scholarly journals Oriented Algebras and the Hochschild Cohomology Group

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 237
Author(s):  
Ali Koam
Keyword(s):  
Associative Algebra ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Chain Complexes ◽  
Hochschild Cohomology Group ◽  
Equivariant Version

Koam and Pirashivili developed the equivariant version of Hochschild cohomology by mixing the standard chain complexes computing group with associative algebra cohomologies to obtain the bicomplex C ˜ G * ( A , X ). In this paper, we form a new bicomplex F ˘ G * ( A , X ) by deleting the first column and the first row and reindexing. We show that H ˘ G 1 ( A , X ) classifies the singular extensions of oriented algebras.

scholarly journals Algebra endomorphisms and derivations of some localized down-up algebras

2014 ◽  
Vol 14 (03) ◽  
pp. 1550034 ◽  
Author(s):  
Xin Tang
Keyword(s):  
Automorphism Group ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Root Of Unity ◽  
Hochschild Cohomology Group ◽  
Algebra Endomorphism

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.

scholarly journals THE FIRST COHOMOLOGY GROUP OF THE TRIVIAL EXTENSION OF A MONOMIAL ALGEBRA

2004 ◽  
Vol 03 (02) ◽  
pp. 143-159 ◽  
Author(s):  
CLAUDE CIBILS ◽  
MARÍA JULIA REDONDO ◽  
MANUEL SAORÍN
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Trivial Extension ◽  
Monomial Algebra ◽  
Finite Dimensional ◽  
Hochschild Cohomology Group ◽  
First Cohomology Group

Given a finite-dimensional monomial algebra A, we consider the trivial extension TA and provide formulae, depending on the characteristic of the field, for the dimensions of the summands HH1(A) and Alt (DA) of the first Hochschild cohomology group HH1(TA). From these a formula for the dimension of HH1(TA) can be derived.

On Simple Connectedness of Minimal Representation-Infinite Algebras

2015 ◽  
Vol 22 (04) ◽  
pp. 639-654
Author(s):  
Hailou Yao ◽  
Guoqiang Han
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Minimal Representation ◽  
Algebraically Closed Field ◽  
Hochschild Cohomology Group ◽  
Simple Connectedness ◽  
Simply Connected ◽  
Equivalent Conditions ◽  
Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

scholarly journals Derivations on Toeplitz Algebras

2014 ◽  
Vol 57 (2) ◽  
pp. 270-276 ◽  
Author(s):  
Michael Didas ◽  
Jörg Eschmeier
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Pseudoconvex Domain ◽  
Cohomology Group ◽  
Hankel Operator ◽  
Hochschild Cohomology ◽  
Toeplitz Algebra ◽  
Strictly Pseudoconvex Domain ◽  
Hochschild Cohomology Group ◽  
Closed Subalgebra

AbstractLet H2(Ω) be the Hardy space on a strictly pseudoconvex domain Ω ⊂ ℂn, and let A ⊂ L∞(∂Ω) denote the subalgebra of all L∞-functions ƒ with compact Hankel operator Hƒ. Given any closed subalgebra B ⊂ A containing C(Ω), we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra 𝒯(B) ⊂ B(H2(Ω). In particular, we show that every derivation on 𝒯(A) is inner. These results are new even for n = 1, where it follows that every derivation on T(H∞ +C) is inner, while there are non-inner derivations on T(H∞ + C(∂ℝn)) over the unit ball Bn in dimension n > 1.

On a Lie Algebraic Approach to Abelian Extensions of Associative Algebras

2020 ◽  
pp. 1-14
Author(s):  
Youjun Tan ◽  
Senrong Xu
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Algebraic Approach ◽  
Abelian Extension ◽  
Associative Algebras ◽  
Abelian Extensions ◽  
Hochschild Cohomology Group ◽  
Obstruction Class

Abstract By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

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