ([Lada-Markl 94](#LadaMarkl94)) it was observed that conversely, such "semifree" [[differential graded coalgebras]] are an equivalent incarnation of $L_\infty$-algebras.

(If one uses unital dg-co-algebras then the $L_\infty$-algbras encoded with way are generally [[curved L-infinity algebras]]. To restrict to the non-curved one one either considers co-augmented unital dg-co-algebras or non-unital coalgebras.)

Notice that this immediately imples that if $\mathfrak{g}$ is degreewise finite dimensional, then passing to [[dual vector spaces]] turns semifree [[differential graded coalgebra]]
into [[semifree dga|semifree]] [[differential graded algebras]], which hence are [[opposite category|opposite]]-equivalent to $L_\infty$-algebras of [[finite type]]. For $\mathfrak{g}$ an ordinary finite dimensional [[Lie algebra]], then this dg-algebras is its [[Chevalley-Eilenberg algebra]], hence we may generally speak of [[Chevalley-Eilenberg algebras]] of $L_\infty$-algebras of [[finite type]] (and also more generally, if one invokes [[pro-objects]], see at [model structure for L-infinity algebras -- Use of pro-dg-algebras](model+structure+for+L-infinity+algebras#OnProAlg) ).

In term of the [[operad|operadic]] definition of $L_\infty$-algebras [above](#InTermsOfAlgebrasOverAnOperad) this equivalence is an incarnation of the [[Koszul duality]] between the [[Lie operad]] and the [[commutative operad]].

We now spell out this dg-coalgebraic incarnation of $L_\infty$-algebras.

A (connected) **$L_\infty$-algebra** is

* an $\mathbb{N}_+$-graded vector space $\mathfrak{g}$;

* equipped with a differential $D : \vee^\bullet \mathfrak{g} \to \vee^\bullet \mathfrak{g}$ of degree $-1$ on the [[free graded co-commutative coalgebra]] over $\mathfrak{g}$ that squares to 0

$$
  D^2 = 0
  \,.
$$

Here the free graded co-commutative co-algebra $\vee^\bullet \mathfrak{g}$ is, as a vector space, the same as the graded [[Grassmann algebra]] $\wedge^\bullet \mathfrak{g}$ whose elements we write as

$$
  3 t_1 \vee t_2 + t_3 + t_3 \vee t_4 \vee t_5
$$

etc (where the $\vee$ is just a funny way to write the wedge $\wedge$, in order to remind us that:...)

but thought of as equipped with the standard coproduct

$$
  \Delta (v_1 \vee v_2 \cdots \vee v_n)
  \propto \sum_i \pm (v_1 \vee \cdots \vee v_i) \otimes (v_{i+1} \vee \cdots \vee v_n)
$$

> (work out or see the references for the signs and prefacors).

Since this is a _free_ graded co-commutative coalgebra, one can see that any differential

$$
  D : \vee^\bullet \mathfrak{g} \to \vee^\bullet \mathfrak{g}
$$

on it is fixed by its value "on cogenerators" (a statement that is maybe unfamiliar, but simply the straightforward dual of the more familar statement to which we come below, that differentials on free graded algebras are fixed by their action on generators) which means that we can decompose $D$ as

$$
  D = D_1 +  D_2 + D_3 + \cdots
  \,,
$$

where each $D_i$ acts as $l_i$ when evaluated on a homogeneous element of the form $t_1 \vee \cdots \vee t_n$ and is then uniquely extended to all of $\vee^\bullet \mathfrak{g}$ by extending it as a _coderivation_ on a coalgebra.

For instance $D_2$ acts on homogeneous elements of word lenght 3 as

$$
  D_2(t_1 \vee t_2 \vee t_3) =
  D_2(t_1, t_2)\vee t_3 \pm permutations
  \,.
$$

> exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below.

Using this, one checks that the simple condition that $D$ squares to 0 is precisely equivalent to the infinite tower of generalized Jacobi identities:

$$
  (D^2 = 0)
  \Leftrightarrow
  \left(
    \forall n :
  \sum_{i+j = n} \sum_{shuffles \sigma}
  \pm l_i (l_j (v_{\sigma(1)}, \cdots, v_{\sigma(j)} )
   , v_{\sigma(j+1)} , \cdots , v_{\sigma(n)} ) = 0
  \right)
  \,.
$$

So in conclusion we have:

+-- {: .standout}

An $L_\infty$-algebra is a [[dg-coalgebra]] whose underlying [[coalgebra]] is cofree and concentrated in negative degree.

=--

### In terms of semifree differential algebra
 {#ReformulationInTermsOfSemifreeDGAlgebra}

The reformulation of an $L_\infty$-algebra as simply a semi-co-free graded-co-commutative coalgebra $(\vee^\bullet \mathfrak{g}, D)$ is a useful repackaging of the original definition, but the coalgebraic aspect tends to be not only unfamiliar, but also a bit inconvenient. At least when the graded vector space $\mathfrak{g}$ is degreewise [[finite number|finite]] [[dimension|dimensional]], we may simply pass to its degreewise dual graded vector space $\mathfrak{g}^*$.

(Fully generally the following works when using not just dg-algebras but [[pro-objects]] in [[dg-algebras]], see at _[model structure for L-infinity algebras -- Use of pro-dg-algebras](model+structure+for+L-infinity+algebras#OnProAlg)_).

Its [[Grassmann algebra]] $\wedge^\bullet \mathfrak{g}^*$ is then naturally equipped with an ordinary differential $d = D^*$ which acts on $\omega \in \wedge^\bullet \mathfrak{g}^*$ as

$$
  (d \omega) (t_1 \vee \cdots \vee t_n) =
  \pm \omega(D(t_1 \vee \cdots \vee t_n))
  \,.
$$

When the grading-dust has settled one finds that with

$$
  \wedge^\bullet \mathfrak{g}^* =
  k \oplus \mathfrak{g}^*_1 \oplus (\mathfrak{g}^*_1 \wedge \mathfrak{g}^*_1 \oplus \mathfrak{g}^*_2) \oplus \cdots
$$

with the ground field in degree 0, the degree 1-elements of $\mathfrak{g}^*$ in degree 1, etc, that $d$ is of degree +1 and of course squares to 0

$$
  d^2 = 0
  \,.
$$

This means that we have a [[semifree dga]]

$$
  CE(\mathfrak{g}) := (\wedge^\bullet \mathfrak{g}^*, d)
  \,.
$$

In the case that $\mathfrak{g}$ happens to be an ordinary [[Lie algebra]], this is the ordinary [[Chevalley-Eilenberg algebra]] of this Lie algebra. Hence we should generally call $CE(\mathfrak{g})$ the [[Chevalley-Eilenberg algebra]]
of the $L_\infty$-algebra $\mathfrak{g}$.

One observes that this construction is bijective: every (degreewise finite dimensional) cochain [[semifree dga]] generated in positive degree comes from a (degreewise finite dimensional) $L_\infty$-algebra this way.

This means that we may just as well _define_ a (degreewise finite dimensional) $L_\infty$-algebra as an object in the [[opposite category]] of (degreewise finite dimensional) commutative [[dg-algebra]]s that are [[semifree dga]]s and generated in positive degree.

(In general this corresponds to [[curved L-infinity algebra]]. The flat $L_\infty$-algebras $\mathfrak{g}$ dually correspond to the dg-algebras which are [[augmented algebra|augmented]] over $\mathbb{R}$, i.e for which the canonical projection $CE(\mathfrak{g}) \longrightarrow \mathbb{R}$ is a homomorphism of dg-algebras.)

And this turns out to be one of the most useful perspectives on $L_\infty$-algebras.

In particular, if we simply drop the condition that the dg-algebra be generated in positive degree and allow it to be generated in non-negative degree over the algebra in degree 0, then we have the notion of the (Chevalley-Eilenberg algebra of) an [[L-infinity-algebroid]].

#### Details
  {#DGAlgebraDetails}

We discuss in explit detail the computation that shows that an $L_\infty$-algebra structure on $\mathfrak{g}$ is equivalently a [[dg-algebra]]-structure on $\wedge^\bullet \mathfrak{g}^*$.

Let $\mathfrak{g}$ be a degreewise finite-dimensional $\mathbb{N}_+$[[graded vector space]] equipped with multilinear graded-symmetric maps

$$
  [-,\cdots,-]_k : Sym^k \mathfrak{g} \to \mathfrak{g}
$$

of degree -1, for each $k \in \mathbb{N}_+$.

Let $\{t_a\}$ be a [[basis]] of $\mathfrak{g}$ and $\{t^a\}$ a dual basis of the degreewise dual $\mathfrak{g}^*$. Equip the [[Grassmann algebra]] $Sym^\bullet \mathfrak{g}^*$ with a [[derivation]]

$$
  d : Sym^\bullet \mathfrak{g}^* \to Sym^\bullet \mathfrak{g}^*
$$

defined on generators by

$$
  d : t^a
    \mapsto
    -
   \sum_{k = 1}^\infty
    \frac{1}{k!}
   [t_{a_1}, \cdots, t_{a_k}]^a_k
   \,
   t^{a_1} \wedge \cdots \wedge t^{a_k}
  \,.
$$

Here we take $t^a$ to be of the same degree as $t_a$. Therefore this derivation has degree +1.

We compute the square $d^2 = d \circ d$:

$$
  \begin{aligned}
    d d t^a
      &=
    d (-1)\sum_{k = 1}^\infty
    \frac{1}{k!}
    [t_{a_1}, \cdots, t_{a_k}]^a_k
   \,
   t^{a_1} \wedge \cdots \wedge t^{a_k}
    \\
    & =
    \sum_{k,l = 1}^\infty
    \frac{1}{(k-1)! l!}
    [[t_{b_1}, \cdots, t_{b_l}], t_{a_2}, \cdots, t_{a_k}]^a
    \,
    t^{b_1} \wedge \cdots \wedge t^{b_l}
    \wedge
    t^{a_2} \wedge \cdots \wedge t^{a_{k}}
  \end{aligned}
  \,.
$$

Here the wedge product on the right projects the nested bracket onto its graded-symmetric components. This is produced by summing over all [[permutation]]s $\sigma \in \Sigma_{k+l-1}$ weighted by the Koszul-[[signature]] of the permutation:

$$
  \cdots =
    \sum_{k,l = 1}^\infty
    \frac{1}{(k+l-1)!}
    \sum_{\sigma \in \Sigma_{k+l-1}}
    (-1)^{sgn(\sigma)}
    \frac{1}{(k-1)! l!}
    [[t_{b_1}, \cdots, t_{b_l}], t_{a_2}, \cdots, t_{a_k}]^a
    \,
    t^{b_1} \wedge \cdots \wedge t^{b_l}
    \wedge
    t^{a_2} \wedge \cdots \wedge t^{a_{k}}
  \,.
$$

The sum over all permutations decomposes into a sum over the $(l,k-1)$-[[unshuffle]]s and a sum over permutations that act inside the first $l$ and the last $(k-1)$ indices.
By the graded-symmetry of the bracket, the latter  do not change the value of the nested bracket. Since there are $(k-1)! l!$ many of them, we get

$$
  \cdots
    =
    \sum_{k,l = 1}^\infty
    \frac{1}{(k+l-1)!}
    \sum_{\sigma \in Unsh(l,k-1)}
    (-1)^{sgn(\sigma)}
    \left[\left[t_{a_1}, \cdots, t_{a_l}\right], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}\right]
    \,
    t^{a_1} \wedge \cdots \wedge t^{a_{k+l-1}}
  \,.
$$

Therefore the condition $d^2 = 0$ is equivalent to the condition

$$
    \sum_{k+l = n-1}
    \sum_{\sigma \in Unsh(l,k-1)}
    (-1)^{sgn(\sigma)}
    \left[\left[t_{a_1}, \cdots, t_{a_l}\right], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}\right]
  = 0
$$

for all $n \in \mathbb{N}$ and all $\{t_{a_i} \in \mathfrak{g}\}$. This is equation (eq:LInfinityJacobiIdentity) which says that $\{\mathfrak{g}, \{[-,\dots,-]_k\}\}$ is an $L_\infty$-algebra.

