Cluster algebras with Grassmann variables

MichaeL Shapiro,,Department Of Mathematics, Michigan State University, East Lansing, Mi 48824, Usa ,Valentin Ovsienko,,Cnrs, Laboratoire De Mathématiques U.f.r. Sciences Exactes Et Naturelles Moulin De La Housse - Bp 1039 51687 Reims Cedex 2, France

doi:10.3934/era.2019.26.001

MichaeL Shapiro, ,Department Of Mathematics, Michigan State University, East Lansing, Mi 48824, Usa

Open Access

https://doi.org/10.3934/era.2019.26.001

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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 towards understanding the notion of cluster superalgebra.

Highlights

Cluster algebras, discovered by Fomin and Zelevinsky [3], are a special class of commutative associative algebras
Cluster algebras naturally appear in algebra, geometry and combinatorics, they are closely related to integrable systems
A cluster algebra is a subalgebra of the algebra of Laurent polynomials in Z[x±1 1, . . . , x±n 1] generated by certain polynomials with positive integer coefficients

Summary

Introduction

Cluster algebras, discovered by Fomin and Zelevinsky [3], are a special class of commutative associative algebras. A cluster algebra is usually defined with the help of a quiver (an oriented graph) with no loops and no 2-cycles; the generators of the algebra are defined with the help of exchange relations and quiver mutations. We can only treat mutations of even variables leaving the odd ones frozen. In this sense, the correct notion of cluster superalgebra is still out of reach. We believe that the correct notion of mutations of odd variables should extend the coordinate transformations considered in [15] and [18, 10]. This paper is based on the unpublished preprint [16], we modify the exchange relations suggested by [16] in such a way the restrictions on quiver mutations disappear. In our opinioin the expressions in [12] has some flaws the most evident of which is that the transcendence degree of the cluster algebra is generally speaking not mutationally invariant

Extended quivers and their mutations

Exchange relations

Laurent phenomenon and invariant presymplectic form

Reduction to the classical case: the monomial transform

The main example: superfriezes