The equality of Hochschild cohomology group and module cohomology group for semigroup algebras

‎Let $S$ be a commutative inverse semigroup with idempotent set $E$‎. ‎In this paper‎, ‎we show that for every $n\in \mathbb{N}_0$‎, ‎$n$-th Hochschild cohomology group of semigroup algebra $\ell^1(S)$ with coefficients in $\ell^\infty(S)$ and its $n$-th $\ell^1(E)$-module cohomology group‎, ‎are equal‎. ‎Indeed‎, ‎we prove that‎ ‎\[ \HH^{n}(\ell^1(S),\ell^\infty(S))=\HH^{n}_{\ell^1(E)}(\ell^1(S),\ell^\infty(S)),\] for all $n\geq 0$‎.

Hochschild cohomology related to graded down-up algebras with weights (1,n)

Let [Formula: see text] be a graded down-up algebra with [Formula: see text] and [Formula: see text], and let [Formula: see text] be the Beilinson algebra of [Formula: see text]. If [Formula: see text], then a description of the Hochschild cohomology group of [Formula: see text] is known. In this paper, we calculate the Hochschild cohomology group of [Formula: see text] for the case [Formula: see text]. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of [Formula: see text] is different depending on whether (10) [Formula: see text] [Formula: see text] is zero or not. Moreover, it turns out that there is a difference between the cases [Formula: see text] and [Formula: see text] in the context of Grothendieck groups.

On a Lie Algebraic Approach to Abelian Extensions of Associative Algebras

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.

Oriented Algebras and the Hochschild Cohomology Group

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.

On Simple Connectedness of Minimal Representation-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.