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 towards understanding the notion of cluster superalgebra.

An Introduction to Supersymmetric Cluster Algebras

“…Surprisingly, although our defintion of cluster superalgebra is quite different from that of [16], we are able to prove some results similar to [16] in our setting too. For example, we are able to show that the supercommutative superalgebra generated by all the entries of a superfrieze is a subalgebra of a cluster superalgebra which is the main result of [16] and [17].…”

“…Recently, Ovsienko [16] and Ovsienko-Shapiro [17] have made an inspiring attempt to define cluster superalgebras, which includes several important examples such as the supergroup OSp(1|2), superfriezes, the extended Somos-4 sequence, etc. In these papers, they consider an extension of a quiver by adding odd vertices and make it an oriented hypergraph.…”

Solution of tetrahedron equation and cluster algebras

“…Such product (outside of the roots of unity) was also noted in [45]. Another promising direction of research is construction of new solution for the tetrahedron equation using cluster algebras with fermionic variables [43], as suggested by recent appearance of quivers with fermionic nodes in representations theory of affine algebras [6,32,38,39,55] and approach of [47] to super-algebras using tetrahedron equation.…”

An Expansion Formula for Decorated Super-Teichmüller Spaces

“…Steps towards defining super cluster algebras appeared in work of Ovsienko [15] and separately in the work of Li, Mixco, Ransingh, and Srivastava [12]. These initial steps were followed up by related work such as [16,17,21].…”

“…In particular, in [16], Ovsienko and Shapiro define a type of super cluster algebra, motivated by super-frieze patterns. In their setup, some of the frozen vertices of the quiver correspond to odd variables θ 1 , .…”