Gauss-Lusztig decomposition for positive quantum groups and representation by q-tori
Gauss-Lusztig decomposition for positive quantum groups and representation by q-tori
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正量子群的 Gauss-Lusztig 分解和 q-tori 表示
DOI：
10.1016/j.jpaa.2015.05.038
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发表时间：
2015
影响因子：
0.8
通讯作者：
Ivan Ip
中科院分区：
文献类型：
--
作者：
永井哲郎;高橋卓也;Todor Milanov;Todor Milanov;渡邉究;岩井良祐，永井哲郎，髙橋 卓也;Ivan Ip
给出了正量子群GL Q+(N，R)及其模二重GL Q Q˜+(N，R)由正本质自伴算子表示的一个显式构造.推广了Lusztig的参数化法，得到了由最长元素w0∈W=S N−1的标准分解所参数化的全正量子群GL q+(N，R)的Gauss型分解.在此参数化下，我们明确地得到了标准量子变量之间的关系，以及量子团簇变量之间的关系，并用Q-Tori v=Q2 v u的非紧生成元用正本质自伴算符实现了它们.模偶是由超越关系自然产生的，在von Neumann情形下还可以定义L 2(GL Q Q˜+(N，R))空间。
We found an explicit construction of a representation of the positive quantum group GL q+(N, R) and its modular double GL q q˜+(N, R) by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group GL q+(N, R) parametrized by the standard decomposition of the longest element w 0∈ W= S N− 1. Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the q-tori u v= q 2 v u by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an L 2 (GL q q˜+(N, R)) space in the von Neumann setting can also be defined.