On the first Hochschild cohomology group of a cluster-tilted algebra. (English) Zbl 1360.16007
Summary: Given a cluster-tilted algebra \(B\), we study its first Hochschild cohomology group \(\mathrm {HH}^{1}(B)\) with coefficients in the \(B-B\)-bimodule \(B\). If \(C\) is a tilted algebra such that \(B\) is the relation-extension of \(C\), then we show that if \(B\) is tame, then \(\mathrm {HH}^{1}(B)\) is isomorphic, as a \(k\)-vector space, to the direct sum of \( \mathrm {HH}^{1}(C)\) with \(k^{n_{B,C}}\), where \(n_{B,C}\) is an invariant linking the bound quivers of \(B\) and \(C\). In the representation-finite case, \(\mathrm {HH}^{1}(B)\) can be read off simply by looking at the quiver of \(B\).
MSC:
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 13F60 | Cluster algebras |
| 16G10 | Representations of associative Artinian rings |
| 16G20 | Representations of quivers and partially ordered sets |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16S70 | Extensions of associative rings by ideals |
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