Monomial Bases of Quantized Enveloping Algebras

We construct a monomial basis of the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of U. 0. Introduction In [L1], Lusztig showed that the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie algebra g, had a remarkable basis called the canonical basis. The main idea in proving its existence and uniqueness was the following. Corresponding to every reduced expression i of the longest element w0 of the Weyl group of g one constructs a Poincare–Birkhoff–Witt basis Bi of U . Lusztig proved in [L1] that the Z[q]–lattice L spanned by Bi is independent of the choice of i and that the image of Bi in the Z–module L/q L is a basis B independent of i. Let L be the image of L under the bar map (a certain Q–algebra involution) of U. The canonical basis B is the preimage of B in L∩L. In [K1], Kashiwara introduced the notion of crystal bases for the quantized algebras of classical type and later generalized it to quantized algebras associated to an arbitrary symmetrizable Kac–Moody algebra. The crystal basis is a base ‘ at q = 0’ of U with certain properties. Later, he proved that the crystal bases could be ‘melted’ to give a basis of U itself, which is called the global crystal base. The main tool used here was a certain bilinear form on the algebra, and the global crystal basis can be characterized as a bar–invariant quasi–orthonormal basis with respect to this form. In [X2], Xi proved that the bases Bi are quasi–orthonormal with respect to this form. The quantized enveloping algebra also admits another symmetric bilinear form introduced by Drinfeld, and Lusztig proved in [L2] that the bases Bi are quasi– orthonormal and that the canonical basis can be characterized as the bar–invariant, quasi–orthonormal basis of U with respect to the Drinfeld form. It is now proved [GL] that in fact the global crystal basis and the canonical basis are the same. In this paper we construct a basis of U whose terms are monomials in the Chevalley generators and hence is bar–invariant but not quasi–orthonormal. We are then able to give a very simple proof of the existence and uniqueness of the canonical basis. Our construction of the monomial basis depends on picking a specific reduced expression for w0. We conjecture that in fact there exists a monomial basis corresponding to every reduced expression. In view of [BCP] we expect also that similar results should be true for the quantized affine algebras. 1991 Mathematics Subject Classification. 17B.