### In terms of algebras over an operad
 {#InTermsOfAlgebrasOverAnOperad}

$L_\infty$-algebras are precisely the  [[algebras over an operad]] of the cofibrant resolution of the [[Lie operad]].

## Examples

### Special cases

* An $L_\infty$-algebra for which $V$ is concentrated in the first $n$ degree is a **Lie $n$-algebra** (sometimes also: "$L_n$-algebra").

* An $L_\infty$-algebra for which only the unary operation and the binary bracket are non-trivial is a **[[dg-Lie algebra]]**: a [[Lie algebra]] [[internalization|internal to]] the [[category]] of [[dg-algebras]]. From the point of view of higher Lie theory this is a **strict $L_\infty$-algebra**: one for which the Jacobi identity does happen to hold "on the nose", not just up to nontrivial coherent isomorphisms.

* So in particular

  * an $L_\infty$-algebra generated just in degree 1 is an ordinary **[[Lie algebra]]** ;

  * an $L_\infty$-algebra generated just in degree 1 and 2 is a **[[Lie 2-algebra]]** ;

    * an $L_\infty$-algebra generated just in degree 1 and 2 and with at most binary brackets is a **[[strict Lie 2-algebra]]** , equivalently encoded in a [[differential crossed module]].

  * an $L_\infty$-algebra generated just in degree 1, 2 and 3 is a **[[Lie 3-algebra]]** ;

* if $\mathfrak{g}$ is a Lie algebra over $\mathbf{K}$, and $b^{k-1}\mathbb{K}$ is the complex consisting of the field $\mathbb{K}$ in degree $1-k$, then an $L_\infty$-algebra morphism from $\mathfrak{g}$ to $b^{k-1}\mathbb{K}$ is precisely a degree $k$ [[Lie algebra cohomology|Lie algebra cocycle]].

* The skew-symmetry of the Lie bracket is retained strictly in $L_\infty$-algebras. It is expected that weakening this, too, yields a more general [[vertical categorification]] of Lie algebras. For $n=2$ this has been worked out by Dmitry Roytenberg: [On weak Lie 2-algebras](http://arxiv.org/abs/0712.3461).

* The [[horizontal categorification]] of $L_\infty$-algebras are $L_\infty$-[[Lie infinity-algebroid|algebroid]]s.

* An $L_\infty$-algebra with only $D_n$ non-vanishing is called an **[[n-Lie algebra]]** -- to be distinguished from a _Lie $n$-algebra_ ! However, in large parts of the literature $n$-Lie algebras are considered for which $D_n$ is _not_ of the required homogeneous degree in the grading, or in which no grading is considered in the first place. Such $n$-Lie algebras are not special examples of $L_\infty$-algebras, then. For more see [[n-Lie algebra]].

* An $L_\infty$-algebra internal to [[super vector space]]s is a [[super L-∞ algebra]].

### Classes of examples

* [[automorphism Lie 2-algebra]]

* [[inner derivation Lie 2-algebra]]

* [[automorphism ∞-Lie algebra]]

* For every $\infty$-Lie algebra $\mathfrak{g}$ there is its [[automorphism ∞-Lie algebra]]. In terms of [[rational homotopy theory]] this models the rational automorphism group of the [[rational space]] corresponding to $\mathfrak{g}$.

* [[Poisson-bracket Lie n-algebra]]

* [[Heisenberg Lie n-algebra]] of an [[n-plectic manifold]] or more generally of an [[n-plectic smooth infinity-groupoid]]

* some classes of [[W-algebras]] are claimed to induce $L_\infty$-algebras in [Blumenhagen-Fuchs-Traube 17](W+algebra#BlumenhagenFuchsTraube17)

### Specific examples

* [[line Lie n-algebra]]

* [[string Lie 2-algebra]]

* [[fivebrane Lie 6-algebra]]

* [[supergravity Lie 3-algebra]]

* [[supergravity Lie 6-algebra]]

## Properties

### Ind-Conilpotency
 {#IndConilpotency}

+-- {: .num_remark #IndConilpotency}
###### Remark
**([Pridham 10, remark 3.15, remark 3.13](#Pridham))**

For $\mathfrak{g}$ an $L_\infty$-algebra, then its CE chain [[dgc-coalgebra]] $CE_\bullet(\mathfrak{g})$ ([above](#ReformulationInSemifreeDgCoalgebra)) is _ind-conilpotent_.

This means that $CE_\bullet(\mathfrak{g})$ is a [[filtered colimit]] of sub-dg-coalgebras which are conilpotent, in that for each of them there is $n \in \mathbb{N}$ such that their $n$-fold coproduct vanishes. As such these are like "co-[[local Artin algebras]]".

Moreover, since every dg-coalgebra is the [[union]] of its finite-dimensonal subalgebras (see at _[[dg-coalgebra]]_ the section _[As filtered colimits of finite-dimensional pieces](differential+graded+coalgebra#AsFilteredColimits)_), this means that $CE_\bullet(\mathfrak{g})$ is a [[filtered colimit]] of finite dimensional conilpotent coalgebras.

This implies that the dual Chevalley-Eilenberg cochain algebra $CE^\bullet(\mathfrak{g})$ is a [[filtered limit]] of finite-dimensional nilpotent [[dgc-algebras]] (actual [[local Artin algebras]]).

=--

### Model category structure

See [[model structure for L-∞ algebras]].

### Relation to dg-Lie algebras

Every [[dg-Lie algebra]] is in an evident way an $L_\infty$-algebra. Dg-Lie algebras are precisely those $L_\infty$-algebras for which all $n$-ary brackets for $n \gt 2$ are trivial. These may be thought of as the _strict_ $L_\infty$-algebras: those for which the [[Jacobi identity]] holds on the nose and all its possible higher coherences are trivial.

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

Let $k$ be a [[field]] of [[characteristic]] 0 and write $L_\infty Alg_k$ for the [[category]] of $L\infty$-algebras over $k$.

Then every object of $L_\infty Alg_k$ is [[quasi-isomorphism|quasi-isomorphic]] to a [[dg-Lie algebra]].

Moreover, one can find a functorial replacement: there is a [[functor]]

$$
  W : L_\infty Alg_k \to L_\infty Alg_k
$$

such that for each $\mathfrak{g} \in L_\infty Alg_k$

1. $W(\mathfrak{k})$ is a [[dg-Lie algebra]];

1. there is a [[quasi-isomorphism]]

   $$
     \mathfrak{g} \stackrel{\simeq}{\to}
     W(\mathfrak{g})
     \,.
   $$

=--

This appears for instance as ([Kriz & May 1995, Cor. 1.6](#KrizMay95)).

For more see at *[[relation between L-∞ algebras and dg-Lie algebras]]*.

### Relation to $\infty$-Lie groupoids

In generalization to how a [[Lie algebra]] integrates to a [[Lie group]], $L_\infty$-algebras integrate to [[∞-Lie group]]s.

See

[[Lie integration]]

and

<a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#LieIntegrated">Lie integrated ∞-Lie groupoids</a>.

## Related concepts

* [[A-∞ algebra]]

* [[E-∞ algebra]]

* **$L_\infty$-algebra**

  * [[homomorphism of L-∞ algebras]]

  * [[infinity-Lie algebra cohomology]]

  * [[universal enveloping E-n algebra]]

  * [[nilpotent L-∞ algebra]]

  * [[L-∞ algebroid]]

  * [[derived L-∞ algebroid]]

  * [[sheaf of L-∞ algebras]]

  * [[quantum L-∞ algebra]]

* [[relation between L-∞ algebras and dg-Lie algebras]]

* [[EL-∞ algebra]]

## References
 {#References}

[[!include L-infinity algebras in physics]]

[[!redirects L-infinity-algebras]]
[[!redirects Lie infinity-algebra]]
[[!redirects L-infinity algebra]]
[[!redirects Lie infinity-algebras]]
[[!redirects L-infinity algebras]]
[[!redirects L-∞ algebra]]
[[!redirects L-∞ algebras]]
[[!redirects L-∞-algebra]]
[[!redirects L-∞-algebras]]

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

[[!redirects infinity-Lie algebra]]
[[!redirects infinity-Lie-algebra]]
[[!redirects infinity-Lie algebras]]
[[!redirects infinity-Lie-algebras]]

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

[[!redirects strong homotopy Lie algebra]]
[[!redirects strong homotopy Lie algebras]]

[[!redirects higher Lie algebra]]
[[!redirects higher Lie algebras]]


L-infinity-algebroid > history

< [[L-infinity-algebroid]]
[[!redirects L-infinity-algebroid -- history]]


L-infinityalgebra


L-parameter

#Contents#
* table of contents
{:toc}

## Idea

A notion in the context of the [[local Langlands correspondence]].

## Definition

See [Silberger-Zink 14, Sec. 3, esp. 3.3 (p. 18)](#SilbergerZink14)

## Related concepts

* [[moduli stack of L-parameters]]

* [[local Langlands correspondence]]

## References

* {#SilbergerZink14} [[Allan J. Silberger]], [[Ernst-Wilhelm Zink]], Section 3 of: *Langlands Classification for L-Parameters*, Journal of Algebra Volume 511, 1 October 2018, Pages 299-357 ([arXiv:1407.6494](https://arxiv.org/abs/1407.6494), [doi:10.1016/j.jalgebra.2018.06.012](https://doi.org/10.1016/j.jalgebra.2018.06.012))

Talk notes:

* [[Ernst-Wilhelm Zink]], with [[Allan J. Silberger]], *Langlands classification for L-parameters* ([pdf](https://www.math.hu-berlin.de/~zyska/zink/Lvortrag1.pdf))

[[!redirects L-parameters]]


L-theory

#Contents#
* table of contents
{:toc}

## Idea

The [[K-theory]] of ([[Hermitian form|Hermitian]]) [[quadratic forms]]. See at: *[[Hermitian K-theory]]*.

## References

* Wikipedia, _[L-theory](http://en.wikipedia.org/wiki/L-theory)_

[[!redirects L theory]]


L. Gaunce Lewis, Jr.

* [webpage](http://math.syr.edu/bios/lewis-gaunce.html)

## Selected writings

On [[compactly generated topological spaces]]:

* {#Lewis78} [[Gaunce Lewis]], _Compactly generated spaces_ ([pdf](http://www.math.uchicago.edu/~may/MISC/GaunceApp.pdf)), appendix A of _The Stable Category and Generalized Thom Spectra_ PhD thesis Chicago, 1978

On [[stabilization]] and [[simplicial topological spaces]]:

* {#Lewis} [[L. Gaunce Lewis, Jr.]], _When is the natural map $X \to \Omega \Sigma X$ a cofibration?_ , Trans. Amer. Math. Soc. 273 (1982), 147--155.

