∞-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 $𝔤={𝔤}_{\mathrm{even}}\oplus {𝔤}_{\mathrm{odd}}$;

2. equipped with a bilinear bracket

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

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

3. that satisfied the ${ℤ}_{2}$-graded Jacobi identity:

$\left[x,\left[y,z\right]\right]=\left[\left[x,y\right],z\right]+\left(-1{\right)}^{\mathrm{deg}x\mathrm{deg}y}\left[y,\left[x,z\right]\right]\phantom{\rule{thinmathspace}{0ex}}.$[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 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)