Simply connected algebras and Hochschild cohomologies. (English) Zbl 0806.16012
Dlab, Vlastimil (ed.) et al., Representations of algebras. Proceedings of the sixth international conference on representations of algebras, Carleton University, Ottawa, Ontario, Canada, August 19-22, 1992. Providence, RI: American Mathematical Society. CMS Conf. Proc. 14, 431-447 (1993).
Let \(A\) be a basic finite dimensional algebra over an algebraically closed field \(K\). It is called triangular if its ordinary quiver \(Q\) has no oriented cycle, and simply connected if moreover, for any presentation \(A \cong KQ/I\) of \(A\) as a bound quiver algebra, the fundamental group \(\Pi_ 1(Q,I)\) is trivial [see the reviewer and the author: Proc. Lond. Math. Soc., III. Ser. 56, 417-450 (1988; Zbl 0617.16018)]. The theme of this paper is the relation between the simple connectedness of a triangular algebra \(A\), the vanishing of its Hochschild cohomology spaces \(H^ n(A)\) (with coefficients in the bimodule \({_ AA_ A}\)) and the separation property of R. Bautista, F. Larrión and L. Salmerón [J. Lond. Math. Soc., II. Ser. 27, 212-220 (1983; Zbl 0511.16022)].
The author shows that, if \(A\) satisfies the separation property, it is simply connected. Also, if \(A\) is triangular with \(H^ 1(A) = 0\), every source in \(Q\) is separating. He gives an example of a simply connected algebra \(A\) with \(H^ 1(A) \neq 0\).
He then defines an algebra to be strongly simply connected if every full convex subcategory is simply connected. He shows that a triangular algebra is strongly simply connected if and only if every full convex subcategory of \(A\) (or, equivalently, of \(A^{op}\)) satisfies the separation property, or if and only if every full convex subcategory \(C\) satisfies \(H^ 1(C) = 0\).
Special attention is given to the polynomial growth case. The author has indeed shown [in “Loop-finite algebras with the separation property”, in preparation] that a strongly simply connected algebra is of polynomial growth if and only if it is a multicoil algebra (hence its indecomposable modules can be described). Here, he gives a necessary and sufficient condition for a strongly simply connected algebra of polynomial growth to be rigid (that is, to have only trivial one-parameter deformations). In particular, all domestic strongly simply connected algebras are rigid.
The author also states some open problems. The results of this paper have already been applied in many other ones [see, for instance, the reviewer and J. A. de le Peña: “The fundamental groups of a triangular algebra”, submitted for publication].
For the entire collection see [Zbl 0786.00022].
The author shows that, if \(A\) satisfies the separation property, it is simply connected. Also, if \(A\) is triangular with \(H^ 1(A) = 0\), every source in \(Q\) is separating. He gives an example of a simply connected algebra \(A\) with \(H^ 1(A) \neq 0\).
He then defines an algebra to be strongly simply connected if every full convex subcategory is simply connected. He shows that a triangular algebra is strongly simply connected if and only if every full convex subcategory of \(A\) (or, equivalently, of \(A^{op}\)) satisfies the separation property, or if and only if every full convex subcategory \(C\) satisfies \(H^ 1(C) = 0\).
Special attention is given to the polynomial growth case. The author has indeed shown [in “Loop-finite algebras with the separation property”, in preparation] that a strongly simply connected algebra is of polynomial growth if and only if it is a multicoil algebra (hence its indecomposable modules can be described). Here, he gives a necessary and sufficient condition for a strongly simply connected algebra of polynomial growth to be rigid (that is, to have only trivial one-parameter deformations). In particular, all domestic strongly simply connected algebras are rigid.
The author also states some open problems. The results of this paper have already been applied in many other ones [see, for instance, the reviewer and J. A. de le Peña: “The fundamental groups of a triangular algebra”, submitted for publication].
For the entire collection see [Zbl 0786.00022].
Reviewer: I.Assem (Sherbrooke)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 16R40 | Identities other than those of matrices over commutative rings |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
