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
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