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Super-Algebra and Super-Geometry



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

See also supersymmetry.


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


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

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

  2. a homomorphism

    [,]:𝔤 k𝔤𝔤 [-,-] \;\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

    𝔤 k𝔤 τ 𝔤,𝔤 super 𝔤 k𝔤 [,] [,] 𝔤 1 𝔤 \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 τ super\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

    𝔤 k𝔤 k𝔤 τ 𝔤,𝔤 super kid 𝔤 k𝔤 k𝔤 [,[,]][[,],] [,[,]] 𝔤. \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{[-,[-,-]]} - [[-,-],-] }\searrow && \swarrow_{\mathrlap{[-,[-,-]]}} \\ && \mathfrak{g} } \,.

As super-graded Lie algebras

Externally this means the following:


A super Lie algebra according to def. 1 is equivalently

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

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

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

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

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

    [x,[y,z]]=[[x,y],z]+(1) σ xσ y[y,[x,z]]. [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 sLieAlg

for the resulting category of super Lie algebras.

As formal duals of a Chevalley-Eilenberg super-algebras


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

𝔤 * \wedge^\bullet \mathfrak{g}^\ast

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

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

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


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

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

α 1α 2=(1) n 1n 2(1) σ 1σ 2α 2α 1. \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(𝔤)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:


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

  1. an nn-cocycle on 𝔤\mathfrak{g} (with coefficients in \mathbb{R}) is an element of degree (n,even)(n,even) in its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) (def. 2) which is d 𝕘d_{\mathbb{g}} closed.

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

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

    H (𝔤,)=H (CE(𝔤)). 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:


The functor

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

that sends a finite dimensional super Lie algebra 𝔤\mathfrak{g} to its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) (def. 2) 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 internal to the cohesive (∞,1)-topos Super∞Grpd over the site of super points (Sachse 08).

See the discussion at superalgebra for details on this.



(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)A(m,n)

      2. B(m,n)=B(m,n) = osp(2m+1,2n)(2m+1,2n) m0m\geq 0, n>0n \gt 0

      3. C(n)C(n)

      4. D(m,n)=D(m,n) = osp(2m,2n)(2m,2n) m2m \geq 2, n>0n \gt 0

    2. two exceptional ones

      1. F(4)F(4)

      2. G(3)G(3)

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

    4. two “strange” series

      1. P(n)P(n)

      2. Q(n)Q(n)

  2. Cartan type


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

𝔤\mathfrak{g}𝔤 even\mathfrak{g}_{even}𝔤 even\mathfrak{g}_{even} rep on 𝔤 odd\mathfrak{g}_{odd}
B(m,n)B(m,n)B mC nB_m \oplus C_nvector \otimes vector
D(m,n)D(m,n)D mC nD_m \oplus C_nvector \otimes vector
D(2,1,α)D(2,1,\alpha)A 1A 1A 1A_1 \oplus A_1 \oplus A_1vector \otimes vector \otimes vector
F(4)F(4)B 3A 1B_3\otimes A_1spinor \otimes vector
G(3)G(3)G 2A 1G_2\oplus A_1spinor \otimes vector
Q(n)Q(n)A nA_nadjoint

For type 1 the /2\mathbb{Z}/2\mathbb{Z}-grading lifts to an \mathbb{Z}-grading with 𝔤=𝔤 1𝔤 0𝔤 1\mathfrak{g} = \mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1.

𝔤\mathfrak{g}𝔤 even\mathfrak{g}_{even}𝔤 even\mathfrak{g}_{even} rep on 𝔤 1\mathfrak{g}_{{-1}}
A(m,n)A(m,n)A mA nCA_m \oplus A_n \oplus Cvector \otimes vector \otimes \mathbb{C}
A(m,m)A(m,m)A mA nA_m \oplus A_nvector \otimes vector
C(n)C(n) 1\mathbb{C}_{-1} \oplus \mathbb{C}vector \otimes \mathbb{C}

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


Some obvious but important classes of examples are the following:


every /2\mathbb{Z}/2-graded vector space VV becomes a super Lie algebra (def. 1, prop. 1) by taking the super Lie bracket to be the zero map

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

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


Every ordinary Lie algebras becomes a super Lie algebra (def. 1, prop. 1) concentrated in even degrees. This constitutes a fully faithful functor

LieAlgsLieAlg. LieAlg \hookrightarrow sLieAlg \,.

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

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

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

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


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)

The original references on the classification of super Lie algebras are

  • Victor Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96.

  • Victor Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. (EUCLID)

See also

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

  • 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) (arXiv:hep-th/9712074).

Discussion in the topos over superpoints is in

Discussion of Lie algebra extensions for super Lie algebras includes

Revised on January 11, 2017 08:55:19 by Urs Schreiber (