On the comparison of the Hochschild cohomology and the fundamental groups of and algebra

Patrick Le Meur (1)

1 CMLA - Centre de Mathématiques et de Leurs Applications

Let A be a basic connected finite dimensional algebra over an algebraically closed field k and with ordinary quiver Q without oriented cycle. To any presentation of A by quiver and admissible relations, R. Martinez-Villa and J. A. de La Pena have associated the fundamental group of the presentation. I. Assem and J. A. de La Pena have related the space of group morphisms from this fundamental group to the additive underlying group of k to the Hochschild cohomoloy HH^1(A) through an injective mapping from the first group to the second one. We compare these morphisms associated to the different presentations of A. Following a work by D. Farkas, E. Green and E. N. Marcos, we characterise the image of these morphisms in terms of diagonalisable subaglebras of the Lie algebra HH^1(A). Finally, we characterise the maximal diagonalisable subalgebras of HH^1(A) when A is monomial and Q has no multiple arrows and also when car(k)=0 and Q has no double bypass.

Mots clés: groupe fondamental; présentation admissible; cohomologie de Hochschild; algèbre de Lie

Domaines: Théorie des représentations [math.RT]

Patrick Le Meur. On the comparison of the Hochschild cohomology and the fundamental groups of and algebra. 2007. ⟨hal-00123561v1⟩