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
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 =
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


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).
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