The First Cohomology Group of an Algebra with Coefficients in a Bimodule

The first Hochschild cohomology group of a schurian cluster-tilted algebra

“…We then also study the case where B is cluster-tilted of type Ã. In this case, B is not necessarily schurian, but it is gentle (see [3] or also [14]), hence we can apply the results of [16].…”

Coverings of laura algebras: The standard case

“…Then π 1 (Γ ) is free of rank 3. A straightforward computation gives dim HH 1 (A) = 7 (see also [16,Thm. 1]).…”

The First Cohomology Group of the Trivial Extension of a Monomial Algebra

“…Using [10][Theorem 5.5], the combination of the formulae in [11][Theorem 1 and Proposition 2] and our Theorems 2.18 (resp. 3.4) and 4.6 gives a precise formula for the dimension of HH 1 (Λ), when Λ = T A is the trivial extension of the monomial algebra A.…”

“…Although these groups are not easy to compute in general, some approaches have been successful when the algebra Λ is given by a quiver with relations. For instance, explicit formulae for the dimensions of HH i (Λ) in terms of those combinatorial data have been found in [5,6,8,11,12]. The first Hochschild cohomology group plays an important role in the representation theory of algebras since it is related to the separation properties of the vertices of the quiver of Λ, and to the notion of (strong) simple connectedness (see [2,3,13,17,20]).…”

“…As an immediate consequence we have that HH 1 (T A) never vanishes. The first and second summands have been studied in [11], when A is a monomial algebra, and explicit formulae for their dimensions are obtained. The Lie algebra structure of HH 1 (T A) with respect to this decomposition is described in [19].…”