# nLab super Lie algebra

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

supersymmetry

# Contents

## Idea

A super Lie algebra is the analog of a Lie algebra in superalgebra/supergeometry.

## Definition

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

### As Lie algebras internal to super vector spaces

###### Definition

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

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

2. a homomorphism

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

of super vector spaces (the super Lie bracket)

such that

1. the bracket is skew-symmetric in that the following diagram commutes

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

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

2. the Jacobi identity holds in that the following diagram commutes

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

Externally this means the following:

###### Proposition

A super Lie algebra according to def. is equivalently

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

2. equipped with a bilinear map (the super Lie bracket)

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

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

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

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

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

$sLieAlg$

for the resulting category of super Lie algebras.

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

###### Definition

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

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

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

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

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

$\,$

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

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

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

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

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

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

###### Definition

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

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

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

3. hene the super-Lie algebra cohomology of $\mathfrak{g}$ (with coefficients in $\mathbb{R}$) is the cochain cohomology of its Chevalley-Eilenberg algebra

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

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

###### Proposition

The functor

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

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

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

Equivalently, a super Lie algebra is a “super-representable” Lie algebra object internal to the cohesive (∞,1)-topos Super∞Grpd over the site of super points (Sachse 08, Section 3.2, towards cor. 3.3).

See the discussion at superalgebra for details on this.

## Properties

### Classification

(Kac 77a, Kac 77b) states a classification of super Lie algebras which are

1. finite dimensional

2. simple

3. over a field of characteristic zero.

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

1. classical type

1. four infinite series

1. $A(m,n)$

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

3. $C(n)$

4. $D(m,n) =$osp$(2m,2n)$ $m \geq 2$, $n \gt 0$

2. two exceptional ones

1. $F(4)$

2. G(3)

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

4. two “strange” series

1. $P(n)$

2. $Q(n)$

2. Cartan type

(…)

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

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

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

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

reviewed e.g. in (Farmer 84, p. 25,26, Minwalla 98, section 4.1).

### Relation to dg-Lie algebras

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

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

equipped with

1. a lift of the $\mathbb{Z}_2$-grading of the underlying vector space to a $\mathbb{Z}$-graded vector space through the projection $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} = \mathbb{Z}_2$, hence

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

such that

$[-,-] \;\colon\; \mathfrak{g}_{n_1} \otimes \mathfrak{g}_{n_2} \longrightarrow \mathfrak{g}_{n_1 + n_2}$
2. an element $Q \in \mathfrak{g}_{-1}$

such that

(2)$[Q,Q] = 0$

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

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

That this differential squares to 0 follows by the super-Jacobi identity (1) and by the nilpotency (2):

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

and the derivation-property of the differential over the bracket follows again with the super Jacobi identity (1):

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

## Examples

### Basic examples

Some obvious but important classes of examples are the following:

###### Example

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

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

These may be called the “abelian” super Lie algebras.

###### Example

Every ordinary Lie algebras becomes a super Lie algebra (def. , prop. ) concentrated in even degrees. This constitutes a fully faithful functor

$LieAlg \hookrightarrow sLieAlg \,.$

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

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

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

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

### Embedding tensors and tensor hierarchy

The following example is highlighted in Palmkvist 13, 3.1 (reviewed more clearly in Lavau-Palmkvist 19, 2.4) where it is attributed to I. L. Kantor (1970).

###### Definition

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

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

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

concentrated in degrees $\leq 1$, recursively as follows:

For $n =1$ we set

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

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

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

Hence:

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

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

On this super vector space consider a super Lie bracket defined recusively as follows:

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

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

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

(4)$[f, v] \;\coloneqq\; f(v)$

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

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

By (4) the definition (5) is equivalent to

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

Hence (5) is already implied by (4) if the bracket is to satisfy the super Jacobi identity (1).

It remains to show that:

###### Proposition

Def. indeed gives a super Lie algebra in that the bracket (5) satisfies the super Jacobi identity (1).

###### Proof

We proceed by induction:

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

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

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

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

###### Example

For $f,g \in \widehat V_0 = Hom_k(V,V)$ (3) we have that the bracket on $\widehat V$ in Def. restricts to

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

(by combining (5) with (4)).

This is the Lie bracket of the general linear Lie algebra $\mathfrak{gl}(V)$, as indicated on the right in (3).

###### Proposition

(embedding tensors are square-0 elements in $\widehat{V}$)

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

An element in degree -1 of the super Lie algebra $\widehat V$ from Def. ,

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

which by Example is identified with a linear map

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

from $V$ to the general linear Lie algebra on $V$, is square-0 (2) precisely if it is an embedding tensor, in that:

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

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

###### Proof

By unwinding of the definition (4) and (5) and using again Example we compute as follows:

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

(embedding tensors induce tensor hierarchies)

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

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

According to Kac77b the definition of super Lie algebra is originally due to

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

• Isaiah Kantor, Graded Lie algebras, Trudy Sem. Vektor. Tenzor. Anal 15 (1970): 227-266.

The original references on the classification of super Lie algebras are

• Werner Nahm, V. Rittenberg, Manfred Scheunert, The classification of graded Lie algebras , Physics Letters B Volume 61, Issue 4, 12 April 1976, Pages 383–384 (publisher)

• M. Parker, Classification Of Real Simple Lie Superalgebras Of Classical Type, J.Math.Phys. 21 (1980) 689-697 (spire)

Further discussion of classification related specifically to classification of supersymmetry is due to

Introductions and surveys include

• Richard Joseph Farmer, Orthosymplectic superalgebras in mathematics and science, PhD Thesis (1984) (web, pdf)

• L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras (arXiv:hep-th/9607161)

• Groeger, Super Lie groups and super Lie algebras, lecture notes 2011 (pdf)

• L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras (arXiv:hep-th/9607161)

• D. Leites, Lie superalgebras, J. Soviet Math. 30 (1985), 2481–2512 (web)

• Manfred Scheunert, The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979)

• D. Westra, Superrings and supergroups (pdf)

• Shiraz Minwalla, Restrictions imposed by superconformal invariance on quan tum field theories Adv. Theor. Math. Phys. 2, 781 (1998)

Discussion in the topos over superpoints is in

Discussion of Lie algebra extensions for super Lie algebras includes

On Lie algebra weight systems arising from super Lie algebras:

On Lie algebra cohomology of super Lie algebras (see also the brane scan) in relation to integrable forms of coset supermanifolds: