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 Solutions of associative Yang–Baxter equation and D−equation in low dimensions and associated Frobenius algebras and Connes cocycles

2018 ◽  
Vol 17 (01) ◽  
pp. 1850010
Author(s):  
Mahouton Norbert Hounkonnou ◽  
Gbêvèwou Damien Houndedji
Keyword(s):  
Baxter Equation ◽  
Low Dimensions ◽  
Frobenius Algebras ◽  
Dendriform Algebras

In this work, we compute solutions of the Yang–Baxter associative equation in dimensions one and two. For these solutions, we describe the double constructions of the associated Frobenius algebras, following Bai’s definitions. Besides, we determine related compatible dendriform algebras and solutions of their [Formula: see text]equations. Finally, using symmetric solutions of these equations, we build the double constructions of related Connes cocycles.

scholarly journals Ternary and n-ary f-distributive structures

2018 ◽  
Vol 16 (1) ◽  
pp. 32-45 ◽  
Author(s):  
Indu R. U. Churchill ◽  
M. Elhamdadi ◽  
M. Green ◽  
A. Makhlouf
Keyword(s):  
Cohomology Theory ◽  
Extension Theory ◽  
Low Dimensions

AbstractWe introduce and study ternary f-distributive structures, Ternary f-quandles and more generally their higher n-ary analogues. A classification of ternary f-quandles is provided in low dimensions. Moreover, we study extension theory and introduce a cohomology theory for ternary, and more generally n-ary, f-quandles. Furthermore, we give some computational examples.

scholarly journals The tangent complex and Hochschild cohomology of -rings

2012 ◽  
Vol 149 (3) ◽  
pp. 430-480 ◽  
Author(s):  
John Francis
Keyword(s):  
Lie Algebras ◽  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Homology Theory ◽  
Configuration Spaces ◽  
Algebra Structure ◽  
Koszul Duality ◽  
Derived Algebraic Geometry ◽  
Factorization Homology ◽  
Infinitesimal Automorphisms

AbstractIn this work, we study the deformation theory of${\mathcal {E}}_n$-rings and the${\mathcal {E}}_n$analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence$A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the${\mathcal {E}}_n$-tangent complex and${\mathcal {E}}_n$-Hochschild cohomology of an${\mathcal {E}}_n$-ring$A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups,$B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here${\mathfrak B}^n\!A$is an enriched$(\infty ,n)$-category constructed from$A$, and${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of${\mathfrak B}^n\!A$. These groups are associated to moduli problems in${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex$T_A[-n]$is a nonunital${\mathcal {E}}_{n+1}$-algebra. The${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known${\mathcal {E}}_{n+1}$-algebra structure on${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed$n$-manifolds with coefficients given by${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

scholarly journals Hochschild cohomology of Frobenius algebras

2003 ◽  
Vol 132 (5) ◽  
pp. 1241-1250 ◽  
Author(s):  
Jorge A. Guccione ◽  
Juan J. Guccione
Keyword(s):  
Hochschild Cohomology ◽  
Frobenius Algebras

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.

HOCHSCHILD COHOMOLOGY OF POLYNOMIAL ALGEBRAS

2001 ◽  
Vol 03 (03) ◽  
pp. 393-402 ◽  
Author(s):  
MICHAEL PENKAVA ◽  
POL VANHAECKE
Keyword(s):  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Differential Operators ◽  
Polynomial Algebra ◽  
Cohomology Groups ◽  
Polynomial Algebras ◽  
Arbitrary Polynomial

In this paper we investigate the Hochschild cohomology groups H2(A) and H3(A) for an arbitrary polynomial algebra A. We also show that the corresponding cohomology groups which are built from differential operators inject in H2(A) and H3(A) and we give an application to deformation theory.

scholarly journals G-Algebra Structure on the Higher Order Hochschild Cohomology HS2∗(A,A)

2022 ◽  
Vol 29 (01) ◽  
pp. 113-124
Author(s):  
Samuel Carolus ◽  
Mihai D. Staic
Keyword(s):  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Higher Order ◽  
Algebra Structure

We present a deformation theory associated to the higher Hochschild cohomology [Formula: see text]. We also study a [Formula: see text]-algebra structure associated to this deformation theory.

scholarly journals Hochschild cohomology, the characteristic morphism and derived deformations

2008 ◽  
Vol 144 (6) ◽  
pp. 1557-1580 ◽  
Author(s):  
Wendy Lowen
Keyword(s):  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Abelian Category ◽  
Derived Category ◽  
Missing Link ◽  
First Order ◽  
Order Deformation

AbstractA notion of Hochschild cohomology HH*(𝒜) of an abelian category 𝒜 was defined by Lowen and Van den Bergh (Adv. Math. 198 (2005), 172–221). They also showed the existence of a characteristic morphism χ from the Hochschild cohomology of 𝒜 into the graded centre ℨ*(Db(𝒜)) of the bounded derived category of 𝒜. An element c∈HH2(𝒜) corresponds to a first-order deformation 𝒜c of 𝒜 (Lowen and Van den Bergh, Trans. Amer. Math. Soc. 358 (2006), 5441–5483). The problem of deforming an object M∈Db(𝒜) to Db(𝒜c) was treated by Lowen (Comm. Algebra 33 (2005), 3195–3223). In this paper we show that the element χ(c)M∈Ext𝒜2(M,M) is precisely the obstruction to deforming M to Db(𝒜c). Hence, this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a ‘derived deformation theory’.

scholarly journals Computing Noncommutative Deformations of Presheaves and Sheaves of Modules

2010 ◽  
Vol 62 (3) ◽  
pp. 520-542 ◽  
Author(s):  
Eivind Eriksen
Keyword(s):  
Elliptic Curve ◽  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Forgetful Functor ◽  
Open Cover ◽  
Small Category ◽  
Obstruction Theory ◽  
Algebraic Structures ◽  
Coherent Sheaves

AbstractWe describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures and the noncommutative deformation theory of modules over algebras due to Laudal.In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases.In the last part of the paper, we describe a noncommutative deformation functor for quasi-coherent sheaves of modules on a ringed space (X,𝒜). We show that for any good A-affine open cover U of X, the forgetful functor QCoh𝒜 → PreSh(U,𝒜) induces an isomorphism of noncommutative deformation functors.Applications. We consider noncommutative deformations of quasi-coherent 𝒜-modules on X when (X,𝒜) = (X,𝒪X) is a scheme or (X,𝒜) = (X,𝒟) is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of X closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left 𝒟X-module 𝒟X when X is an elliptic curve as an example.

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