Chapter 1
Introduction
Welcome to the world of super mathematics! In this short chapter we give
some of the basic definitions concerning Lie superalgebras. We also state
the Classification Theorem for finite dimensional classical simple Lie super-
algebras.
1.1. Basic Definitions
Let K be a field. We assume that the characteristic of K is different from
2, 3. Unless otherwise stated, all vector spaces, Lie algebras, etc., are defined
over K. A Z2-graded vector space is merely a direct sum of vector spaces
V = V0 V1. We call elements of V0 (resp. V1) even (resp. odd). Nonzero
elements of V0 V1 are homogeneous and for homogeneous v Vi, we set
v = i, the degree of v. First we mention an important convention which we
use throughout this book.
Degree Convention 1.1.1. If v is an element of a Z2-graded vector space
and v appears in some formula or expression, then v is assumed to be homo-
geneous.
This convention simplifies the notation in many formulas beginning with the
very definition of a Lie superalgebra. Background on Z2-graded structures
is contained in Section A.1 of Appendix A.
A Lie superalgebra is a Z2-graded vector space g = g0 g1 together with a
bilinear map [ , ] : g × g g such that
(a) [gα, gβ] gα+β for α, β Z2 (Z2-grading),
(b) [a, b] = −(−1)ab[b, a] (graded skew-symmetry),
1
http://dx.doi.org/10.1090/gsm/131/01
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