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The derived non-commutative Poisson bracket on Koszul Calabi–Yau algebras
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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.
European Mathematical Society - EMS - Publishing House GmbH
Title: The derived non-commutative Poisson bracket on Koszul Calabi–Yau algebras
Description:
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.
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