Deformation quantization of double Poisson algebras

Abstract: Double Poisson brackets, introduced by M. Van den Bergh in 2004, are noncommutative analogs of the usual Poisson brackets in the sense of the Kontsevich–Rosenberg principle: for any $N$, they induce Poisson structures on the space of $N$-dimensional representations $\mathrm{Rep}_N(A)$ of an associative algebra $A$. The problem of deformation quantization of double Poisson brackets was raised by D. Calaque in 2010 and has remained open since then. In the talk, I will discuss a solution to this problem and present a structure on $A$ that induces a star-product under the representation functor and therefore, according to the Kontsevich–Rosenberg principle, can be viewed as an analog of star-products in noncommutative geometry. I will also discuss an analog of the famous formality theorem of M.Kontsevich in this context.

The talk is based on arXiv:2506.00699.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( paper | slides | video )

---

BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

---