2019/04/23 by Li, Yu
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1904.10141
In his work on crystal bases \citeKas, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra \mathfrak g, which he called a quantum boson algebra. In this paper, 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 ≃ \mathbb C[\mathfrak n∗], \~~,~~\P), where \mathfrak n is the positive part of \mathfrak g and \~~,~~\P is a Poisson bracket that has the same rank as, but is different from, the Kirillov-Kostant bracket \~~,~~\KK on \mathfrak n∗. Furthermore, we prove that, in the special case of type A, any linear combination a1 \~~,~~\P + a2 \~~,~~\KK, a1, a2 ∈ \mathbb 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 \mathfrak n∗ defined by a root vector for the highest root.