What is a double star-product?

par Nikita Safonkin (Leipzig)

vendredi 20 juin 2025, 14:00 → 15:00 Europe/Paris

112 (Bat. Braconnier)

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: they induce Poisson
structures on the space of N-dimensional representations of an
associative algebra A  for any N. The problem of deformation
quantization of double Poisson brackets was raised by D. Calaque in
2010, and had remained open since then.

 In the talk, I plan to present a possible answer to the question in
the title. Namely, I will discuss a structure on an associative
algebra 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. If
time permits, I will also discuss a way to invert the
Kontsevich-Rosenberg principle by introducing the notion of a double
algebra over an arbitrary operad. The talk is based on
arXiv:2506.00699.