From the Potential to the First Hochschild Cohomology Group of a Cluster Tilted Algebra
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From the Potential to the First Hochschild Cohomology Group of a Cluster Tilted Algebra Ibrahim Assem1 · Juan Carlos Bustamante2 · Sonia Trepode3 · Yadira Valdivieso4 Dedicated to Jos´e Antonio de la Pe˜na for his 60th birthday Received: 21 May 2019 / Accepted: 28 July 2020 / © The Author(s) 2020
Abstract The objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential. Keywords Hochschild cohomology · Cluster tilted algebras · Triangulated surfaces Mathematics Subject Classification (2010) MSC 16E40 · MSC 13F60 · MSC 16G20
Presented by: Christof Geiss Yadira Valdivieso
[email protected] Ibrahim Assem [email protected] Juan Carlos Bustamante [email protected] Sonia Trepode [email protected] 1
D´epartement de Math´ematiques, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, Canada
2
Departament of Mathematics, Champlain College, Sherbrooke, Qu´ebec, Canada
3
CEMIM, FCEYN, Universidad Nacional de Mar del Plata. CONICET, Provincia de Buenos Aires, Argentina
4
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
I. Assem et al.
Introduction In this paper, we present a concrete computation of the first Hochschild cohomology group of cyclically oriented and tame cluster tilted algebras. Cluster tilted algebras were introduced in [21] and independently in [22] for type A, as an application of the categorification of the cluster algebras of Fomin and Zelevinsky, see [19]. Cyclically oriented cluster tilted algebras were defined in [16] where it is shown to be the largest known class of cluster tilted algebras for which a system of top relations, called minimal relations in [20], is known. Our motivation is twofold. First, it can be argued that any (co)-homology theory has for object to detect and even compute cycles. In cluster tilted algebras, there are cycles which naturally occur: indeed, any cluster tilted algebra can be represented as the jacobian algebra of a quiver with potential, the latter being a linear combination of cycles in its quiver [29]. We are interested here in the relation between the cycles appearing in the potential and the first Hochschild cohomology group of the given cluster tilted algebra. Our second motivation, more ad hoc, comes from the very simple formula given in [10, Theorem 1.2], for a representation-finite cluster tilted algebra, allowing to read the dimension of the first Hochschild cohomology space directly in the ordinary quiver of the algebra. It is natural to ask for which class of (cluster tilted) algebras does this dimension depend only on the quiver. Because representation-finite cluster tilted algebras are cyclically oriented, the latter class is a natural candidate. However, cyclically oriented cluster tilted algebras are generally representation-infinite, they may be tame or wild. For tame, not necessarily cyclically oriented cluster tilted algeb
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