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

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

= 0,

= 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
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