nLab
super Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Super-Algebra and Super-Geometry

Contents

Idea

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

See also supersymmetry.

Definition

Definition

A super Lie algebra over a field kk is a Lie algebra internal to the symmetric monoidal kk-linear category SVect of super vector spaces.

Note

This means that a super Lie algebra is

  1. a super vector space 𝔤=𝔤 even𝔤 odd\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd};

  2. equipped with a bilinear bracket

    [,]:𝔤𝔤𝔤 [-,-] : \mathfrak{g}\otimes \mathfrak{g} \to \mathfrak{g}

    that is graded skew-symmetric: is is skew symmetric on 𝔤 even\mathfrak{g}_{even} and symmetric on 𝔤 odd\mathfrak{g}_{odd}.

  3. that satisfied the 2\mathbb{Z}_2-graded Jacobi identity:

    [x,[y,z]]=[[x,y],z]+(1) degxdegy[y,[x,z]]. [x, [y, z]] = [[x,y],z] + (-1)^{deg x deg y} [y, [x,z]] \,.
Note

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.

See the discussion at superalgebra for details on this.

Examples

References

One of the original references (or the original reference?) is

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

A review of the classification is in

Surveys:

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

A useful survey with more pointers to the literature is

Another useful survey is

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

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

A useful PhD thesis covering Lie superalgebras and superalgebras more generally is

  • D. Westra, Superrings and supergroups ([pdf][http://www.mat.univie.ac.at/~michor/westra_diss.pdf])

Revised on September 6, 2013 15:59:10 by Bruce Bartlett (146.232.117.101)