与q-类似 Virasoro-like 李代数相关的李代数的结构和表示

V. Arnold has defined and studied the Virasoro-like Lie algebra in order to study a simplified model of atmospheric motion. The q-analog Virasoro-like Lie algebra is the q-deformation of the Virasoro-like Lie algebra, it was defined by Kirkman. The q-analog Virasoro-like Lie algebra is the Lie algebra of inner derivations of the quantum torus. The B, C and D-series of the q-analog Virasoro-like Lie algebra are defined by Kutuzova and Lebedev in 1992 when they study the vertex operator representations of some quantum tori Lie algebras. We will study the structure and representations of the C,D Lie subalgebras of the q-analog Virasoro-like. We will study the wsymmetric invariant bilinear form, the derivation algebra and the universal central extension of the C, D-series subalgebras of the q-analog Virasoro-like Lie algebra. We will study the highest weight representations and Verma module of those two Lie algebras, and construct some operator representations.

Arnold在研究流体动力学时为研究大气运动简化模型给出了Virasoro-like李代数的定义，q-类似Virasoro-like李代数可以看成是Virasoro-like李代数的q形变，其定义是由Kirkman等给出来的，其是量子环面的内导子做成的李代数的泛中心扩张。Kutuzova和Lebedev在1992年给出了q-类似Virasoro-like李代数的B、C、D型子代数的定义。本课题拟对q-类似Virasoro-like李代数的C、D型子代数的结构和表示进行研究。我们将研究这两类李代数的对称不变双线性型，外导子代数和泛中心扩张，在表示方面研究其最高权模，特别是Verma模的性质，并构造一些性质好的算子表示。

本项目主要研究了q-类似 Virasoro-like 李代数的C、D型子李代数的结构及表示。我们证明了C、D型子李代数都是单李代数，它们都有唯一的对称不变双线性型，它们的外导子代数都是一维的，它们的二阶上同调群是一维的。本项目完成一篇论文，已经发表在SCI收录期刊 Algebra Colloquium 上。