报告时间：3月29日 9:00
报 告 人：Alexander V.Mikhailov（英国利兹大学数学学院教授）
报告地点：包玉书9号楼113
报告题目：Commutative Poisson algebras from deformations of noncommutative algebras
报告摘要：By a well-known procedure, usually referred to as ``taking the classical limit'', quantum systems become classical systems, equipped with a Hamiltonian stucture (symplectic or Poisson). From the deformation quantisation theory we know that a formal deformation of a commutative algebra 𝒜 leads to a Poisson bracket on 𝒜 and that the classical limit of a derivation on the deformation leads to a Hamiltonian derivation on 𝒜 defined by the Poisson bracket. In this talk I present a generalisation of it for formal deformations of an arbitrary noncommutative algebra $\cA$ [4]. The deformation leads in this case to a commutative Poisson algebra structure on Π(𝒜):=Z(𝒜)⨉ (𝒜/Z(𝒜)) and to the structure of a Π(𝒜)-Poisson module on 𝒜, where Z(𝒜) denotes the centre of 𝒜. The limiting derivations are then still derivations of 𝒜, but with the Hamiltonians belong to Π(𝒜), rather than to 𝒜. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantised Grassmann algebra.
This talk is based on a joint work with Pol Vanhaecke [4].
References
[1] A. V. Mikhailov. Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020.
[2] S. Carpentier, A. V. Mikhailov and J. P. Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.
[3] S. Carpentier, A. V. Mikhailov, and J. P. Wang. Hamiltonians for the quantised Volterra hierarchy. arXiv:2312.12077, 2023. (Submitted to Nonlinearity)
[4] A. V. Mikhailov and P. Vanhaecke. Commutative Poisson algebras from deformations of noncommutative algebras. arXiv:2402.16191v2, 2024. (Submitted to CMP)
报告人简介：Alexander V.Mikhailov，英国利兹大学数学学院教授。可积系统的研究活动，特别是可积系统、非对易可积系统分类和量化问题。1978年在朗道理论物理研究所获得理论和数学物理博士学位，1987年全博士学位。剑桥大学克莱尔·霍尔学院终身院士。组织了多个会议和研讨会，并在国际期刊上发表了100多篇论文，引用次数超过3400次。
