Multiplicative structures for Koszul algebras

Let $\mathrm{\Lambda }=kQ\mathrm{/}I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $\mathrm{\Lambda }$ is given in Green-Solberg. This resolution is shown to have a “comultiplicative” structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $\mathrm{\Lambda }$ over the enveloping algebra ${\mathrm{\Lambda }}^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $\text{L}$ relative to the resolution $\mathbb{P}$ 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 $\text{L}$ with respect to a canonical basis over $k$ associated to the resolution $\mathbb{F}$. The natural map from the Hochschild cohomology to the Koszul dual of $\mathrm{\Lambda }$ is shown to be surjective onto the graded centre of the Koszul dual.