学 术 报 告

报告题目：Poisson structure and second quantization of quantum cluster algebras

报告人：李方(浙江大学)

报告时间：2020年12月7号下午3:00

报告地点：5楼数研中心

44118太阳成城集团

2020.12.3

报告人简介：李方, 浙江大学教授, 博士生导师，高等数学研究所所长, 中国数学会理事. 2000年至今已培养出22位博士生，有的已成为国内有一定学术影响的青年学者。在Adv. Math., J. Algebra等国内外重要期刊杂志上发表论文130余篇. 先后主持国家自然科学基金六项和浙江省自然科学基金重大和重点项目各一项。曾获浙江省高校科技进步一等奖等奖项，是国家教育部新世纪人才和浙江省151人才入选者。

报告摘要：Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the second quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its secondly quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses dual quantum cluster algebras such that their second quantization is essentially the same.

As an example, we give the secondly quantized cluster algebra A_{p,q}(SL(2)) of Fun_{\C}(SL_{q}(2)) and show that it is a non-trivial second quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. Furthermore, we obtain a class of quantum cluster algebras with coefficients which possess a non-trivial second quantization. Its one special kind is quantum cluster algebras with almost principal coefficients with an additional condition.

Finally, we prove that the compatible Poisson structures of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the second quantization of a quantum cluster algebra without coefficients is in fact trivial.