标题： 什么是双星积？ 显示英文标题

标题： What is a double star-product?

摘要： 双泊松括号是由M. Van den Bergh在2004年引入的，它们在Kontsevich-Rosenberg原理的意义上是通常泊松括号的非交换类似物：对于任何$N$，它们在结合代数$A$的$N$维表示空间$\operatorname{Rep}_N(A)$上诱导泊松结构。 D. Calaque在2010年提出了双泊松括号的形变量子化问题，从那时起这个问题一直悬而未决。 在本文中，我们通过回答标题中的问题来解决这个问题。 我们提出了一个$A$上的结构，它在表示函子下诱导一个乘积，并因此根据Kontsevich-Rosenberg原理，可以看作是非交换几何中乘积的类似物。 我们还为$A=\Bbbk\langle x_1,\ldots,x_d\rangle$提供了一个显式例子，并在此情况下证明了一个双形式性定理。 在过程中，我们通过引入任意操作子上的双代数概念来反转Kontsevich-Rosenberg原理。 显示英文摘要

摘要： 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.