On Tight Monomials in Quantized Enveloping Algebras

In this paper, the author studies when some monomials are in the canonical basis of the quantized enveloping algebra corresponding to a simply laced semisimple finite dimensional complex Lie algebra. 0. Introduction To any graph Γ, Lusztig has associated in [L1] and [L2] an algebra U− over Z[v, v−1] provided with a canonical basis B. In the case that Γ is the Dynkin graph of a simply laced semisimple finite dimensional complex Lie algebra g, then U− is the negative part of the corresponding quantized enveloping algebra U and B, the canonical basis (or crystal basis). The simplest elements in U− are certain elements F (a) i , where i is a vertex of Γ and a ∈ N. In this paper, we study when some monomials in the F (a) i ’s are in B. These monomials are said to be tight in that case. In Section 1, we first recall the approach of Lusztig to this question as presented in [L2]. This comes down to studying a quadratic form Q̄Ω,i where Ω is a quiver whose graph is Γ and i = (i1, i2, . . . , im), a sequence of vertices of Γ. This is explored in more detail in Sections 2 and 3. The nicest case is when Γ is loop free. This is studied in Section 3. In Section 4, we give criteria for tightness and semi-tightness. In Sections 5 and 6, we give many examples of tight and semi-tight monomials. Some of these were already studied by Lusztig in [L2] and by Marsh in [M]. We present these examples using our approach. Finally in Section 7, we consider the case where Γ is a Dynkin graph of a simply laced semisimple finite dimensional complex Lie algebra g of small rank and i is the reduced expression for the longest element of the Weyl group of g. Some of these were also studied by Lusztig in [L2] and by Marsh in [M]. In our approach, there is a unit form Q+i that we need to study. We do this using results of the theory of representations of algebras. They are presented in Section 6. Our motivation was a question of Lusztig presented in Section 16 of [L2]. We recall this in the first part of Section 7. We still don’t know the answer to this question, but our hope is that this article will be useful in the search of a solution. Received by the editors July 1, 2003 and, in revised form, April 27, 2004. 2000 Mathematics Subject Classification. Primary 17B37; Secondary 20G99.