On [[equivariant stable homotopy theory]]:

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], [[Mark Steinberger]] (with contributions by [[James McClure|J.E. McClure]]), _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics  **1213** (1986) [[pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf), [doi:10.1007/BFb0075778](https://link.springer.com/book/10.1007/BFb0075778)]

On [[Eilenberg-MacLane objects]] and [[Seifert-van Kampen theorem]] in [[equivariant homotopy theory]]:

* {#Lewis92} [[L. Gaunce Lewis, Jr.]], _Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems_, Topology Appl., 48 (1992), no. 1, pp. 25–61.

On [[equivariant stable homotopy theory]]:

* {#Lewis00} [[L. Gaunce Lewis, Jr.]], section 10 of _Splitting theorems for certain equivariant spectra_, Memoirs of the AMS, number 686, March 2000, Volume 144 ([pdf](http://hopf.math.purdue.edu/LewisG/spltspec.pdf))

On the [[Picard group]] of [[equivariant stable homotopy theory]] and the notion of [[RO(G)-grading]]:

* {#FauskLewisMay01} [[Halvard Fausk]], [[L. Gaunce Lewis]], [[Peter May]],  _The Picard group of equivariant stable  homotopy theory_, Advances in Mathematics Volume 163, Issue 1, 15 October 2001, Pages 17–33  ([pdf](http://www.math.uchicago.edu/~may/PAPERS/FLMJan01.pdf))

On cases where [[prime field|$\mathbb{Z}/p$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]] groups  are [[free modules]] over the Bredon cohomology of the point:

* [[Kevin K. Ferland]], [[L. Gaunce Lewis, Jr.]], *The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G=\mathbb{Z}/p$*, Memoirs of the AMS **167** (2004) [[ams:memo-167-794](https://bookstore.ams.org/memo-167-794), 978-ISBN:1-4704-0392-8]

## Related entries

* [[compactly generated topological space]]

* [[nice simplicial topological space]]

* [[equivariant stable homotopy theory]]

* [[fixed point spectrum]]

category: people

[[!redirects Gaunce Lewis]]
[[!redirects L. Gaunce Lewis]]


L. J. Landau

* [ACM page](https://dl.acm.org/profile/81430660895/publications?Role=author)

## Selected writings

On [[extremal quantum channels]]:

* [[L. J. Landau]], [[Raymond F. Streater]],  *On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras*, Linear Algebra and its Applications **193** (1993) 107-127 [doi:10.1016/0024-3795(93)90274-R]

category: people


L. J. Romans

## Selected writings

Introducing [[massive type IIA supergravity]] ("[[Romans supergravity]]"):

* {#Romans86} [[L. J. Romans]], _Massive $N = 2a$ supergravity in ten dimensions_ , Physics Letters B Volume 169, Issue 4, 3 April 1986, Pages 374-380 (doi:10.1016/0370-2693(86)90375-8, [spire:17727](http://inspirehep.net/record/17727))

On [[gauged supergravity|gauged]] [[D=5 supergravity]]:

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Gauged $N = 8$ Supergravity in Five Dimensions_, Phys. Lett. 154B, (1985) 268 ([spire:207663](http://inspirehep.net/record/207663), doi:10.1016/0370-2693(85)90361-2)

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Compact and Non–Compact Gauged Supergravity Theories in Five Dimensions_, Nucl. Phys. B272 (1986) 598 ([spire:219727](http://inspirehep.net/record/219727), doi:10.1016/0550-3213(86)90237-3)

category: people


L.E.J. Brouwer

Luitzen Egbertus Jan Brouwer (1881--1966) was an early worker in [[algebraic topology]] and the founder of [[intuitionism]].

## Important results:

*  [[topological invariance of dimension]],
*  [[Brouwer fixed point theorem]],
*  [[simplicial approximation theorem]],
*  [[intuitionistic mathematics]],
*  [[fan theorem]] and [[bar theorem]].

* [[Brouwer-Heyting-Kolmogorov interpretation]]

## Selected writings

On [[intuitionism]]:

* [[L. E. J. Brouwer]], *Points and Spaces*, Canadian Journal of Mathematics **6** (1954) 1-17 [[doi:10.4153/CJM-1954-001-9](https://doi.org/10.4153/CJM-1954-001-9)]

## Brouwer's intuitionism
 {#BrouwerIntuitionism}

From p. 6 of [[Errett Bishop]]'s *[[Foundations of Constructive Analysis]]* (McGraw-Hill, 1967):

<img src="/nlab/files/BishopOnBrouwer.jpg" width="570">

category: people

[[!redirects L.E.J. Brouwer]]
[[!redirects L. E. J. Brouwer]]
[[!redirects LEJ Brouwer]]
[[!redirects L E J Brouwer]]
[[!redirects Luitzen Egbertus Jan Brouwer]]
[[!redirects Luitzen Brouwer]]
[[!redirects Egbertus Brouwer]]
[[!redirects Bertus Brouwer]]
[[!redirects Jan Brouwer]]
[[!redirects Brouwer]]


l.m.c.-C*-algebra

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of [[C*-algebras]]. Used in ([Joachim-Johnson 07](#JoachimJohnson07)) order to have a [[model category]] presentation of [[KK-theory]] instead of just a [[category of fibrant objects]]-structure (as described at [[homotopical structure on C*-algebras]]).

## References

* {#JoachimJohnson07} [[Michael Joachim]], [[Mark Johnson]], _Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category_ ([arXiv:0705.1971](http://arxiv.org/abs/0705.1971))

[[!redirects l.m.c.-C*-algebras]]

[[!redirects l.m.c-C-star-algebra]]
[[!redirects l.m.c-C-star-algebras]]


lab elf > history


Lack's coherence observation

[[!redirects Lack's coherence theorem]]

+-- {: .rightHandSide}
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###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One form of the [[bicategory#Coherence|coherence theorem for bicategories]] states that every [[bicategory]] is [[equivalence of bicategories|equivalent]] to a [[strict 2-category]].  **Lack's coherence observation** (named for [[Steve Lack]]) states that naturally occurring bicategories tend to be [[equivalence of bicategories|equivalent]] to naturally occurring strict 2-categories.

## Examples

Instances of Lack's coherence observation include:

* The bicategory [[Prof]] of [[small categories]] and [[profunctors]] is equivalent to the strict 2-category of [[presheaf categories]] and [[cocontinuous functors]].

* The bicategory [[Span]](C) of [[spans]] in a category $C$ with [[pullbacks]] is equivalent to the strict 2-category of [[slice categories]] of $C$ and [[linear polynomial functors]]. More generally, the bicategory of [[polynomial functor|polynomials]] in a [[locally cartesian closed category]] is equivalent to the strict 2-category of slices and [[polynomial functors]].

Both examples are instances of a more general result. Given a [[relative pseudomonad]] $T$ on a strict 2-category $K$, the [[Eilenberg--Moore bicategory]] of $T$ will also be a strict 2-category, whereas the [[Kleisli bicategory]] will usually not be. However, the Kleisli bicategory embeds fully faithfully into the Eilenberg--Moore bicategory, and so will be [[biequivalent]] to the full sub-2-category of the Eilenberg--Moore bicategory on the free algebras.

It is not clear whether there are other examples that are not special cases of the observation above.

## References

This observation first appears in Example 1.5(k) of

*  [[Steve Lack]], _A 2-categories companion_, in: John C. Baez and J. Peter May (Eds) Towards Higher Categories, The IMA Volumes in Mathematics and its Applications, vol 152, Springer, 2009, doi:[10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4), [arXiv:math/0702535](https://arxiv.org/abs/math/0702535).

[[!redirects Lack's coherence theorem]]


lacuna

* wikipedia [Petrovsky lacuna](http://en.wikipedia.org/wiki/Petrovsky_lacuna)
* И. Г. Петровский, _О диффузии волн и лакунах для систем гиперболических уравнений_, Изв. АН СССР. Сер. матем., 8:3 (1944), 101–106 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=3720&what=fullt&option_lang=rus) with French summary and title _Sur la diffusion des ondes et les lacunes pour les systèmes d'équations hyperboliques_, [MR11880](http://www.ams.org/mathscinet-getitem?mr=11880)
* I. G. Petrovsky, _Dependence of solutions of Cauchy's problem on initial conditions_, C. R. (Doklady) Acad. Sci. URSS (N.S.) 38, (1943). 151–153 [MR8450](http://www.ams.org/mathscinet-getitem?mr=8450); _On the diffusion of waves and the lacunas for hyperbolic equations_, Recueil Mathématique (Matematicheskii Sbornik) 17 (59)1945 (3): 289–368, [MR16861](http://www.ams.org/mathscinet-getitem?mr=16861)
* [[M. F. Atiyah]], [[R. Bott]], L. Gårding, _Lacunas for hyperbolic differential operators with constant coefficients. I_, Acta Mathematica 124 (1970) 109–189, [doi](http://dx.doi.org/10.1007/BF02394570), [MR0470499]; _Lacunas for hyperbolic differential operators with constant coefficients. II_, Acta Mathematica __131__ (1973) 145–206, [doi](http://dx.doi.org/10.1007/BF02392039), [MR0470500](http://www.ams.org/mathscinet-getitem?mr=0470500)
* [[Yuri Berest]], _The problem of lacunas and analysis on root systems_, Trans. Amer. Math. Soc. __352__ (2000), 3743–3776 [doi](http://dx.doi.org/10.1090/S0002-9947-00-02543-5) [pdf](http://www.ams.org/journals/tran/2000-352-08/S0002-9947-00-02543-5/S0002-9947-00-02543-5.pdf)
* Yuri Yu. Berest, _The theory of lacunas and quantum integrable systems_,  Calogero-Moser-Sutherland models (Montréal, QC, 1997), 53–64, CRM Ser. Math. Phys., Springer 2000.

category: analysis


Lagrange inversion

+-- {: .rightHandSide}
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### Context
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
=--
=--

Lagrange inversion denotes a formula for the power series coefficients of the compositional inverse of an analytic function $f$ around $0$ satisfying $f(0)=0$, or of a formal power series in one or several variables.

Wikipedia: [Lagrange inversion theorem](http://en.wikipedia.org/wiki/Lagrange_inversion_theorem), [formal power series](http://en.wikipedia.org/wiki/Formal_power_series#The_Lagrange_inversion_formula)

Related entries: [[Faà di Bruno formula]], [[noncommutative symmetric function]], [[Lambert W-function]]

* I.M. Gessel, _A noncommutative generalization and a $q$-analog of the Lagrange inversion formula_,  Trans. Amer. Soc. __257__ (1980), pp. 455–482, _A combinatorial proof of the multivariable Lagrange inversion formula_, J. Combin. Theory Ser. A __45__ (1987), pp. 178–196
* Ira Gessel, Gilbert Labelle, _Lagrange inversion for species_, J. Combin. Theory Ser. A 72 (1995), 95–117.

