scholarly journals On (co)homology of Frobenius Poisson algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 371-386 ◽  
Author(s):  
Can Zhu ◽  
Fred Van Oystaeyen ◽  
Yinhuo Zhang
Keyword(s):  
Bilinear Form ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Poisson Algebra ◽  
Hochschild Homology ◽  
Poisson Algebras ◽  
Poisson Homology ◽  
Poisson Cohomology ◽  
Frobenius Algebras

AbstractIn this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.

Cohomology structure for a Poisson algebra: I

2015 ◽  
Vol 15 (02) ◽  
pp. 1650034 ◽  
Author(s):  
Yan-Hong Bao ◽  
Yu Ye
Keyword(s):  
Spectral Sequence ◽  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Algebra I ◽  
Poisson Algebra ◽  
Cohomology Groups ◽  
Lie Algebra Cohomology ◽  
Poisson Cohomology

We introduce quasi-Poisson cohomology groups for a Poisson algebra, which can be computed by its quasi-Poisson complex. Moreover, there exists a Grothendieck spectral sequence relating quasi-Poisson cohomology to Hochschild cohomology and Lie algebra cohomology.

scholarly journals Residues and homology for pseudodifferential operators on foliations

2004 ◽  
Vol 94 (1) ◽  
pp. 75 ◽  
Author(s):  
M.-T. Benameur ◽  
V. Nistor
Keyword(s):  
Diophantine Approximation ◽  
Pseudodifferential Operators ◽  
Hochschild Homology ◽  
Foliated Manifold ◽  
Homology Groups ◽  
General Properties ◽  
Poisson Homology ◽  
Foliated Manifolds

We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras of pseudodifferential operators along the leaves of a foliation.

scholarly journals Quantization of a Poisson algebra and polynomials associated to links

1990 ◽  
Vol 120 ◽  
pp. 113-127 ◽  
Author(s):  
Tetsuya Ozawa
Keyword(s):  
Riemann Surface ◽  
Lie Algebra ◽  
Poisson Algebra ◽  
Homotopy Classes ◽  
Poisson Algebras

A formal quantization of Poisson algebras was discussed by several authors (see for instance Drinfel’d [D]). A formal Lie algebra generated by homotopy classes of loops on a Riemann surface ∑ was obtained by W. Goldman in [G], and its Poisson algebra was quantized, in the sense of Drinfel’d, by Turaev in [T].

scholarly journals COHOMOLOGY OF FROBENIUS ALGEBRAS AND THE YANG-BAXTER EQUATION

2008 ◽  
Vol 10 (supp01) ◽  
pp. 791-814 ◽  
Author(s):  
J. SCOTT CARTER ◽  
ALISSA S. CRANS ◽  
MOHAMED ELHAMDADI ◽  
ENVER KARADAYI ◽  
MASAHICO SAITO
Keyword(s):  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Cohomology Theory ◽  
Baxter Equation ◽  
Low Dimensions ◽  
Frobenius Algebras

A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions, in analogy with Hochschild cohomology of bialgebras, based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation, using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.

scholarly journals Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit

2012 ◽  
Vol 149 (4) ◽  
pp. 658-678 ◽  
Author(s):  
Julien Bichon
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Bilinear Form ◽  
Hopf Algebras ◽  
Betti Numbers ◽  
Free Resolution ◽  
Hochschild Homology ◽  
Invertible Matrix ◽  
Tensor Categories ◽  
Orthogonal Case

AbstractWe show that if$A$and$H$are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of$A$to the same kind of resolution for the counit of$H$, exhibiting in this way strong links between the Hochschild homologies of$A$and$H$. This enables us to obtain a finite free resolution of the counit of$\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix$E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case$E=I_n$. It follows that$\mathcal {B}(E)$is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when$E$is an antisymmetric matrix, the$L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of$\mathcal {B}(E)$in the cosemisimple case.

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