Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras
Yuri Berest, Xiaojun Chen, Farkhod Eshmatov, Ajay Ramadoss
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Source: Crossref
Published: Jan 1, 2012
DOI: 10.1090/conm/583/11570
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In this paper, we introduce and study the notion of a derived Poisson structure on an associative algebra A A . This structure is characterized by the property of being the weakest structure on A A that induces natural (graded commutative) Poisson structures on the derived moduli spaces of finite-dimensional representations of A A . A derived Poisson structure is represented by a graded (super) Lie algebra bracket on the cyclic homology H C ∙ ( A ) \mathrm {HC}_\bullet (A) and can be viewed as a higher homological extension of the H 0 H_0 -Poisson structure introduced by W. Crawley-Boevey (2011). In the second part of the paper, we construct a large class of examples of derived Poisson structures arising from finite-dimensional n n -cyclic coalgebras. These examples include linear duals of finite-dimensional n n -cyclic algebras which are n n -Calabi-Yau categories in the sense of Kontsevich and Soibelman (2009).
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