A homotopical algebra proof of the Lagrange inversion formula is exhibited in

* Vladimir Dotsenko, _A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula_, [arxiv/1606.08222](http://arxiv.org/abs/1606.08222)

There are relations to [[Faà di Bruno algebra]]:

* Christian Brouder, Alessandra Frabetti, Christian Krattenthaler, _Non-commutative Hopf algebra of formal diffeomorphisms_, Adv. Math. __200__:2 (2006) 479-524 [pdf](http://www.sciencedirect.com/science/article/pii/S0001870805000265/pdf?md5=cb2367d0f3f8b07bf9799d61be34a595&pid=1-s2.0-S0001870805000265-main.pdf)
* Jean-Paul Bultel, _Combinatorial properties of the noncommutative Faà di Bruno algebra_, J. of Algebraic Combinatorics 38:243–273 (2013) [MR3081645](http://www.ams.org/mathscinet-getitem?mr=3081645)

An approach to Lagrange inversion using Heisenberg-Weyl algebra is in

*  O. V. Viskov, _A random walk with a skip-free component and the Lagrange inversion formula_, Theory Probab. Appl., 45(1), 164–172. (9 pages); O. V. Viskov, _Obrasčenie stepennyh rjadov i formula Lagranža_, Dokl. AN SSSR, 254:2 (1980) 269–271

category: analysis, combinatorics

[[!redirects Lagrange inversion formula]]
[[!redirects Lagrange inversion theorem]]


Lagrange multiplier

#Contents#
* table of contents
{:toc}

## Idea following Loomis-Sternberg

$X$ -- finite dimensional real vector space

$U\subset X$ open

$F: U\to \mathbf{R}$ differentiable function

$S\subset U$ a smooth submanifold, which can be represented as a zero set of a differentiable map $G: U\to Y$, whre $Y$ is a real vector space and such that $d G_x$ is surjective for each $x\in S$.

We want to minimize $F(x)$ for $x\in S$. It won't work to set $d F_x = 0$ and solving for $x$ as $x$ will not be a critical point of $F$ in general. The Lagrange multipliers are used to define another function $L$ such that solving $d L_x = 0$ gives extrema of the constrained extremization problem.

__Theorem__ (Loomis-Sternberg 3.12.2) Suppose $F$ has a maximum on $S$ at $x$. Then there is a function(al) $l$ in $Y^*$ such that $x$ is a critical point of the function $F - l\circ G$.

The proof uses implicit function theorem and the usual extremization arguments.

To get to a more familiar form of Lagrange multipliers, one uses the local coordinates $(x_1,\ldots,x_n)$ on $U$ and sets $Y = \mathbf{R}^m$, so that $G = (g^1,\ldots, g^n)$. Now $l: Y\to\mathbf{R}$ will be of the form $l(y_1,\ldots,y_m) = \sum_{i = 1}^m \lambda_i y_i$ and $F - l\circ G = F - \sum_{i = 1}^m \lambda_i g^i$ and $d (F - l\circ G) = 0$ gives

$$
\frac{\partial F}{\partial x_j} - \sum_{i = 1}^m\lambda_i\frac{\partial g^i}{\partial x_j} = 0,\,\,\,\,\,\,\,\,j = 1,\ldots, n.
$$

This is $n$ equations, which together with $m$ equations $G = (g^1,\ldots, g^n) = 0$ for $S$ give $m+n$ equations for $m+n$ unknowns $x_1,\ldots, x_n, \lambda_1,\ldots,\lambda_m$. The last $m$ variables here are the Lagrange multipliers.

## Applications

### To spectral theory
 {#ApplicationToSpectralTheory}

The method of Lagrange multipliers affords an elementary proof of the [[spectral theorem]] for finite-dimensional real [[vector spaces]], one which does not involve passage to the [[complex numbers]] and the [[fundamental theorem of algebra]].

+-- {: .un_prop}
######Proposition
Let $A$ be a real symmetric $n \times n$ matrix. Then $A$ is diagonalizable over the real numbers.
=--

+-- {: .proof}
######Proof
Consider the problem of maximizing the function $f(x) = \langle x \vert A \vert x \rangle$ where $x \in \mathbb{R}^n$ is subject to the constraint $\langle x \vert x \rangle = 1$. (Such an extreme point exists, say by compactness.) By the symmetry of $A$, the [[gradient]] of $f$ is easily calculated to be $\nabla f (x) = 2 A x$, whereas the gradient of the [[Euclidean norm]] $\langle x \vert x \rangle$ is $2 x$. At a point $x$ where a maximum is attained, we have $\nabla f(x) = 2 A x = \lambda (2 x)$ for some Lagrange multiplier $\lambda$. Thus $x$ is an eigenvector of $A$ with eigenvalue $\lambda$. The usual arguments show that $A$ restricts to a [[self-adjoint operator]] on the hyperplane orthogonal to $x$; by picking an orthonormal basis of this hyperplane, we may represent this restriction of $A$ by a real symmetric matrix of size $(n-1) \times (n-1)$, and the argument repeats.

=--

## Related concepts

* [[auxiliary field]]

* [[gauge fixing]]

## References

Named after [[Joseph-Louis Lagrange]].

* wikipedia [Lagrange multiplier](http://en.wikipedia.org/wiki/Lagrange_multiplier)

* Springer eom: [Lagrange multipliers](http://eom.springer.de/L/l057190.htm), [Pontrjagin maximum principle](http://eom.springer.de/p/p073780.htm)

* Lynn H. Loomis, [[S. Sternberg]], _Advanced calculus_, section 3.12

"Submanifolds and Lagrange multipliers" [[LoomisSternberg13s2.djvu:file]]

* Robert Hermann, _Some differential-geometric aspects of the Lagrange variational problem_, Illinois J. Math. __6__, 1962, 634–673, [MR145457](http://www.ams.org/mathscinet-getitem?mr=145457),[euclid](http://projecteuclid.org/euclid.ijm/1255632711)
* Juan Carlos Marrero, David Martín de Diego, Ari Stern, _Lagrangian submanifolds and discrete constrained mechanics_, slides, [pdf](http://ccom.ucsd.edu/~astern/pdfs/dspde20100601.pdf)
* Manuel de León, David Martín Diego, _Solving non-holonomic Lagrangian dynamics in terms of almost product structures_, Extracta Math. 11 (1996), no. 2, 325–347, [pdf](http://digital.csic.es/bitstream/10261/2232/1/solving.pdf) [MR97m:58079](http://www.ams.org/mathscinet-getitem?mr=1437457)

[[!redirects Lagrange multipliers]]


Lagrange's theorem

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

#Contents#
* table of contents
{:toc}

## Idea

In [[group theory]], *Lagrange's theorem* asserts that for $G$ a [[finite group]] and $H \,\subset\, G$ a [[subgroup]], the [[order of a group|order]] $\left\vert H\right\vert$ of $H$ divides the order $\left\vert G \right\vert$ of $G$, in that the [[quotient]]

$$
  [G : H]
  \;\coloneqq\;
  \frac
    { \left\vert G \right\vert }
    { \left\vert H \right\vert }
  \;\;\;
  \in
  \;
  \mathbb{N}
$$

is a [[natural number]], called the [[index of a subgroup|index]] of $H$ in $G$.

## Related statements

A partial converse to Lagrange's theorem is

* [[Sylow's theorem]]

## References

Named after:

* [[Joseph-Louis Lagrange]], *Réflexions sur la résolution algébrique des équations*,  Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin, années 1770 et 1771, Œuvres complètes, tome 3, 205-421 ([mathdoc](http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAGRANGE__3_205_0))

More on the history:

* Richard L. Roth, *A History of Lagrange's Theorem on Groups*, Mathematics Magazine **74** 2 (2001) 99-108 ([doi:10.2307/2690624](https://doi.org/10.2307/2690624), [jstor:2690624](https://www.jstor.org/stable/2690624))

See also:

* Wikipedia, *Lagrange's theorem (group theory)*


Lagrangian cobordism

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###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Lagrangian cobordism_ is roughly a [[cobordism]] between [[Lagrangian submanifolds]] which is itself a [[Lagrangian submanifold]] in a suitable sense.

More in detail, given an ambient [[symplectic manifold]] $(X, \omega)$, a _Lagrangian cobordism_ in $X$ is a [[cobordism]] with an [[embedding]] into $([0,1] \times \mathbb{R} \times X )$ as a [[Lagrangian submanifold]] (where $[0,1] \times \mathbb{R} \hookrightarrow \mathbb{R}^2$ is equipped with its canonical [[symplectic structure]]) and such that the [[boundary]] components are [[Lagrangian submanifolds]] of $X$. (for details see e.g. [Biran-Cornea 11, 2.1.1](#BiranCornea11)).

Lagrangian cobordisms in $(X,\omega)$ arrange into a [[stable (infinity,1)-category]] which is supposed to be at least closely related to the [[Fukaya category]] of $(X,\omega)$ ([Nadler-Tannaka 11](#NadlerTannaka11)).

## Related concepts

* [[Fukaya category]]

## References

The notion was introduced in

* [[Vladimir Arnold]], _Lagrange and Legendre cobordisms. I, II_, Funkts. Anal. Prilozh.  14:3, 1–13 (1980) [article](http://www.mathnet.ru/eng/faa1827); 14:4 (1980), 8–17 [article](http://mi.mathnet.ru/eng/faa1848)

Further developments are in

* {#BiranCornea11} Paul Biran, Octav Cornea, _Lagrangian cobordism I, ([arxiv/1109.4984](http://arxiv.org/abs/1109.4984))

* {#BiranCornea13} Paul Biran, Octav Cornea, _Lagrangian cobordism III_([arxiv/1304.6032](http://arxiv.org/abs/1304.6032))

A [[stable (infinity,1)-category|stable $(\infty,1)$-category]] of Lagrangian cobordisms is discussed in

* {#NadlerTannaka11} [[David Nadler]], [[Hiro Tannaka]], _A stable infinity-category of Lagrangian cobordisms_, Adv. Math. __366__ 3 (2020) 107026 [[arXiv:1109.4835](https://arxiv.org/abs/1109.4835), [doi:10.1016/j.aim.2020.107026](https://doi.org/10.1016/j.aim.2020.107026)]

* Wenyuan Li, _Lagrangian cobordism functor in microlocal sheaf theory I_, [arXiv:2108.10914](https://arxiv.org/abs/2108.10914)

[[!redirects Lagrangian cobordisms]]


Lagrangian correspondence

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###Context###
#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _Lagrangian correspondence_ is a [[correspondence]] between two [[symplectic manifolds]] $(X_i,\omega_i)$ given by a [[Lagrangian submanifold]] of their [[product]] $(X_1 \times X_2, p_1^\ast \omega_1 - p_2^\ast \omega_2)$. The [[graph of a function|graph]] of any [[symplectomorphism]] induces a Lagrangian correspondence.

Lagrangian correspondences are supposed to form, subject to some technicalities, the [[morphisms]] of a [[category]] to be called the _[[symplectic category]]_.

When [[symplectic geometry]] is used to model [[mechanics]] in [[physics]], then a [[symplectic manifold]] $(X,\omega)$ encodes the [[phase space]] of a [[mechanical system]] and a [[symplectomorphism]]

$$
  \phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)
$$

encodes a process undergone by this system, for instance the time evolution induced by a [[Hamiltonian vector field]]. In particular if this is a [[Hamiltonian symplectomorphism]] then this is traditionally called a _[[canonical transformation]]_ in physics. Therefore Lagrangian correspondence have also been called _canonical relations_ ([Weinstein 83, p. 5](#Weinstein83)).

## Definition
 {#Definition}

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

For $(X_j, \omega_j)$ two [[symplectic manifold]]s, a **Lagrangian correspondence** is a [[correspondence]] $Z \to X^-_0 \times X_1$ which is a  [[submanifold]] of $X^-_0 \times X_1$

$$
  \iota : L_{0,1} \hookrightarrow X^-_0 \times X_1
$$

with $dim(L_{0,1}) = \frac{1}{2}(dim(X_0) + dim(X_1))$

and

$$
  \iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0
  \,,
$$

where $\pi_i$ are the two projections out of the [[product]].

=--

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

The **composition** of two Lagrangian correspondences is

$$
  L_{01} \circ L_{12} :=
    \pi_{02}(L_{01} \times_{X_1} L_{12})
$$

which is itself a Lagrangian correspondence in $X^-_0 \times X_2$ if everything is suitably smoothly embedded by $\pi_{02}$.

