Quantum Boson Algebra and Poisson Geometry of the Flag Variety

In his work on crystal bases [13], Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semisimple Lie algebra g, which he called a quantum boson algebra. In this work, we construct Kashiwara operators associated to all positive roots and use them to define a variant of Kashiwara's quantum boson algebra. We show that a quasi-classical limit of the positive half of our variant is a Poisson algebra of the form (P=C[n∗], {,}P), where n is the positive part of g and {,}P is a Poisson bracket that has the same rank as, but is different from, the Kirillov-Kostant bracket {,}KK on n∗. Furthermore, we prove that, in the special case of type A, any linear combination a1{,}P+a2{,}KK, a1,a2∈C, is again a Poisson bracket. In the general case, we establish an isomorphism of P and the Poisson algebra of regular functions on the open Bruhat cell in the flag variety. In type A, we also construct a Casimir function on the open Bruhat cell, together with its quantization, which may be thought of as an analog of the linear function on n∗ defined by a root vector for the highest root.