MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS
By R. Buchweitz, Edward L. Green, N. Snashall, Oyvind Solberg. 2005
67 citations recorded by the source corpus.
Let $\Lambda=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $\Lambda$ 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 $\Lambda$ over the enveloping algebra $\Lambda^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $\L$ 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 $\L$ 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 $\Lambda$ is shown to be surjective onto the graded centre of the Koszul dual.
Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/67391b4bf9b7bac00abc7aca954a1ecc4c7eb2ee
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Citations in the corpus
- The hochschild cohomology ring of a class of special biserial algebras (2008, methodology): “A general description of the multiplicative structure of the Hochschild cohomology ring of a Koszul algebra was given in [ 2 ], but it remains open as to whether the conjecture of [18] holds for all Koszul algebras.”