On maximal diagonalizable Lie subalgebras of the first Hochschild cohomology

Abstract

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, Martinez-Villa and de La Pena have associated the fundamental group of the presentation. Assem and de La Pena have constructed an injective mapping from the additive characters of this fundamental group (with values in the ground field) to the first Hochschild cohomology group HH^1(A). We study the image of these mappings associated to the different presentations of A in terms of diagonalizable Lie subalgebras of HH^1(A). Then 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.