Calabi-Yau algebras and the shifted noncommutative symplectic structure

Abstract

In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg [14], on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss [3], have a shifted symplectic structure in the sense of Pantev-Toën-Vaquié-Vezzosi [28].1 2 XIAOJUN CHEN AND FARKHOD ESHMATOV justify this question, let us remind a version of noncommutative symplectic structure introduced by Crawley-Boevey, Etingof and Ginzburg in [14], which they called the bi-symplectic structure in the paper. For an associative algebra, a bi-symplectic structure on it is a closed 2-form in its Karoubi-de Rham complex that induces an isomorphism between the space of noncommutative vector fields (more precisely, the space of double derivations) and the space of noncommutative 1-forms. In [38, Appendix], Van den Bergh showed that any bi-symplectic structure naturally gives rise to a double Poisson structure, which is completely analogous to the classical case. However, the reverse is in general not true. Nevertheless, it is still interesting to ask if this is true in the special case of Calabi-Yau algebras.The second motivation of the paper comes from the 2012 paper [28] of Pantev, Toën, Vaquié and Vezzosi, where they introduced the notion of shifted symplectic structure for derived stacks. This not only generalizes the classical symplectic geometry to a much broader context, but also reveals many new features on a lot of geometric spaces, especially on the various moduli spaces that mathematicians are now studying. As remarked by the authors, the shifted symplectic structure, if it exists, always comes from the Poincaré duality of the corresponding source spaces; they also outlined how to generalize the shifted symplectic structure to noncommutative spaces, such as Calabi-Yau categories.Note that Calabi-Yau algebras are highly related to Calabi-Yau categories. For example, a theorem of Keller (see [21, Lemma 4.1]) says that the bounded derived category of a Calabi-Yau algebra is a Calabi-Yau category. Applying the idea of [28], we would expect that the noncommutative Poincaré duality of a Calabi-Yau algebra shall also play a role in the corresponding shifted noncommutative symplectic structure if it exists.The main results of the current paper may be summarized as follows. Let A be a Calabi-Yau algebra of dimension n. Assume A is also Koszul, and denote its Koszul dual coalegbra by A ¡ . LetR = Ω(à ¡ ) be the cobar construction of the co-unitalizationà ¡ of A ¡ . In this paper, we show thatR, rather than Ω(A ¡ ) as studied in [1,9], has a (2 − n)-shifted bi-symplectic structure. Such bi-symplectic structure comes from the volume form of the noncommutative Poincaré duality of A, and naturally induces a (2 − n)-shifted symplectic structure on the DG representation schemes Rep V (R), for all vector spaces V (see Theorem 4.6). By taking the corresponding trace maps we obtain a commutative di...

Commutativity of Quantization and Reduction for Quiver Representations

“…Quantization of the necklace Lie algebra. In this subsection, we go over the quantization of the non-commutative Poisson structure on KQ, which is due to Schedler [18]; the DG algebras case has recently been studied by Chen and Eshmatov in [4].…”