Cluster algebras with Grassmann variables

Abstract

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 of understanding the notion of cluster superalgebra

Final version, to appear in Electron. Res. Announc. Math. Sci., 14 pages. arXiv admin note: substantial text overlap with arXiv:1503.01894