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Cluster algebras with Grassmann variables

We are grateful to Sophie Morier-Genoud, Gregg Musiker and Sergei Tabachnikov for a number of fruitful discussions

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  • We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of "extended quivers," which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step towards understanding the notion of cluster superalgebra.

    Mathematics Subject Classification: Primary: 13F60.

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