Chapter 2

The Classical Simple

Lie Superalgebras. I

2.1. Introduction

For the remainder of this book the ground field K is assumed to be alge-

braically closed, unless otherwise noted. In this chapter we give an explicit

construction for each nonexceptional classical simple Lie superalgebra fol-

lowing [Kac77a] and [Sch79]. The exceptional cases will be handled in

Chapter 4. First we establish some notation. Let g be any finite dimen-

sional Lie superalgebra such that g0 is reductive and

g1

is a semisimple

g0-module.

Let h0 be a Cartan subalgebra of g0. For α ∈ h0,

∗

set

gα

= {x ∈ g|[h, x] = α(h)x for all h ∈ h0},

and let

Δ = {α ∈ h0|α

∗

= 0,

gα

= 0}

be the set of roots of g. Since the action of h0 on any finite dimensional

simple g0-module is diagonalizable, it follows that the adjoint action of h0

on g is diagonalizable. Thus there is a root space decomposition

(2.1.1) g = h ⊕

α∈Δ

gα,

where h =

g0

is the centralizer of h0 in g. In each case we describe the

roots Δ for a simple choice of h0. By checking each case we can see that the

following holds.

Lemma 2.1.1. If

g

is a classical simple Lie superalgebra and α, β, α + β

are roots of g, then

[gα, gβ]

=

gα+β.

11

http://dx.doi.org/10.1090/gsm/131/02