Séminaires : Séminaire d'Algèbre

The aim of my talk (based on joint work in progress with Arkady Berenstein and Vladimir Retakh) is to introduce and study certain noncommutative cluster algebras A from marked orbifolds. They are non-commutative versions of the generalized cluster algebras defined by Chekhov and Shapiro. These algebras admit noncommutative clusters, i.e., embeddings of a given group G which is either free or one-relator (we call it a triangle group) into the multiplicative monoid A×. The clusters are parametrized by triangulations of the orbifold and exhibit a noncommutative Laurent Phenomenon, which asserts that generators of the algebra can be written as sums of the images of elements of G for any noncommutative cluster. In particular, if the surface is unpunctured, then our algebra A can be specialized to the ordinary quantum cluster algebra, and the noncommutative Laurent Phenomenon becomes the (positive) quantum one. 

This talk will take place on Zoom only.