MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Let A = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution IF of the graded simple modules over A is given in [E. L. Green and O. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012-1033]. This resolution is shown to have a 'comultiplicative' structure in [E. L. Green, G. Hartman, E. N. Marcos and O. Solberg, Resolutions over Koszul algebras, Arch. Math. 85 (2005), 118-127.], and this is used to find a minimal projective resolution P of A over the enveloping algebra A(e). Using these results, we show that the multiplication in the Hochschild cohomology ring of A relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of A with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of A is shown to be surjective onto the graded centre of the Koszul dual.