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

Abstract

Given a finite-dimensional monomial algebra A we consider the trivial extension T A 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 HH 1 (T A). From these a formula for the dimension of HH 1 (T A) can be derived.

On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras

“…Then the local configurations in Figure 3 (2.b), and Figures 4,5,6,7,8,9,10 and 11 give rise to a basis of HH 1 (Λ).…”

“…Alt A (DA) of a gentle algebra A. We use the notation and results of [7] to find the following basis of Alt A (DA). (1), ψ (e,p) and ψ (p,p) with p as in (2) is a basis of Alt A (DA).…”

“…The aforementioned correspondence allows us to use results by Cibils, Marcos, Redondo and Solotar [6] and Cibils, Redondo and Saorín [7] that provide an explicit decomposition of the first cohomology space of a trivial extension algebra A in terms of the centre of the algebra, its Hochschild cohomology in degree 1, its Hochschild homology in degree 1 and another space denoted Alt A (DA) that we will recall in Section 3.4. We will also use the description of the bracket given by Strametz [33].…”

“…and that, in particular, we have HH 1 (T A) = 0, since Z(A) is never zero. In [7], the authors provide the dimensions of HH 1 (A), HH 1 (A) and Alt A (DA) in terms of the quiver of A.…”

The first Hochschild (co)homology when adding arrows to a bound quiver algebra

“…For a bound quiver algebra B = kQ/I, given some hypotheses on I, J. A. de la Peña and M. Saorín in [14] obtained formulas computing the dimension of HH 1 (B), see also [9,10,12,20]. For several families of algebras, results concerning the first cohomology vector space are given for instance in [2,3,4,25,26,27,31,32].…”