Quantum boson algebra and Poisson geometry of the flag variety
李昱（德国马普所）
ZOOM Meeting：653 2000 5749
上课时间：2022年03月04日 下午 15:00
摘要：
In his work on crystal bases, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semisimple Lie algebra $\mathfrak g$, which he called a quantum boson algebra. In this talk, 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 $(P \simeq \mathbb C[\mathfrak n^{\ast}], {~,~}_P)$, 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 $\mathfrak n^{\ast}$ . Furthermore, we prove that, in the special case of type $A$, any linear combination $a_1{~,~}_P + a_2{~,~}_{KK}, a_1, a_2 \in \mathbb C$, is again a Poisson bracket, i.e. ${~,~}_P$ and ${~,~}_{KK}$ are compatible Poisson brackets. 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 $\mathfrak n^{\ast}$ defined by a root vector for the highest root.