=--

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

The category of Lagrangian correspondences is a [[full subcategory]] of that of correspondence of the [[slice topos]] $SmoothSpaces_{/\Omega^2_{cl}}$ of [[smooth spaces]] over the [[moduli space]] $\Omega^2_{cl}$ of closed [[differential 2-forms]]:

a [[symplectic manifold]] $(X,\omega)$ is given by a map of [[smooth spaces]] $\omega \colon X \to \Omega^2_{cl}$ (generally this is a [[presymplectic manifold]]) and a correspondence in $SmoothSpaces_{/\Omega^2_{cl}}$ is a [[commuting diagram]] in [[smooth spaces|SmoothSpaces]] of the form

$$
  \array{
    && Z
    \\
    & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
    \\
    X_1 && {i_1^\ast \omega_1 = i_2^\ast \omega_2} && X_2
    \\
    & {}_{\mathllap{\omega_1}}\searrow && \swarrow_{\mathrlap{\omega_2}}
    \\
    && \Omega^2_{cl}
  }
  \,.
$$

If here $(i_1, i_2) \colon Z \to X \times Y$ is a manifold maximal with the property of fitting into the above diagram, then this is a Lagrangian correspondence.

From this is naturally induced the notion of a _[[prequantized Lagrangian correspondence]]_. See there for more details.

=--

## Examples

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

For $\phi : (X_0, \omega_0) \to (X_1, \omega_1)$ a [[symplectomorphism]] we have that its [[graph of a function|graph]] $graph(\phi) \subset X_0^- \times X_1$ is a Lagrangian correspondence and  composition of syplectomorphisms corresponds to composition of  Lagrangian correspondences.

=--

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

For a [[Hamiltonian action]] on a [[symplectic manifold]] $(X,\omega)$ of a [[Lie group]] $G$ given by a [[moment map]] $\mu$, the [[zero locus]] $\mu^{-1}(0)$ consitutes a Lagrangian correspondence between $(X,\omega)$ and its [[symplectic reduction]] $\mu^{-1}(0)/G$.

=--

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

Let $X$ be a [[manifold]], $G= U(n)$ the [[unitary group]], $P \to X$ a $G$-[[principal bundle]] and $D \to X$ a $U(1)$-bundle with [[connection on a bundle|connection]].

Then there is the [[moduli space]] $M(X) = M(P,D)$  of connections on $P$ with central curvature and given determinant.

For example if $X$ has [[genus]] $g$ then

$$
  M(X) = \{ (A,B, \cdots, A_g, B_g) \in G^{2g}\}
$$

such that $\prod_{j=1}^g A_j B_j A_j^{-1} B_j^{-1} = diag(e^{2\pi i d/})/G$

Let $Y_{01}$ be a [[cobordism]] from $X_0$ to $X_1$ with extension

$$
  L(Y_{01}) = Image(M(Y_{01}) \stackrel{restr.}{\to} M(X_0)^- \times M(X_1) )
$$

is a Lagrangian correspondence if $Y_{01}$ is sufficiently simple. Further assuming this we have for composition that

$$
   L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12})
   \,.
$$

=--

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

Given symplectic manifolds $(X_1, \omega_1)$ and $(X_2, \omega_2)$ and a [[symplectomorphism]] $(X_1 \times X_2 , p_1^\ast \omega_1 - p_2^\ast \omega_2) \stackrel{\simeq}{\longrightarrow} (X,\omega)$, then certain Lagrangian correspondences between $(X_1, \omega_1)$ and $(X_2, \omega_2)$ are identified with functions on $X$. This is identified with the calculus of generating functions for [[canonical transformations]] as used in [[classical mechanics]]. ([Weinstein 83, p. 5](#Weinstein83))

=--

## Related entries

* [[symplectic dual pair]]

* [[Weinstein symplectic category]]

* [[prequantized Lagrangian correspondence]]

* [[Lagrangian correspondences and category-valued TFT]]

* [[cosmic Galois group]]

## References

The notion originates somewhere around

* [[Lars Hörmander]], _Fourier Integral Operators I._, Acta Math. __127__ (1971)
 {#Hoermander71}

* [[Alan Weinstein]], _Symplectic manifolds and their lagrangian submanifolds_, Advances in Math. 6 (1971) doi {#Weinstein71}

The use of Lagrangian correspondences for encoding [[symplectomorphisms]] was further highlighted in

* [[Alan Weinstein]], _The symplectic "category"_, In Differential geometric methods in mathematical physics (Clausthal, 1980), volume 905 of Lecture Notes in Math., pages 45–51. Springer, Berlin, 1982.

and on p. 5 and then section 3

*  {#Weinstein83} [[Alan Weinstein]], _Lectures on Symplectic Manifolds_, volume 29 of CBMS Regional Conf. Series in Math. Amer. Math. Soc., 1983. third  printing.

Review of the way Lagrangian correspondences encode [[symplectomorphisms]] induced by Hamiltonian time evolution in the context of [[field theory]] and generalization to the broader context of [[BV-BRST formalism]]:

* {#CattaneoMnevReshetikhin12} [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical and quantum Lagrangian field theories with boundary_ ([arXiv:1207.0239](http://arxiv.org/abs/1207.0239))

More on the construction of [[symplectic category|symplectic categories]] with Lagrangian correspondences as morphisms:

* [[Katrin Wehrheim]], [[Chris T. Woodward]], _Functoriality for Lagrangian correspondences in Floer theory_, Quantum Topology **1** 2 (2010) 129-170 [[doi:10.4171/qt/4](https://doi.org/10.4171/qt/4), [arXiv:0708.2851](https://arxiv.org/abs/0708.2851)]

and on its [[(∞,1)-category]]-theoretic version:

* [[Damien Calaque]], _Three lectures on derived symplectic geometry and topological field theories_, Indagationes Mathematicae **25** 5 (2014) 926–947 [[doi:10.1016/j.indag.2014.07.005](https://doi.org/10.1016/j.indag.2014.07.005)]

* [[Rune Haugseng]], _Iterated spans and classical topological field theories_, Mathematische Zeitschrift **289** 3 (2018) 1427–1488 [[arXiv](https://arxiv.org/abs/1409.0837), [doi:10.1007/s00209-017-2005-x](https://doi.org/10.1007/s00209-017-2005-x)]

See also:

* [[Katrin Wehrheim]], [[Chris Woodward]], _Floer Cohomology and Geometric Composition of Lagrangian Correspondences_ [[arXiv:0905.1368](http://arxiv.org/abs/0905.1368)]

[[!redirects Lagrangian correspondences]]
[[!redirects lagrangian correspondence]]
[[!redirects canonical relation]]
[[!redirects canonical relations]]


Lagrangian correspondences and category-valued TFT

This entry describes classes of examples of [[A-∞ category]]-valued  [[FQFTs]] defined on a version of the [[symplectic category]].

#Contents#
* table of contents
{:toc}

## Overview

Let $(X.\omega)$ be a [[compact space|compact]] [[symplectic manifold]].
At least in good cases to this is associated a [[Fukaya category]]
$Fuk(X)$ of
[[Lagrangian submanifold]]s and an enlarged version $Fuk^#(X)$.

Write $X^-$ for the symplectiv manifold $(X,-\omega)$.

Now if $(X_j, \omega_j)$ for $j = 0,1$ are two Lagrangian submanifolds and
$L_{01} \subset  X^-_0 \times X_1$ a Lagrangian correspondence then we get
an [[A-∞ category|A-∞ functor]]  $\phi(L_{01}) : Fuk^#(X_0) \to Fuk^#(X_1)$

+-- {: .num_theorem}
###### Theorem
**(Wehrheim, Woodward)**

For  $L_{01} \subset X^{-}_0 \times X_1$ and $L_{12} \subset X^{-}_1 \times X_2$ Lagrangian submanifolds,  assuming monotonicity and Maslov conditions we have an $A_\infty$-homotopy

$$
  \Phi(L_{01}) \circ \Phi(L_{12}) \simeq \Phi(L_{01} \circ L_{12})
  \,,
$$

where on the right we have a natural notion of composition of Lagrangian submanifolds.

=--

([Wehrheim-Woodward 07, section 3.2](WehrheimWoodward07))

This is the symplectic version of [[Mukai functors]].

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

For $X_0$ and $X_1$ a compact [[Riemann surfaces]] and
$M(X_0), M(X_1)$

their [[moduli spaces]] of fixed determinant rank $n$-bundles, and for $Y_{01}$ a [[cobordism]] (compact, oriented) from $X_0$ to $X_1$
then consider

$$
  L(Y_{01}) := Image(
      M(Y_{01}) \stackrel{restriction}{\to} M(X_0)^- \times M(X_1)
  )
$$

If $Y_{01}$ is _elementary_ in that there exists a [[Morse function]] $Y \to \mathbb{R}$ with $\leq 1$ critical points then $L(Y_{01})$ is a Lagrangian correspondence.

=--

+-- {: .num_cor}
###### Corollary

The assignment

$$
  Y_{01} \mapsto \Phi(L(Y_{01}))
$$

defines a 2+1-dimensional [[FQFT]] for connected cobordisms with values in [[A-∞ categories]].

=--

This is supposed to be the 2+1-dimensional part of [[Donaldson theory]].

Other theories that fit into this framework:

1. symplectic [[Khovanov theory]] (Seidel-Smith and Rezazodegab)

1. [[Heegard-Floer theory]]

   $X$ a surface $\mapsto$ $Fuk^# sym X$

   elementary cobordisms $\mapsto$ $\Phi($vanishing cycle$)$

## Lagrangian correspondences

Write $X^-j = (X_j , -\omega_j)$ for a [[symplectic manifold]] with its symplectic form
reversed.

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

For $(X_j, \omega_j)$ two [[symplectic manifold]]s, a **[[Lagrangian correspondence]]** is a Lagrangian submanifold of $X^-_0 \times X_1$, that is

$$
  \iota : L_{0,1} \hookrightarrow X^-_0 \times X_1
$$

with $dim(L_{0,1}) = \frac{1}{2}(dim(x_0) + dim(X_1))$

and

$$
  \iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0
  \,,
$$

where $\pi_i$ are the two projections out of the [[product]].

The **composition** of two Lagrangian submanifolds is

$$
  L_{01} \circ L_{12} :=
    \pi_{02}(L_{01} \times_{X_1} L_{12})
$$

which is a Lagrangian correspondence in $X^-_0 \times X_2$ if everything is suitably smoothly embedded by $\pi_{02}$.

=--

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

 For $\phi : X_0 \to X_1$ a [[symplectomorphism]] we have

   $graph(\phi) \subset X_0^- \times X_1$ is a Lagrangian correspondence and
   composition of syplectomorphisms corresponds to composition of
   Lagrangian correspondences.

=--

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

Let $X$ be a [[manifold]], $G= U(n)$ the [[unitary group]],
   $P \to X$ a $G$-[[principal bundle]]
   and $D \to X$ a $U(1)$-bundle with [[connection on a bundle|connection]].

   Then there is the [[moduli space]] $M(X) = M(P,D)$
   of connections on $P$ with central curvature and given determinant.

