A PoIncare–BIrkhoff–Witt Theorem for Quantized Universal Enveloping Algebras of Type $A_N$

Abstract

In this paper, we construct an explicit basis of the quantized universal enveloping algebra Uq(slN + 1(C)). Let A = (%-)i<jj<;v be a symmetrizable generalized Cartan matrix, and &(A) the Kac-Moody Lie algebra of A. Motivated by studies of YangBaxter equations, Jimbo [6] and Drinfeld [2, 3] introduced a Hopf algebra Uq(&(A)) with a nonzero complex parameter q. This Hopf algebra, which is also called [3] a quantum group, can be considered as a natural ^-analogue of the universal enveloping algebra U(&(A)) of &(A). For example, it is known that the representation theory of Uq(&(A)) is quite analogous to that of U(&(A)). See Lusztig [9] and Rosso [11]. The purpose of this paper is to show that, if &(A) is of type AN, and q 8 =£ 1, then Uq(&(A)) has a PoincareBirkhoff-Witt type basis. Let R be a commutative ring with 1. Denote by slN+l(R), the Lie algebra of (N + 1) x (N + 1) matrices over R of trace 0. It has the standard J^-basis consisting of the elements