Alexandre Mikhailov (University of Leeds) – Commutative Poisson algebras from deformations of noncommutative algebras and non-Abelian Hamiltonian systems

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 $\mathcal{A}$ leads to a Poisson bracket on $\mathcal{A}$ and that the classical limit of a derivation on the deformation leads to a Hamiltonian derivation on $\mathcal{A}$ defined by the Poisson bracket. In this talk I present a generalisation of it for formal deformations of an arbitrary noncommutative associative algebra $\mathcal{A}$ [1]. I will show that a deformation leads to a commutative Poisson algebra structure on $\Pi(\mathcal{A}) := Z(\mathcal{A}) × (\mathcal{A}/Z(\mathcal{A}))$ and to the structure of a $\Pi(\mathcal{A})$-Poisson module on $\mathcal{A}$, where $Z(\mathcal{A})$ denotes the centre of $\mathcal{A}$. The limiting derivations are then still derivations of $\mathcal{A}$, but with the Hamiltonians belong to $\Pi(A)$, 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 Kontsevich integrable map, the quantum plane, the quantised Grassmann algebra and quantisations of the Volterra hierarchy [2, 3, 4].

This talk is based on a joint work with Pol Vanhaecke [1].

References

[1] Alexander V. Mikhailov and Pol Vanhaecke. Commutative Poisson algebras from deformations of noncommutative algebras. Lett. Math. Phys., 114(5), 1-51, 2024, arXiv:2402.16191v2.
[2] Alexander V. Mikhailov Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020, (arXiv:2009.01838), 2020).
[3] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022, (arXiv:2204.03095).
[4] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Hamiltonians for the quantised Volterra hierarchy. Nonlinearity, 37(9), 095033 2024, arXiv:2312.12077