[Paper Review] The derived non-commutative Poisson bracket on Koszul Calabi-Yau algebras

Xiaojun Chen, Alimjon Eshmatov+2 more|arXiv (Cornell University)|Apr 11, 2015

Algebraic structures and combinatorial models24 references3 citations

TL;DR

This paper establishes a derived non-commutative Poisson structure on Koszul and N-Koszul Calabi-Yau algebras, showing it induces a graded Lie algebra structure on cyclic homology and a Lie module structure on Hochschild homology. The Connes long exact sequence is shown to be a sequence of Lie modules, and the Leibniz-Loday bracket maps naturally to the Gerstenhaber bracket on the Hochschild cohomology of the Koszul dual algebra, providing a deep link between non-commutative Poisson geometry and homological algebra.

ABSTRACT

Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie algebra structure on the cyclic homology of $A$; moreover, we show that the Hochschild homology of $A$ is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz-Loday bracket associated to the derived non-commutative Poisson structure on $A$ is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of $A$ itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.

Motivation & Objective

• To extend the notion of non-commutative Poisson structures to the derived setting for Koszul and N-Koszul Calabi-Yau algebras.
• To show that the derived non-commutative Poisson structure induces a graded Lie algebra structure on the cyclic homology of such algebras.
• To establish that Hochschild homology is a Lie module over cyclic homology, with the Connes long exact sequence being a sequence of Lie modules.
• To demonstrate that the Leibniz-Loday bracket on the derived Poisson structure maps naturally to the Gerstenhaber bracket on the Hochschild cohomology of the Koszul dual algebra.
• To provide explicit examples, including the polynomial algebra, and clarify relations with other brackets in the literature.

Proposed method

• The authors use the cobar construction of a cyclic coalgebra to obtain a cofibrant resolution of the algebra, enabling the definition of a derived non-commutative Poisson structure.
• They define a derived trace map from cyclic homology to the homology of derived representation schemes, which preserves Lie structures.
• The construction relies on the isomorphism between the bimodule of one-forms and differential forms via the BV operator and volume form.
• The Gerstenhaber bracket on the Koszul dual algebra is shown to be isomorphic to the image of the Leibniz-Loday bracket under the derived trace map.
• The paper uses the isomorphism between polyvector fields and differential forms via the volume form to express the Lie bracket on cyclic homology in terms of contraction and exterior derivative.
• Explicit computations are performed for the polynomial algebra using the volume form and the Hodge star isomorphism to relate forms to polyvector fields.

Experimental results

Research questions

• RQ1 How can a derived non-commutative Poisson structure be defined on Koszul Calabi-Yau algebras, and what are its homological properties?
• RQ2 Does the derived non-commutative Poisson structure induce a graded Lie algebra structure on the cyclic homology of the algebra?
• RQ3 Is the Hochschild homology of a Koszul Calabi-Yau algebra a Lie module over its cyclic homology, and how does this relate to the Connes long exact sequence?
• RQ4 How is the Leibniz-Loday bracket on the derived Poisson structure related to the Gerstenhaber bracket on the Hochschild cohomology of the Koszul dual algebra?
• RQ5 What is the explicit form of the Lie bracket on cyclic homology for the polynomial algebra, and how does it relate to differential forms and contraction?

Key findings

• The derived non-commutative Poisson structure on a Koszul Calabi-Yau algebra induces a graded Lie algebra structure on its cyclic homology.
• The Hochschild homology of the algebra is a Lie module over the cyclic homology, and the Connes long exact sequence is a sequence of Lie modules.
• The Leibniz-Loday bracket on the derived Poisson structure maps naturally to the Gerstenhaber bracket on the Hochschild cohomology of the Koszul dual algebra.
• For the polynomial algebra Sym(V), the Lie bracket on cyclic homology is given by $\{α,\u03b2\} = (-1)^{(m-|\u03b1|-1)(m-|\u03b2|)} \iota_{\eta} d\alpha$, where $\eta = \Psi^{-1}(d\beta)$.
• The bracket on cyclic homology does not satisfy the Jacobi identity, so it does not define a Lie algebra on $\mathrm{HH}_\bullet(A)$, but only a Lie module structure.
• In the case of the polynomial algebra, the cyclic dual coalgebra admits a unique cyclic form of degree $-m$, determined by its value on the top-degree monomial.