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 [15], 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].

Introduction

The notion of Calabi-Yau algebras was introduced by Ginzburg in 2007. It is a noncommutative generalization of affine Calabi-Yau varieties, and due to Van den Bergh, admits the so-called “noncommutative Poincaré duality”. In a joint paper with Berest and Ramadoss [1] we showed that for a Koszul Calabi-Yau algebra, say A, there is a version of noncommutative Poisson structure, called the shifted double Poisson structure, on its cofibrant resolution, and hence induces a shifted Poisson structure on the derived representation schemes of A, a notion introduced by Berest, Khachatryan and Ramadoss in [3]. Here “shifted” means there is a degree shifting, depending on the dimension of the Calabi-Yau algebra, on the Poisson bracket. Such a shifted Poisson structure was later further studied in [9] in much detail.

A natural question that arises now is, given a Koszul Calabi-Yau algebra, whether or not the noncommutative Poisson structure associated to it is noncommutative symplectic. To justify this question, let us remind the reader of a version of noncommutative symplectic structure introduced by Crawley-Boevey, Etingof and Ginzburg in [15], which they called the bi-symplectic structure in the paper. A bi-symplectic structure on an associative algebra 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 for 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 classical symplectic geometry to a much broader context, but also reveals many new features on a lot of geometric spaces, especially on 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 shifted symplectic structures 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 [22, 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 that A is also Koszul, and denote its Koszul dual coalgebra by $A^{\mathrm{\text{!‘}}}$. Let $\tilde{R}=\mathrm{\Omega }({\tilde{A}}^{\mathrm{\text{!‘}}})$ be the cobar construction of the co-unitalization ${\tilde{A}}^{\mathrm{\text{!‘}}}$ of $A^{\mathrm{\text{!‘}}}$. In this paper, we show that $\tilde{R}$, rather than $\mathrm{\Omega }(A^{\mathrm{\text{!‘}}})$ as studied in [1], [9], has a $(2-n)$-shifted bi-symplectic structure. Such a 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 ${\mathrm{Rep}}_V(\tilde{R})$, for all vector spaces V (see Theorem 4.6). By taking the corresponding trace maps we obtain a commutative diagram (see (38) for more details)

where the left hand side are the Hochschild cohomology and homology of A, with the isomorphism being Van den Bergh's noncommutative Poincaré duality, and the right hand side are the cohomology and homology of the $\mathrm{GL}(V)$-invariant complexes of vector fields and 1-forms on ${\mathrm{Rep}}_V(\tilde{R})$ respectively.

By Van den Bergh's result mentioned above, the shifted bi-symplectic structure induces a shifted double Poisson structure on $\tilde{R}$; therefore there is a shifted Poisson structure on ${\mathrm{Rep}}_V(\tilde{R})$. In this paper, we will study the deformation quantization of such a shifted Poisson structure, and show that it comes from the quantization of the “functions” ${\tilde{R}}_{♮}:=\tilde{R}/[\tilde{R},\tilde{R}]$ of $\tilde{R}$ (see Theorem 7.5). By Koszul duality, the homology of ${\tilde{R}}_{♮}$ minus the unit is isomorphic to the cyclic homology of A, and thus we obtain a quantization of the latter as well. This construction is inspired by the quantization of quiver representations.

This paper is organized as follows. In §2 we collect some basics of noncommutative geometry, such as the noncommutative 1-forms and vector fields, and some operations, such as the contraction and Lie derivative between them, then we recall the definition of bi-symplectic structure introduced by Crawley-Boevey et al. in [15].

In §3 we recall the notion of Koszul algebras and some of their basic properties. Let A be a Koszul algebra over a field k. We give explicit formulas for the double derivations and 1-forms of $\tilde{R}$. The commutator quotient spaces of them are identified with the Hochschild cohomology and homology of A respectively.

In §4 we first recall the definition of Calabi-Yau algebras, and then show that if the Calabi-Yau algebra, say A, is Koszul, then its noncommutative volume form gives a shifted bi-symplectic structure on $\tilde{R}$, where $\tilde{R}$ is given as above.

In §5 we first recall the DG representation schemes of a DG algebra, which was introduced by Berest et al. in [3]. Following the works [3], [15], we see that if a DG algebra admits a shifted bi-symplectic structure, then its DG representation schemes have a shifted symplectic structure. We then apply it to the Koszul Calabi-Yau algebra case.

In §6 and §7 we study the shifted Poisson structure on the representation schemes of $\tilde{R}$ and their quantization. We show that such quantizations are induced by the quantization of ${\tilde{R}}_{♮}$ as a Lie bialgebra. This is completely analogous to the papers of Schedler [32] and Ginzburg-Schedler [19], where the quantization of the representation spaces of doubled quivers is constructed, which is compatible with the quantization of the necklace Lie bialgebra of the quivers via the canonical trace map.

In §8, we briefly discuss some relationships of the current paper with the series of papers by Berest and his collaborators [1], [2], [3], [4], where the derived representation schemes of associative algebras were introduced and studied.

In the last section, §9, we give the two examples of Calabi-Yau algebras, namely, the 3- and 4-dimensional Sklyanin algebras, and study the corresponding shifted bi-symplectic structure in some detail.

This paper is a sequel to [1], [9], where the shifted noncommutative Poisson structure associated to Calabi-Yau algebras was studied; however, in the current paper we try to be as self-contained as possible. When we were in the final stage of the paper, we learned that Y.T. Lam in his thesis [24] as well as W.-K. Yeung in his paper [41] have obtained several results which are similar to ours. Nevertheless, the main results and methods of theirs and ours are quite different.

Convention 1.1

Throughout the paper, k is a field of characteristic zero, though in a lot of cases it need not necessarily to be so. All morphisms and tensors are over k unless otherwise specified. DG algebras (respectively DG coalgebras) are unital and augmented (respectively co-unital and co-augmented), with the degree of the differential being −1. For a chain complex, its homology is denoted by ${\mathrm{H}}_{•}(-)$, and its cohomology is given by ${\mathrm{H}}^{•}(-):={\mathrm{H}}_{-•}(-)$.