What is a double star-product?

TL;DR

This paper addresses the open problem of deformation quantization of double Poisson brackets by introducing a structure on associative algebras that induces star-products on their representation spaces, aligning with noncommutative geometry principles.

Contribution

It presents a new structure on associative algebras that induces star-products on representation spaces and introduces the concept of double algebra over an operad, with explicit examples and a formality theorem.

Findings

Provides an explicit example for free associative algebras.

Proves a double formality theorem in the case of free associative algebras.

Introduces a notion of double algebra over an operad that inverts the Kontsevich-Rosenberg principle.

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: they induce Poisson structures on the space of $N$-dimensional representations $\operatorname{Rep}_N(A)$ 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 this paper, we address this problem by answering the question in the title. We 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. We also provide an explicit example for $A=\Bbbk\langle x_1,\ldots,x_d\rangle$ and prove a double formality theorem in this case. Along the way, we invert the Kontsevich-Rosenberg principle by introducing a notion of double algebra over an arbitrary operad.