   For example if $X$ has [[genus]] $g$ then

   $$
     M(X) = \{ (A,B, \cdots, A_g, B_g) \in G^{2g}\}
   $$

   such that $\prod_{j=1}^g A_j B_j A_j^{-1} B_j^{-1} = diag(e^{2\pi i d/})/G$

   Let $Y_{01}$ be a [[cobordism]] from $X_0$ to $X_1$ with extension

   $$
     L(Y_{01}) = Image(M(Y_{01}) \stackrel{restr.}{\to}
     M(X_0)^- \times M(X_1) )
   $$

   is a Lagrangian correspondence if $Y_{01}$ is sufficiently simple.
   Further assuming this
   we have for composition that

   $$
      L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12})
      \,.
   $$

=--

## References

* [[Katrin Wehrheim]], [[Chris Woodward]], _Functoriality for Lagrangian correspondences in Floer theory _ ([arXiv:0708.2851](http://arxiv.org/abs/0708.2851))
 {#WehrheimWoodward07}

* [[Katrin Wehrheim]], [[Chris Woodward]], _Floer Cohomology and Geometric Composition of Lagrangian Correspondences_ ([arXiv:0905.1368](http://arxiv.org/abs/0905.1368))


Lagrangian density

[[!redirects Lagrangian]]

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#### Variational calculus
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#### Physics
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[[!include physicscontents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

For a [[classical mechanics|classical mechanical]] system, the [[equations of motion|laws of motion]] can be expressed in terms of an [[action]] principle: the actual paths must be the (locally) extremal paths of the [[action functional]].

In one of the formulations of the classical mechanics, called Lagrangean formalism, every mechanical system is characterized by its configuration space and a single function called Lagrangian which determines the laws of motion (the initial configuration should be given independently).

The **Lagrangian**, Lagrangian function or Lagrangean $L = L(q,\stackrel{\cdot}q,t)$ is a real valued function of the points in [[configuration space]] and their time derivatives (for some sytems also depending on time), such that the corresponding action principle can be expressed as [[Euler-Lagrange equation|Euler-Lagrange equations]]: for all $i$,

$$
\frac{d}{dt} \left( \frac{\partial L}{\partial \stackrel{\cdot}{q}_i} \right) - \frac{\partial L}{\partial {q}_i}  = 0
$$

Here $q = (q_1,\ldots, q_n)$ is the coordinate in the configuration space.

For continuum systems satisfying reasonable locality, Lagrangians can be expressed in terms of integrating a local quantity, so-called Lagrangian density.

## Definition

For $X$ a ([[spacetime]]/[[worldvolume]]) [[smooth manifold]] of [[dimension]] $n$, let $E \to X$ be a [[vector bundle]],
to serve as the _[[field bundle]]_ for the $n$-dimensional [[field theory]] Lagrangian to be defined.

Denote the [[jet bundle]] by $j_\infty E \to X$ and write $\Omega^{\bullet, \bullet}(j_\infty E)$ be the corresponding [[variational bicomplex]].

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

A **local Lagrangian** on [[field (physics)|fields]] given by the [[field bundle]] $E \to X$ is given by an element

$$
  L \in \Omega^{n,0}(j_\infty E)
  \,,
$$

hence a [[horizontal differential form]] of degree $n$ on the [[jet bundle]] of $E$.

The local Lagrangian itself is the [[pullback of differential forms|pullback]] of this along the [[jet prolongation]] map
$j_\infty \colon \Gamma_X(E) \longrightarrow \Gamma(j_\infty E)$,
hence the [[differential form]]-valued [[functional]] on the
[[space of sections]] of $E$ given by

$$
  L : (\phi \in \Gamma(E)) \mapsto L(j_\infty \phi) \in \Omega^n(X)
  \,.
$$

The [[integral]] (for [[compact topological space|compact]] $X$)

$$
  \int_X L(j_\infty(-)) \;\colon\;
  \Gamma(E) \longrightarrow \mathbb{R}
$$

is the corresponding [[local action functional]].

=--

## Related concepts

* [[action functional]]

* [[extended Lagrangian]],

* [[BV-Lagrangian density]], [[gauge fixing Lagrangian density]]

* traditional [[Lagrangian mechanics]] and [[Hamiltonian mechanics]] are naturally embedding into [[local prequantum field theory]] by the notion of [[prequantized Lagrangian correspondences]]

* [[Lagrangian quantum field theory]]

* [[non-Lagrangian field theory]]

[[!include Hamiltonian and Lagrangian -- table]]

## References

Named after [[Joseph-Louis Lagrange]].

Textbook account in the context of [[gauge theory]]:

* [[Marc Henneaux]], [[Claudio Teitelboim]], §1.1.1 in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

Discussion in the convenient context of [[smooth sets]]:

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §3 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

See also

* Wikipedia, *[Lagrangian mechanics](https://en.wikipedia.org/wiki/Lagrangian_mechanics)*

* [[Daniel Freed]], _Lecture 2 of [[Five lectures on supersymmetry]]_

[[!redirects Lagrangean]]
[[!redirects Lagrangian function]]
[[!redirects Lagrangian density]]

[[!redirects Lagrangian mechanics]]

[[!redirects Lagrangians]]

[[!redirects local Lagrangian]]
[[!redirects local Lagrangians]]

[[!redirects Lagrangian densities]]
[[!redirects local Lagrangian density]]
[[!redirects local Lagrangian densities]]


Lagrangian distribution

## Related concepts

* [[Fourier integral operator]]

## References

* [[Michael Forger]], Sandra Z. Yepes, _Lagrangian Distributions and Connections in Symplectic Geometry_ ([arXiv:1202.5054](http://arxiv.org/abs/1202.5054))

[[!redirects Lagrangian distributions]]


Lagrangian Grassmanian > history

< [[Lagrangian Grassmanian]]


Lagrangian Grassmannian

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### Context
#### Symplectic geometry
+-- {: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

For $V$ a [[symplectic vector space]], its __Lagrangian Grassmannian__ $LGrass(V)$ is the space of its [[Lagrangian subspace|Lagrangian]] (maximal [[isotropic subspace|isotropic]]) subspaces.

## Properties

### As a coset space

The [[symplectic group]] of $V$ naturally acts on $LGrass(V)$.

The unitary group $U_J(V)$ associated to any fixed compatible [[complex structure]] $J$ on $V$ [[transitive action|acts transitively]] on $LGrass(V)$.  In fact, $LGrass(V)$ is [[diffeomorphism|diffeomorphic]] to the [[coset space]] $U(n)/O(n)$ of the [[unitary group]] by the [[orthogonal group]], where $n$ is the complex [[dimension]] of $(V,J)$ (so the real dimension of $V$ is $2n$).

### Cohomology and Maslov index

The first [[ordinary cohomology]] of the stable Lagrangian Grassmannian

$$
LGrass = \lim_{n \to \infty} LGrass(\mathbb{C}^n)
$$

with [[integer]] [[coefficients]] is isomorphic to the [[integers]]

$$
  H^1(LGrass, \mathbb{Z})
  \cong
  \mathbb{Z}
  \,.
$$

[[generalized the|The]] generator of this [[cohomology group]] is called the _[[Maslov index|universal Maslov index]]_

$$
  u \in H^1(LGrass, \mathbb{Z})
  \,.
$$

Given a [[Lagrangian submanifold]] $Y \hookrightarrow X$ of a [[symplectic manifold]] $(X,\omega)$, its [[tangent bundle]] is [[classifying space|classified]] by a function

$$
  i \;\colon\; Y \to LGrass
  \,.
$$

The [[Maslov index]] of $Y$ is the universal Maslov index pulled back along this map

$$
  i^\ast u \in H^1(Y,\mathbb{Z})
  \,.
$$

Unstably, a generator of

$$
H^1(\LGrass(\mathbb{C}^n),\mathbb{Z}) \cong [U(n)/O(n), U(1)]
$$

is given by the map

$$
\mathrm{det}^2 \colon U(n) \to U(1)
$$

This map passes to the quotient $U(n)/O(n)$ because the determinant of an element of $O(n)$ can only be $\pm 1$.

## Related concepts

* [[Grassmannian]]

  * [[isotropic Grassmannian]]

  * **Lagrangian Grassmannian**

* [[Maslov index]]

## References

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)

* [[Andrew Ranicki]], _The Maslov Index_ ([pdf](http://www.maths.ed.ac.uk/~aar/maslovnotes.pdf))

* Esteban Andruchow, Gabriel Larotonda, _Lagrangian Grassmannian in Infinite Dimension_ ([arXiv:0808.2270](http://arxiv.org/abs/0808.2270))

* J. Carrillo-Pacheco, F. Jarquín-Zárate, M. Velasco-Fuentes, F. Zaldívar, _An explicit description in terms of Plücker coordinates of the Langrangian-Grassmannian_, [arxiv/1601.07501](http://arxiv.org/abs/1601.07501)

[[!redirects Lagrangian Grassmannian]]
[[!redirects Lagrangian Grassmannians]]
[[!redirects Lagrangian grassmannian]]
[[!redirects Lagrangian grassmannians]]
[[!redirects Lagrangean Grassmannian]]
[[!redirects Lagrangean Grassmannians]]
[[!redirects Lagrangian Grassmanian]]
[[!redirects Lagrangian Grassmanians]]
[[!redirects lagrangian grassmanian]]
[[!redirects lagrangian grassmanians]]
[[!redirects Lagrangean Grassmanian]]
[[!redirects Lagrangean Grassmanians]]


Lagrangian quantum field theory

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###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Functorial Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], a _Lagrangian quantum field theory_ is a _[[quantum field theory]]_ (QFT) which arises via some version of [[quantization]] from a [[Lagrangian density]] on (the [[jet bundle]] of) some [[field bundle]], hence from a [[prequantum field theory]].

Most QFTs that are being considered are Lagrangian quantum field theories. Indeed all of traditional [[perturbative quantum field theory]], subsuming in particular [[QED]], [[EW]], [[QCD]], [[pQG]] and hence the [[standard model of particle physics]], is Lagrangian.

But not all QFTs are Lagrangian; or at least they need not have been explicitly constructed via [[quantization]] from a Lagrangian density, even if they could be constructed this way:

For example [[self-dual higher gauge theory]] (such as the _chiral_ [[WZW model]], which is "one half" of the Lagrangian WZW model) is thought not to be Lagrangian. Still, these [[self-dual higher gauge theories]] are thought be be defined via the [[holographic principle]] as the holographic boundary of a QFT that is Lagrangian: [[higher dimensional Chern-Simons theory]].

Many [[topological quantum field theories]] may be constructed by abstract algebraic means. Famous examples are the [[A-model]]/[[B-model]] [[topological string]] [[worldvolume]] theories, which may abstractly be constructed from [[Calabi-Yau A-∞ categories]]. Still, these were originally conceived of as [[topological twist|topological twists]] of a Lagrangian field theory, namely of the 2-dimensional [[supersymmetric sigma-model]] with [[target space]] a 3-dimensional [[Calabi-Yau manifold]].

Similarly, some classes of [[2d CFTs]] may be constructed by purely algebraic means, in particular the construction of 2d [[rational conformal field theory]] is completely reduced to an algebraic construction in suitable [[modular tensor categories]] by the [[FRS-theorem on rational 2d CFT]]. This includes the (chiral or non-chiral) [[WZW model]] mentioned above, but there are also [[2d CFTs]] that are thought to be genuinely "non-geometric", such as the [[Gepner model]].

