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

3. that satisfies the $\mathbb{Z}_2$-graded Jacobi identity in that for any three element $x,y,z \in \mathbb{g}$ of homogeneous degree $deg(x),deg(y),deg(z))\in \mathbb{Z}$ then

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

## Properties

### Classification

(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. $A(m,n)$

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

3. $C(n)$

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

2. two exceptional ones

1. $F(4)$

2. $G(3)$

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

4. two “strange” series

1. $P(n)$

2. $Q(n)$

2. Cartan type

(…)

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

$\mathfrak{g}$$\mathfrak{g}_{even}$$\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{odd}$
$B(m,n)$$B_m \oplus C_n$vector $\otimes$ vector
$D(m,n)$$D_m \oplus C_n$vector $\otimes$ vector
$D(2,1,\alpha)$$A_1 \oplus A_1 \oplus A_1$vector $\otimes$ vector $\otimes$ vector
$F(4)$$B_3\otimes A_1$spinor $\otimes$ vector
$G(3)$$G_2\oplus A_1$spinor $\otimes$ vector
$Q(n)$$A_n$adjoint

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

$\mathfrak{g}$$\mathfrak{g}_{even}$$\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{{-1}}$
$A(m,n)$$A_m \oplus A_n \oplus C$vector $\otimes$ vector $\otimes$ $\mathbb{C}$
$A(m,m)$$A_m \oplus A_n$vector $\otimes$ vector
$C(n)$$\mathbb{C}_{-1} \oplus \mathbb{C}$vector $\otimes$ $\mathbb{C}$

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

## References

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)

• 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 of Lie algebra extensions for super Lie algebras includes

Revised on April 1, 2015 04:41:43 by Urs Schreiber (195.113.30.252)