nLab super Lie algebra

1. Idea

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

See also at supersymmetry.

2. 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 2.1. 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 2.2. A super Lie algebra according to def. 2.1 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 2.3. 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 2.4. 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. 2.3) 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 2.5. 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. 2.3) 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.

3. 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):

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

4. Examples

Basic examples

Some obvious but important classes of examples are the following:

Example 4.1. every $\mathbb{Z}/2$-graded vector space $V$ becomes a super Lie algebra (def. 2.1, prop. 2.2) by taking the super Lie bracket to be the zero map

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

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

Example 4.2. Every ordinary Lie algebras becomes a super Lie algebra (def. 2.1, prop. 2.2) 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 4.3. 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 4.4. 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 4.5. Def. 4.3 indeed gives a super Lie algebra in that the bracket (5) satisfies the super Jacobi identity (1).

Proof. We proceed by induction:

By Remark 4.4 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 4.6. 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 4.7. (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 4.6 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 4.6 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 4.8. (embedding tensors induce tensor hierarchies)

In view of the relation between super Lie algebras and dg-Lie algebras (above), Prop. 4.7 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.

5. Related entries

• Haag–Łopuszański–Sohnius theorem

• geometry of physics – supersymmetry

6. 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)