## Related concepts

* [[non-Lagrangian field theory]]

* [[local quantum field theory]]

* [[locally variational field theory]]

* [[algebraic quantum field theory]]

* [[perturbative algebraic quantum field theory]]

## References

On [[smooth sets]] as a [[convenient category of spaces|convenient category]] for [[variational calculus]] of [[Lagrangian quantum field theory|Lagrangian]] [[classical field theory]]:

* [[Grigorios Giotopoulos]], [[Hisham Sati]], *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

[[!redirects Lagrangian quantum field theories]]

[[!redirects Lagrangian field theory]]
[[!redirects Lagrangian field theories]]


lagrangian submanifold

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## Idea

A _Lagrangian submanifold_ of a [[symplectic manifold]] is a submanifold which is a maximal [[isotropic submanifold]], hence a [[submanifold]] on which the [[symplectic form]] vanishes, and which is maximal with this property.

In the archetypical example of an even dimensional [[Cartesian space]] $X = \mathbb{R}^{2n}$ equipped with its canonical symplectic form  $\omega = \sum_{i = 1}^n d q_i \wedge d p^i$, standard Lagrangian submanifolds are the submanifolds $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$ of fixed values of the $\{q_i\}_{i = 1}^n$ [[coordinates]]. Indeed _locally_, every Lagrangian submanifold looks like this.

Lagrangian submanifolds are of central importance in [[symplectic geometry]] where they constitute [[leaves]] of [[real polarizations]] and are closely related to [[quantum states]]:

If one thinks of a [[symplectic manifold]] as a [[phase space]] of a [[physical system]], then a Lagrangian submanifold may be thought of (locally) as the space of "all [[canonical momenta]] (= parameterization of a [[leaf]]) at fixed [[canonical coordinate]] (= parameterization of [[leaf space]])".

A Lagrangian submanifold equipped with a [[half-density]] is a model for a [[state]] of the physical system in [[semiclassical approximation]] (see e.g. [Bates-Weinstein, p. 14](#BatesWeinstein)). A [[quantum state]] given by a [[wave function]] (see there) is a refinement of this concept.

## Definition

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

A (**Lagrangean** or) **lagrangian submanifold** of a [[symplectic manifold]] $(X,\omega)$ is a [[submanifold]] $L \hookrightarrow X$ such that the following equivalent conditions hold

* at each point $\ell \in L$ the [[tangent space]] $T_\ell L \hookrightarrow T_\ell X$ is a [[Lagrangian subspace]] (hence a simultanously [[isotropic subspace]] and [[coisotropic subspace]]) of $T_\ell X$ equiiped with the [[symplectic form]] $\omega_x$;

* $L \hookrightarrow X$ is a maximal [[isotropic submanifold]].

=--

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

More generally one can consider Lagrangian submanifolds of symplectic structures in [[higher geometry]], such as [[symplectic Lie n-algebroids]] equipped with their canonical [[invariant polynomials]], thought of as [[dg-manifolds]] (via their [[Chevalley-Eilenberg algebra]]) and equipped with graded symplectic forms. Lagrangian dg-submanifolds of such symplectic dg-manifolds have been called **$\Lambda$-structures** in ([Ševera, section 4](#Severa)).

=--

## Examples in higher differential geometry

We discuss classes of examples of Lagrangian dg-submanifolds, remark \ref{LambdaStructures}, of [[symplectic Lie n-algebroids]].

### Of a Poisson Lie algebroid
 {#OfAPoissonLieAlgebroid}

A [[Poisson Lie algebroid]] $\mathfrak{P}$ is a [[symplectic Lie n-algebroid]] for $n = 1$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

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

A dg-[[Lagrangian submanifold]] of $(\mathfrak{P}, \omega)$ is a Lagrangian dg-submanifold, also called a **$\Lambda$-structure**. ([Ševera, section 4](#Severa)).

=--

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

A [[foliation]] by such [[leaves]] is a Lagrangian _[[foliation of a Lie algebroid]]_.

=--

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

For $(X, \pi)$ the [[Poisson manifold]] underlying a [[Poisson Lie algebroid]] $(\mathfrak{P}, \omega)$, a dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ corresponds to a [[coisotropic submanifold]] of $(X, \pi)$.

=--

([Ševera, section 4](#Severa))

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

As a [[vector bundle]] with bracket structure, the [[Poisson Lie algebroid]] $\mathfrak{P}$ is

$$
  \array{
    T^* X &&\stackrel{\pi}{\to}&& T X
    \\
    & \searrow && \swarrow
    \\
    && X
  }
$$

where the horizontal morphism is given by contraction/pairing with the [[Poisson tensor]].

It is sufficient to consider this locally over a [[coordinate chart]] and hence we set without essential restriction of generality
$X = \mathbb{R}^n$ with the [[invariant polynomial]]/graded [[symplectic form]] on $CE(\mathfrak{P})$ being

$$
  \omega = \mathbf{d} x^i \wedge \mathbf{d} p_i
  \,,
$$

where the $\{q_i\}_{i = 1}^n$ are the canonical [[coordinates]] on $\mathbb{R}^n$ and where the $\{p_i\}$ are the canonical coordinates on $T^*_x \mathbb{R}^n \simeq \mathbb{R}^n$, regarded as being in degree 1.

Consider then a sub-Lie algebroid of $\mathfrak{P}$ over a [[submanifold]] $S \hookrightarrow \mathbb{R}^n$. That the corresponding subbundle

$$
  \array{
    E &\hookrightarrow& T^* X
    \\
    \downarrow && \downarrow
    \\
    S &\hookrightarrow & X
  }
$$

over $S$ is [[Lagrangian subspace|Lagrangian]] with respect to the above $\omega$ means that $E$ consists of precisely those [[cotangent vectors]] to $X$ which vanish when evaluated on [[tangent vectors]] of $S$. Hence

$$
  E = N^* S
$$

is the [[conormal bundle]] to $S \hookrightarrow X$. The inclusion $N^* S \hookrightarrow T^*_S X$ of vector bundles is an inclusion of [[Lie algebroids]] over $S$ precisely if the [[anchor map]] restricts to an anchor on $S$, hence that contraction with the Poisson tensor restricted to conormal vectors  of $S$ lands in tangent vectors of $S$:

$$
 \pi(N^* S) \subset T S
  \,.
$$

This is the standard definition for what it means for $S$ to be a [[coisotropic submanifold]].

=--

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

The dg-Lagrangian submanifolds also correspond to [[branes]] in the [[Poisson sigma-model]] (see there) on $(\mathfrak{P}, \omega)$.

=--

### Of a Courant Lie 2-algebroid
 {#OfACourantLie2Algebroid}

A [[Courant Lie algebroid]] $\mathfrak{C}$ is a [[symplectic Lie n-algebroid]] for $n = 2$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

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

A dg-[[Lagrangian submanifold]] of $(\mathfrak{C}, \omega)$ is also called a **$\Lambda$-structure**. ([Ševera, section 4](#Severa)).

Hence we might say **real polarization** of $(\mathfrak{C}, \omega)$ is a foliation by dg-Lagrangian submanifolds.

=--

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

The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid $(\mathfrak{C}, \omega)$ correspond to [[Dirac structures]] on $(\mathfrak{C}, \omega)$.

=--

([Ševera, section 4](#Severa))

## Related concepts

* [[special Lagrangian submanifold]]

* [[Lagrangian subspace]], [[Lagrangian Grassmannian]]

* [[Lagrangian correspondence]], [[prequantized Lagrangian correspondence]]

* [[Lagrangian cobordism]]

* [[Lagrangian subspace]]

* [[isotropic submanifold]]

* [[polarization]]

* [[Legendrean submanifold]]

[[!include (co)isotropic subspaces - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

The concept of lagrangian submanifold has been defined/named in

* [[Victor Maslov]], _Perturbation Theory and Asymptotic Methods_ (MSU Publ., Moscow, 1965; English translation: Mir, Moscow, 1965).

See also:

* [[Alan Weinstein]], _Symplectic Manifolds and Their Lagrangian Submanifolds_, Advances in Math. **6** (1971) 329346 \[doi:10.1016/0001-8708(71)90020-X\]

An introduction with an eye towards [[geometric quantization]] is for instance in

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)
 {#BatesWeinstein}

(pages 10 and onward and then section 4.3).

Lagrangian submanfolds of symplectic [[dg-manifolds]] are called "$\Lambda$-structures" in

* [[Pavol Ševera]], _Some title containing the words "homotopy" and "symplectic", e.g. this one_ ([arXiv:0105080](http://arxiv.org/abs/math/0105080))
 {#Severa}

[[!redirects lagrangian submanifold]]
[[!redirects lagrangian submanifolds]]
[[!redirects Lagrangian submanifold]]
[[!redirects Lagrangian submanifolds]]
[[!redirects lagrangean submanifold]]
[[!redirects lagrangean submanifolds]]
[[!redirects Lagrangean submanifold]]
[[!redirects Lagrangean submanifolds]]

[[!redirects lagrangian manifold]]
[[!redirects lagrangian manifolds]]
[[!redirects Lagrangian manifold]]
[[!redirects Lagrangian manifolds]]
[[!redirects lagrangean manifold]]
[[!redirects lagrangean manifolds]]
[[!redirects Lagrangean manifold]]
[[!redirects Lagrangean manifolds]]

[[!redirects Lagrangian dg-submanifold]]
[[!redirects Lagrangian dg-submanifolds]]
[[!redirects dg-Lagrangian submanifold]]
[[!redirects dg-Lagrangian submanifolds]]


Lagrangian subspace

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## Definition

In a [[symplectic vector space]] a **Lagrangian subspace** is a maximal [[isotropic subspace]]:

a sub-[[vector space]]

* on which the restriction of the [[symplectic form]] vanishes;

* and which has maximal [[dimension]] with this property.

Similarly for a [[symplectic manifold]]. See _[[Lagrangian submanifold]]_ .

The collection of all Lagrangian subspaces of a given space is called its [[Lagrangian Grassmannian]].

## Related concepts

* [[Lagrangian Grassmannian]]

* [[Lagrangian submanifold]]

* [[Maslov index]]

* [[metaplectic structure]]

[[!include (co)isotropic subspaces - table]]

[[!redirects lagrangian subspace]]

[[!redirects lagrangian subspaces]]
[[!redirects Lagrangian subspaces]]

[[!redirects Lagrangean subspace]]
[[!redirects Lagrangean subspaces]]


Laiachi El Kaoutit

[[!redirects L. El Kaoutit]]
Laiachi El Kaoutit is a _Profesor Titular de Universidad_
at the University of Granada, Department of Algebra and the IEMath-Granada.

His interests are in the general areas of [[representation]] theory and [[homological algebra]].

* [Homepage](http://www.ugr.es/~kaoutit/page_web_English.html)

## Selected papers

* [[Tomasz Brzeziński]], L. El Kaoutit, J. Gómez-Torrecillas, _The bicategories of corings_, J. Pure & Appl. Alg. __205__:3 (2006) 510-541 [doi:10.1016/j.jpaa.2005.07.013](https://doi.org/10.1016/j.jpaa.2005.07.013) [math.RA/0408042](https://arxiv.org/abs/math/0408042)

On [[Morita contexts]] in [[bicategories]]:

* L. El Kaoutit, _Wide Morita contexts in bicategories_, Arab. J. Sci. Eng. __33__, (2008), 153–173

A [[Tannaka duality]]-type theorem relating certain subcategory of
commutative [[Hopf algebroid]]s to discrete [[groupoid]]s is in

* Laiachi El Kaoutit, _Representative functions on discrete groupoids and duality with Hopf algebroids_, [arxiv/1311.3109](http://arxiv.org/abs/1311.3109)

category: people


Lakshya Bhardwaj

* [InSpire page](https://inspirehep.net/authors/1275756)

* [institute page](https://www.maths.ox.ac.uk/people/lakshya.bhardwaj)

## Selected writings

On [[generalized global symmetries]] and their [[gauge symmetry|gauging]]:

* {#BT17} [[Lakshya Bhardwaj]], [[Yuji Tachikawa]]. *On finite symmetries and their gauging in two dimensions*, J. High Energ. Phys. **2018** 189 (2018) [[arXiv:1704.02330](https://arxiv.org/abs/1704.02330), doi:10.1007/JHEP03(2018)189]


Lam Hui

* [webpage](http://user.astro.columbia.edu/~lhui/)

## related $n$Lab entries

* [[fuzzy dark matter]]

* [[axion]]

category: people


Lamb shift

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## Idea

The _Lamb shift_ is a tiny shift in the [[energy]] spectrum of excitations of the [[electron]] orbits in the [[hydrogen atom]].

