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

ASSEM, IBRAHIM; REDONDO, MARIA JULIA

MANUSCRIPTA MATHEMATICA

SPRINGER

Lugar: Berlin; Año: 2009 vol. 128 p. 373 - 388

0025-2611

This paper studies the first Hochschild cohomology group of a Schurian cluster-tilted algebra. Throughout this review, k will be an algebraically closed field and A will be a basic connected finite-dimensional algebra over k. Let T be a tilting module over A, let C denote the tilted algebra End_A(T), and let B denote the algebra C x Ext_C^2(DC,C), where DC denotes the dual module Hom_k(C,k). Then B is also a basic connected finite-dimensional algebra. Furthermore, if A is hereditary, then B is a so-called cluster-tilted algebra, and every cluster-tilted algebra is of this form. Since B is a basic connected finite-dimensional k-algebra, there exists a quiver Q and an admissible ideal I of kQ such that B = kQ/I. For vertices x,y in Q, let kQ(x,y) denote the k-vector space of all linear combinations of paths from x to y, and set I(x,y) = kQ(x,y) cap I. Then B is said to be Schurian if dim_k kQ(x,y)/I(x,y) < 1 for all x,y (this does not depend on the representation B cong kQ/I). Lemma 2.2 of the paper proves that if B is a representation-finite cluster-tilted k-algebra, then B is Schurian. Let P be a quiver and let J be an admissible ideal of kP such that C = kP/J. Then one can assume that P has the same vertices as Q, and that the arrows P_1 of P form a subset of the arrows Q_1 of Q. An equivalence relation is set up on Q_1 P_1, and the number of equivalence classes is denoted n_{B,C}. The main result of this paper is that there is a short exact sequence 0 -> k^{n_{B,C}} -> {HH}^1(B) ->  {HH}^1(C) -> 0. Applications of this theorem are considered. The authors conjecture that for a cluster-tilted algebra B, one has HH^1(B)=0 if and only if B is hereditary with ordinary quiver a tree. This paper proves this conjecture in the case in which B is Schurian. Also, one can define the fundamental group pi_1(Q,I). In general this is not an invariant of B. However, an application of the main theorem proves that cluster-tilted algebras are freely connected, that is, pi_1(Q,I) is free for all presentations B = kQ/I. Finally, HH^1(B) is studied when B is cluster-tilted; in this case, B is not necessarily Schurian.