MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Abstract

Let Λ = 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 𝔽 of the graded simple modules over Λ is given in [E. L. Green and Ø. 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 Ø. Solberg, Resolutions over Koszul algebras, Arch. Math.85 (2005), 118–127.], and this is used to find a minimal projective resolution ℙ of Λ over the enveloping algebra Λe. Using these results, we show that the multiplication in the Hochschild cohomology ring of Λ 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 Λ 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 Λ is shown to be surjective onto the graded centre of the Koszul dual.