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Register a new userConvergent presentations and polygraphic resolutions of associative algebras
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Several constructive homological methods based on noncommutative Grobner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the existence of a quadratic Grobner basis of its ideal of relations. In this article, using a higher-dimensional rewriting theory approach, we give several improvements of these methods. We define polygraphs for associative algebras as higher-dimensional linear rewriting systems that generalise the notion of noncommutative Grobner bases, and allow more possibilities of termination orders than those associated to monomial orders. We introduce polygraphic resolutions of associative algebras, giving a categorical description of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how these resolutions can be linked with the Koszul property.
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We compute coherent presentations of Artin monoids, that is presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squiers and Knuth-Bendixs completions into a homotopical completion-reduction, applied to Artins and Garsides presentations. The main result of the paper states that the so-called Tits-Zamolodchikov 3-cells extend Artins presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.
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We introduce the notion of weakly associative algebra and its relations with the notion of nonassociative Poisson algebras.
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