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Multiplicative structures for Koszul algebras | | Ragnar-Olaf Buchweitz
; Edward L. Green
; Nicole Snashall
; {O}yvind Solberg
; | | Date: |
10 Aug 2005 | | Subject: | Rings and Algebras; Representation Theory MSC-class: 16S37, 16E40 | math.RA math.RT | | Abstract: | 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. | | Source: | arXiv, math.RA/0508177 | | Services: | Forum | Review | PDF | Favorites | |
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