∞-Lie theory

superalgebra

and

supergeometry

# Contents

## Idea

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

## 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 $\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}$;

2. equipped with a bilinear bracket

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

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

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

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

## 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)