This is a "fine structure" effect not seen by the [[quantum mechanics]] of the [[electron]], but resulting from [[radiative corrections]] involving [[photon]] emission/absorption only seen by regarding the electron in [[perturbative quantum field theory]], here in [[quantum electrodynamics]].

The Lamb shift may be attributed to [[loop order|1-loop]] [[Feynman diagram]]-contributions giving "[[radiative corrections]]" to the basic [[tree level]] [[electron-photon vertex]] that describes the basic [[interaction]] of the [[electron]] with the [[background field|background]] [[electromagnetic field]], via the [[electron-photon vertex]], hence called _vertex corrections_ and governed by the [[coupling constant]] of [[quantum electrodynamics]], the _[[fine structure constant]]_.

{#VertexCorrectionDiagrams}

(Here the dot on the right is the external vertex for the [[background field|background]] [[electromagnetic field]].)

The main contribution (often referred to as _the_ vertex correction) is due to the [[radiative correction]] induced by the [[electron]] emitting a [[virtual particle|virtual]] [[photon]], then interacting with the [[background field|background]] [[electromagnetic field]], and and finally re-absorbing the [[virtual particle|virtual]] [[photon]]; described by this [[Feynman diagram]]:

Historically, the observation of the Lamb shift in [[experiment]] catalyzed the theoretic development of [[quantum electrodynamics]] specifically and of [[perturbative quantum field theory]] generally.

## Related concepts

* [[Casimir effect]]

## References:

Named after [[Willis Lamb]], who had worked on war-time radar technology during the WWII, came back to his lab, applied his radar technology to the [[hydrogen atom]] and found the effect:

* [[Willis Lamb]], Robert C. Retherford, _Fine Structure of the Hydrogen Atom by a Microwave Method_, Phys. Rev. 72, 241 – Published 1 August 1947 ([doi:10.1103/PhysRev.72.241](https://doi.org/10.1103/PhysRev.72.241))

Texts discussing the proper derivation in terms of [[Feynman diagrams]] include

* {#EidesGrotchShelyuto00} Michael Eides, Howard Grotch, Valery A. Shelyuto, _Theory of Light Hydrogenlike Atoms_, Phys. Rept. 342:63-261, 2001 ([arXiv:hep-ph/0002158](https://arxiv.org/abs/hep-ph/0002158))

* {#Weinberg95} [[Steven Weinberg]], section 14.3 of _The quantum theory of fields - Volume I: Foundations_, Cambridge 1995

a useful lightning survey is in

* {#Alipour14} [[Mojtaba Alipour]], _Lamb Shift_, 2014 ([[AlipourLambShift14.pdf:file]])

Many lecture notes instead present a heuristic plausibility argument in terms of oscillations of the electron picked up from "vacuum fluctuations" of the background electomagnetic field:

* [[Hitoshi Murayama]], section 1.1 of _Quantum Electrodynamics_ ([pdf](http://hitoshi.berkeley.edu/221B-S02/QED.pdf))

See also

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

Discussion of the Lamb shift in [[experiment]] includes

* _[the Lamb Shift in Hydrogen](http://courses.washington.edu/phys432/lamb_shift/lamb_shift.php)_

[[!redirects Lamb shifts]]


Lambda baryon

[[!redirects Lambda baryons]]

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{:toc}

## Idea

A _$\Lambda$-baryon_ is a [[baryon]] which is a [[bound state]] of an [[up quark]], a [[down quark]] and one quark from a higher [[generation of fundamental particles]]: a [[charm quark]] ($\Lambda_c$), a [[bottom quark]] ($\Lambda_b$) or a [[top quark]] ($\Lambda_t$).

## Related concepts

* [[proton]], [[neutron]]

* [[flavour anomaly]]

## References

### General

See also

* Wikipedia, _[Lambda baryon](https://en.m.wikipedia.org/wiki/Lambda_baryon)_

### Flavour anomaly

* [[LHCb collaboration]], _Test of lepton universality with $\Lambda^0_b\to p K^-\ell^+\ell^-$ decays_ ([arXiv:1912.08139](https://arxiv.org/abs/1912.08139))

  also:

  CERN courier, _[Debut for baryons in flavour puzzle](https://cerncourier.com/a/debut-for-baryons-in-flavour-puzzle/)_, 20 Dec. 2019

  CERN news, _[LHCb explores the beauty of lepton universality](https://home.cern/news/news/physics/lhcb-explores-beauty-lepton-universality)_, 15 Jan 2020

  (first measurement of lepton universality for $\Lambda^0_b$ [[Lambda baryons]], finding the [[flavour anomaly]] at (only) 1 [[standard deviation|sigma]] but pointing in the same direct


lambda theory

## Idea

The typing rules for [[simply typed lambda calculus]]

$$
\frac{\Gamma \vdash t: \sigma\to \tau \quad \Gamma \vdash u:\sigma}{\Gamma \vdash t(u) : \tau}
\qquad
\frac{\Gamma,x:\sigma \vdash t:\tau}{\Gamma \vdash \lambda x.t : \sigma \to \tau}
$$

can be seen as defining a family of sets of [[terms]], indexed by context and by type, while the computation rule $(\lambda x.t)u = t[u/x]$ can be seen as considering the [[quotient]] of this family of sets by the $\beta$ axiom (defined in terms of the [[substitution]] operation).

But suppose that we collapse all of the types into one "universal" type $U$, so that terms need no longer be simply-typed:

$$
\frac{\Gamma \vdash t: U \quad \Gamma \vdash u: U}{\Gamma \vdash t(u) : U}
\qquad
\frac{\Gamma,x: U \vdash t: U}{\Gamma \vdash \lambda x.t:U}
$$

Now, these rules can be seen as defining a family of sets of terms indexed only by the _length_ of the context (corresponding to the number of free variables).  Together with the action of substitution and after quotienting by the $\beta$ axiom, this family of sets can then be considered as a cartesian [[operad]] defining an [[algebraic theory]] of [[pure lambda calculus]] terms.

## Definition

Let $L : Fin \to Set$ be a cartesian operad (i.e., essentially a [[Lawvere theory]]).  A **semi-closed structure** ([Hyland](#Hyland13)) on $L$ is a family of maps

$$\rho : L(n) \to L(n+1) \quad \lambda : L(n+1) \to L(n)$$
$$\rho \circ \lambda = id$$

natural in $n$, which are compatible with the operadic structure in an appropriate sense.  A **lambda theory** is a cartesian operad equiped with a semi-closed structure.

## Scott's representation theorem

Any lambda theory is isomorphic to the endomorphism lambda theory of a [[reflexive object]] in a [[cartesian closed category]].  In particular, let $L$ be a lambda theory, then for the associated reflexive object $U^U \lhd U$ we simply take $U = L$ itself, seen as an object of the [[presheaf]] category $[Fin,Set]$ together with its associated ccc structure (see [[closed monoidal structure on presheaves]]).  We have that the exponential $U^U$ is isomorphic to $L(-+1)$, and thus a retract of $U$ by the semi-closed structure of $L$.  Moreover, we have that $[Fin,Set](U^n,U) \cong L(n)$ naturally in $n$.

## Related concepts

* [[reflexive object]]

## References

* {#Hyland13} [[Martin Hyland]]. _Classical lambda calculus in modern dress_ Mathematical Structures in Computer Science, 27(5) (2017) 762-781. doi:[10.1017/S0960129515000377](https://doi.org/10.1017/S0960129515000377). [arxiv:1211.5762](http://arxiv.org/abs/1211.5762)


lambda-abstraction

# $\lambda$-Abstraction
* table of contents
{: toc}

## Idea

$\lambda$-Abstraction is the process of interpreting a formula for a function (or operation) as defining an actual [[function]].

## Notation

If in a [[context]] with a [[free variable]] $x$ of [[type]] $A$ we have a [[term]] $t$ of type $B$, then
\[ \lambda_{x\colon A} t \label{implicit} \]
is a term (in a context *without* the free variable) of the [[dependent product]] type $\prod_{x\colon A} B$.  If $B$ is defined in the context without the free variable, then this expression is a term of the [[function type]] $A \to B$ or $B^A$.

Often we make the presence of the free variable explicit in the notation, saying that $t[x]$ is a term of type $B[x]$; then
\[ \lambda_{x\colon A} t[x] ,\label{explicit} \]
a term of the dependent product $\prod_{x\colon A} B[x]$.  Or if $t[x]$ is a term of the constant type $B$, then the same expression is a term of the function type $A \to B$.

## In set theory

Although developed for [[type theory]] and commonly used in [[computer science]], $\lambda$-abstraction makes sense in ordinary mathematics usually based on [[set theory]].  If the types $A$ and $B$ are interpreted as [[sets]] and we define [[functions]] as certain [[subsets]] of [[cartesian products]], then the notation (eq:explicit) refers to
$$ \{ (x,y) \in A \times B \;|\; y = t[x] \} $$
(at least in the simple case where $B$ is constant).

[[!redirects lambda abstraction]]
[[!redirects lambda abstractions]]
[[!redirects lambda-abstraction]]
[[!redirects lambda-abstractions]]
[[!redirects ? abstraction]]
[[!redirects ? abstractions]]
[[!redirects ∞-abstraction]]
[[!redirects ∞-abstractions]]

[[!redirects function abstraction]]
[[!redirects function abstractions]]


Lambda-algebra

#Contents#
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{:toc}

## Idea

What is called the _Lambda algebra_ ($\Lambda$-algebra) in [[stable homotopy theory]] is a [[differential graded algebra]] over $\mathbb{F}_2$ whose [[chain homology]] is the second page of the [[classical Adams spectral sequence]].

It has the advantage over the [[May spectral sequence]] that it tends to make the [extension problem](spectral+sequence#ExtensionProblem) for the Adams spectral sequence more tractable ([Kochman 96, corollary 5.2.8](#Kochman96))

## Related concepts

* [[EHP spectral sequence]]

* [[Curtis algorithm]]

## References

The construction originates in

* [[Aldridge Bousfield]], [[Edward Curtis]], [[Daniel Kan]], [[Daniel Quillen]], D.L. Rector, J.W. Schlesinger, _The mod-$p$ lower central series and the Adams spectral sequence_, Topology, Volume 5, Issue 4, November 1966, Pages 331-342

See also

* [[Edward Curtis]], _The Dyer-Lashof algebra and the $\Lambda$-algebra_,     Illinois J. Math. Volume 19, Issue 2 (1975), 231-246. ([Euclid](http://projecteuclid.org/euclid.ijm/1256050812))

Textbook accounts include

* [[Doug Ravenel]], section 3.3 of _[[Complex cobordism and stable homotopy groups of spheres]]_ 1986/2003

* {#Kochman96} [[Stanley Kochman]], section 5.2 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

[[!redirects Lambda algebra]]