nLab
super Lie algebra

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Context

∞-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

∞-Lie groupoids

∞-Lie groups

∞-Lie algebroids

∞-Lie algebras

Super-Algebra and Super-Geometry

superalgebra

and

supergeometry

Formal context

Superalgebra

Supergeometry

Structures

Applications

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 k is a Lie algebra internal to the symmetric monoidal k-linear category SVect of super vector spaces.

Note

This means that a super Lie algebra is

  1. a super vector space 𝔤=𝔤 even⊕𝔤 odd;

  2. equipped with a bilinear bracket

    [−,−]:𝔤⊗𝔤→𝔤[-,-] : \mathfrak{g}\otimes \mathfrak{g} \to \mathfrak{g}

    that is graded skew-symmetric: is is skew symmetric on 𝔤 even and symmetric on 𝔤 odd.

  3. that satisfied the 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 useful survey with more pointers to the literature is

  • L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras (arXiv)

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)

Revised on January 3, 2013 04:55:19 by Urs Schreiber (89.204.153.128)
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