CHAPTER 2

The Serre relations

2.1. The Serre relations

If we take a semisimple Lie algebra as g, we have a more concise presentation of

U(g). (Since we only consider specific examples in these lectures, it is not necessary

to know what semisimple Lie algebras are.) This implies that we do not have to

deform g itself to obtain a deformation of U(g), which is crucial in defining quantum

algebras. In this chapter, we pick up the simplest example of a semisimple Lie

algebra and verify Serre’s presentation by a concrete argument.

We first introduce the notion of a Cartan subalgebra and a root system of this

Lie algebra, and then go on to the statement and the proof of the main theorem

of this chapter. As in Chapter 1, gl(n, C) is the set of n × n complex matrices

which is viewed as a Lie algebra via [ X, Y ] = XY − Y X, and ad is the adjoint

representation.

Definition 2.1. The special linear Lie algebra is the Lie algebra

sl(n, C) = {X ∈ gl(n, C)| tr(X) = 0}.

We denote matrix units by Eij: namely, the unique non-zero entry of Eij is 1

in the (i, j)th entry. Set

h = X =

n

i=1

ciEii

n

i=1

ci = 0

and define xi : h → C by xi(X) = ci.

Then h is a Lie subalgebra of sl(n, C). It is a Cartan subalgebra of sl(n, C).

(All other Cartan subalgebras of sl(n, C) are of the form g−1hg, for some g ∈

GL(n, C).) The Lie algebra sl(n, C) admits a simultaneous eigenspace decomposi-

tion with respect to ad(h) as follows.

sl(n, C) = h

⎛

⎝

i=j

CEij⎠

⎞

This decomposition is called the root space decomposition of sl(n, C). The

simultaneous eigenvalues are 0 on h, and xi − xj on Eij. The latter non-zero

simultaneous eigenvalues are called roots, and the set Φ = {xi − xj}i=j is called

the root system of type An−1. The roots αi = xi − xi+1 (1 ≤ i n) are called

simple roots.

theorem 2.2. Let g = sl(n, C). Then U(g) is isomorphic to the C-algebra U

defined by the following generators and relations.

Generators: ei,fi,hi (1 ≤ i ≤ n − 1).

9

http://dx.doi.org/10.1090/ulect/026/02