Hochschild cohomology of relation extension algebras

Abstract

Abstract. Let B be the split extension of a finite dimensional algebra C by a C-C-bimodule E. We define a morphism of associative graded algebras ϕ * : HH * (B) → HH * (C) from the Hochschild cohomology of B to that of C, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler.In the case of a trivial extension B = C ⋉E, we give necessary and sufficient conditions for each ϕ n to be surjective. We prove the surjectivity of ϕ 1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of ϕ 1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.

Derivations and the First Cohomology Group of Trivial Extension Algebras

“…Obviously, this homomorphism is in fact the identity when the center Z(A) of A has a symmetric action on M. Thus, the following result is a generalization of [2,Theorem 4.4] (compare it also with [12,Theorem 2.5]).…”

“…The main result in this section (Theorem 4.2) relates the restricted first cohomology groups with the first cohomology group of the base ring and the quotient group of the group of all bimodule homomorphisms by the subgroup of all central inner bimodule homomorphisms. This leads to, Corollary 4.3, a generalization of both the classical result [7,Theorem 5.5] and the recent result [2,Theorem 4.4] which uses a purely homological argument (also, compare it with [12,Theorem 2.5]). As a consequence we get a characterization of trivial extension algebras on which every derivation is inner (see Corollaries 4.8 and 4.9).…”

“…can be strict (see assertion (2) in Example 3.10). However, in the next result (Theorem 3.8), we show that, when l.Ann A (M) ∩ r.Ann A (M) = 0, the two groups Innbi A (M) and InnBi A (M) coincide.…”

Representation theory of partial relation extensions

“…The Jacobian algebra J( Q, W ) is the one given by the quiver Q bound by all partial cyclic derivatives ∂ β W of the Keller potential W with respect to each arrow β ∈ Q 1 . Then the relation extension C is isomorphic to J( Q, W )/J where J is the square of the ideal of J( Q, W ) generated by the new arrows, see [5,Lemma 5.2]. If, in particular, C is tilted, so that C is cluster tilted, then C ≃ J( Q, W ), see for instance [18].…”

Hochschild cohomology of partial relation extensions

“…These are finite-dimensional algebras which are endomorphism algebras of tilting objects in the cluster category [10]. This class of algebras was much investigated, see for example [1,2,7,8,11,12,15,16], and [5,6,4] for results on their Hochschild cohomology. Among the main results is that every cluster-tilted algebra can be written as trivial extension of a tilted algebra by a bimodule called the relation bimodule [1].…”