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 On a spectral sequence for the cohomology of a nilpotent Lie algebra

2014 ◽  
Vol 14 (01) ◽  
pp. 1450078
Author(s):  
Viviana del Barco
Keyword(s):  
Spectral Sequence ◽  
Lie Algebra ◽  
Cohomology Groups ◽  
Nilpotent Lie Algebra ◽  
Lie Algebra Cohomology ◽  
Direct Sum ◽  
Differential Complex ◽  
Dimension One

Given a nilpotent Lie algebra 𝔫 we construct a spectral sequence which is derived from a filtration of its Chevalley–Eilenberg differential complex and converges to the Lie algebra cohomology of 𝔫. The limit of this spectral sequence gives a grading for the Lie algebra cohomology, except for the cohomology groups of degree 0, 1, dim 𝔫 - 1 and dim 𝔫 as we shall prove. We describe the spectral sequence associated to a nilpotent Lie algebra which is a direct sum of two ideals, one of them of dimension one, in terms of the spectral sequence of the co-dimension one ideal. Also, we compute the spectral sequence corresponding to each real nilpotent Lie algebra of dimension less than or equal to six.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

2017 ◽  
pp. 1-31
Author(s):  
María Julia Redondo ◽  
Lucrecia Román
Keyword(s):  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Efficient Tool ◽  
Simple Description ◽  
Monomial Algebra ◽  
Cup Product ◽  
Lie Bracket ◽  
Projective Resolutions ◽  
Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

2017 ◽  
pp. 1-31
Author(s):  
María Julia Redondo ◽  
Lucrecia Román
Keyword(s):  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Efficient Tool ◽  
Simple Description ◽  
Monomial Algebra ◽  
Cup Product ◽  
Lie Bracket ◽  
Projective Resolutions ◽  
Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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.

scholarly journals THE VAN EST SPECTRAL SEQUENCE FOR HOPF ALGEBRAS

2004 ◽  
Vol 01 (01n02) ◽  
pp. 33-48 ◽  
Author(s):  
E. J. BEGGS ◽  
TOMASZ BRZEZIŃSKI
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Cohomology Group ◽  
Spectral Sequences ◽  
De Rham Cohomology ◽  
Lie Algebra Cohomology ◽  
De Rham ◽  
Definition Of

Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduced. A definition of Hopf–Lie algebra cohomology is also given.

Automorphism Group of a Special Type Lie Algebra I

2010 ◽  
Vol 17 (spec01) ◽  
pp. 815-828
Author(s):  
Seul Hee Choi ◽  
Ki-Bong Nam
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Algebra I

In an earlier paper, we defined a degree stable Lie algebra, and determined the Lie algebra automorphism group AutLie(S+(2)) of the Lie algebra S+(2). In this paper, we determine the Lie algebra automorphism group AutLie(S(1,0,2)) of the Lie algebra S(1,0,2).

scholarly journals Loop Schrödinger–Virasoro Lie conformal algebra

2016 ◽  
Vol 27 (06) ◽  
pp. 1650057 ◽  
Author(s):  
Haibo Chen ◽  
Jianzhi Han ◽  
Yucai Su ◽  
Ying Xu
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Cohomology Groups ◽  
Conformal Algebras ◽  
Rank One ◽  
Intermediate Series

In this paper, we introduce two kinds of Lie conformal algebras, associated with the loop Schrödinger–Virasoro Lie algebra and the extended loop Schrödinger–Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and [Formula: see text]-graded free intermediate series modules over these two conformal algebras are also classified in the present paper.

scholarly journals A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

2013 ◽  
Vol 112 (2) ◽  
pp. 182 ◽  
Author(s):  
Shoham Shamir
Keyword(s):  
Spectral Sequence ◽  
Loop Space ◽  
Hochschild Cohomology ◽  
Derived Category ◽  
Orientable Manifold ◽  
Support Varieties ◽  
Mod P

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

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