The Finite Dimensional Representations of U _q (sl₂(k))

Abstract

Recall that in the analogous classical setting of the finite dimensional representation theory of U(sl₂ (k)) (when k has characteristic zero), there is a unique irreducible U(s1₂(k))-module of dimension n for each non-negative integer n, [98 Chapter 7]; and every finite dimensional U(sl₂(k))-module is semisimple [98 Theorem 6.3]. We shall see that there are analogous results for U _q (s1 ₂ (k)) when q is not a root of unity. For convenience we’ll write U for U _q (sl ₂ (k)) in this chapter, with corresponding notation for the subalgebras defined in (I.3.1).