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),

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http://dx.doi.org/10.1090/gsm/131/01