Modified Double Poisson Brackets

Arthamonov, Semeon

doi:10.48550/arxiv.1608.08287

Cited by 3 publications

(3 citation statements)

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“…The question on better understanding of the structure needed to produce Poisson bracket on representation spaces recently received much attention. It is discussed, apart from mentioned paper [18], for example, in [3,40]. Our work [16], [18] was also extended to the case when A ≠ A 0 , by means of the same correspondence given by formula (*), with signs added in [12].…”

Section: Double Poisson Bracket and The Maurer-cartan Equationmentioning

confidence: 97%

Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

Iyudu¹,

Kontsevich²

2020

Preprint

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We prove L ∞ -formality for the higher cyclic Hochschild complex HCH over free associative algebra or path algebra of a quiver. The HCH complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them ξδ-monomials. The Lie structure on this subcomplex is combinatorially described in terms of ξδ-monomials. This subcomplex and a basis of ξδ-monomials in combination with arguments from Gröbner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of ξδ-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.

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Section: Double Poisson Bracket and The Maurer-cartan Equationmentioning

confidence: 97%

Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

Iyudu¹,

Kontsevich²

2020

Preprint

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“…There exist modifications of the definition of double brackets. For example, we can relax the antisymmetry rule (2.1) as did Arthamonov [3], or Massuyeau and Turaev using Fox pairings on Hopf algebras [40]. We can also consider graded versions of double brackets [59,41].…”

Section: Basic Definitionsmentioning

confidence: 99%

Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems

Fairon

2020

Preprint

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Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the H 0 -Poisson structures of Crawley-Boevey. We derive from our results a representation theoretic description of action-angle duality for several classical integrable systems.

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“…This differs from the double Poisson bimodules from [8] which are maps A × M → (A ⊗ M ) ⊕ (M ⊗ A) where M is a fixed Abimodule. Another completely different point of view that was undertaken by Arthamonov [2], then extended by Goncharov and Gubarev [14], relies on a modification of the antisymmetry rule of a double bracket. We do not pursue these directions at all.…”

Section: Introductionmentioning

confidence: 99%

Around Van den Bergh's double brackets for different bimodule structures

Fairon¹,

McCulloch²

2022

Preprint

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A double Poisson bracket, in the sense of M. Van den Bergh, is an operation on an associative algebra A which induces a Poisson bracket on each representation space Rep(A, n) in an explicit way. In this note, we study the impact of changing the Leibniz rules underlying a double bracket. This change amounts to make a suitable choice of A-bimodule structure on A ⊗ A. In the most important cases, we describe how the choice of A-bimodule structure fixes an analogue to Jacobi identity, and we obtain induced Poisson brackets on representation spaces. The present theory also encodes a formalisation of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics.

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Modified Double Poisson Brackets

Cited by 3 publications

References 0 publications

Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems

Around Van den Bergh's double brackets for different bimodule structures

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