nLab super Lie algebra

Idea

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

See also at supersymmetry.

Definition

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

As Lie algebras internal to super vector spaces

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

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

As super-graded Lie algebras

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

(See also at “NQ-supermanifold”.)

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

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

That this differential squares to 0 follows by the super-Jacobi identity (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

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

Super-Poincaré super Lie algebras (supersymmetry)

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

• super q-Schur algebra

• For every connected topological space, the Whitehead product makes its homotopy groups into a super Lie algebra over the ring of integers – the Whitehead super Lie algebra.

• higher super Lie algebras

  Just as Lie algebras are categorified to L-infinity algebras and L-infinity algebroids, so super Lie algebras categorifie to super L-infinity algebras. A secretly famous example is the

  • supergravity Lie 3-algebra, supergravity Lie 6-algebra

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.

Related entries

• Haag–Łopuszański–Sohnius theorem

• geometry of physics – supersymmetry

References

According to V. Kac 1977b the definition of super Lie algebra is originally due to:

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

  (in Russian)

See also:

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

• Felix A. Berezin (edited by Alexandre A. Kirillov): Lie Superalgebras, chapters I.5 and II.1 in: Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6_6, doi:10.1007/978-94-017-1963-6_7]

The original references on the classification of super Lie algebras:

• Victor Kac, Lie superalgebras, Advances in Math. 26 1 (1977) 8-96 [doi:10.1016/0001-8708(77)90017-2]

• Victor Kac: A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 1 (1977) 31-64 [euclid:1103900590, doi:10.1007/BF01609166]

See also:

• Werner Nahm, Vladimir Rittenberg, Manfred Scheunert, The classification of graded Lie algebras, Physics Letters B 61 4 (1976) 383-384 [doi:10.1016/0370-2693(76)90594-3]

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

Further discussion specifically of classification of supersymmetry:

• Werner Nahm, Supersymmetries and their Representations, Nucl. Phys. B 135 (1978) 149 [spire, pdf]

• Steven Shnider, The superconformal algebra in higher dimensions, Letters in Mathematical Physics 16 4 (1988) 377-383 [doi:10.1007/BF00402046]

• Victor Kac, Classification of supersymmetries, Proceedings of the ICM, Beijing 2002, vol. 1, 319–344 (arXiv:math-ph/0302016)

Introductions and surveys:

• Vladimir Rittenberg: A guide to Lie superalgebras, in P. Kramers, A. Rieckers (eds.): Group Theoretical Methods in Physics, Lecture Notes in Physics 79 Springer (1978) 3-21 [doi:10.1007/3-540-08848-2_1, pdf]

• Manfred Scheunert, The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979) [doi:10.1007/BFb0070929]

• Dimitry A. Leites, Lie Superalgebras, J. Soviet Math. 30 (1985) 2481-2512 [doi:10.1007/BF02249121]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf]

  (with an eye towards supergravity)

• Richard J. Farmer, Orthosymplectic superalgebras in mathematics and science, PhD Thesis (1984) [eprints:19542, pdf]

• Luc Frappat, Antonio Sciarrino, Paul Sorba: Structure of basic Lie superalgebras and of their affine extensions, Comm. Math. Phys. 121 3 (1989) 457-500 [euclid:cmp/1104178142]

• Luc Frappat, Antonio Sciarrino, Paul Sorba: Dictionary on Lie Superalgebras, Academic Press (2000) [arXiv:hep-th/9607161, ISBN:978-0122653407]

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

• D. Westra, Superrings and supergroups (pdf)

• Shiraz Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [arXiv:hep-th/9712074]

Discussion in the presheaf topos over superpoints:

• Christoph Sachse, Section 3 of: A Categorical Formulation of Superalgebra and Supergeometry [arXiv:0802.4067]

Discussion of Lie algebra extensions for super Lie algebras includes

• Dmitri Alekseevsky, Peter Michor, Wolfgang Ruppert, Extensions of super Lie algebras, J. Lie Theory 15 1 (2005) 125-134 [arXiv:math/0101190]

On Lie algebra weight systems arising from super Lie algebras:

• José Figueroa-O’Farrill, Takashi Kimura, Arkady Vaintrob, The universal Vassiliev invariant for the Lie superalgebra $\mathfrak{gl}(1\vert1)$, Commun. Math. Phys. 185 (1997) 93-127 (arXiv:q-alg/9602014)

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

• Roberto Catenacci, C. A. Cremonini, Pietro Antonio Grassi, Simone Noja, Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces (arXiv:2012.05246)