An Introduction to Supersymmetric Cluster Algebras

In this paper we propose the notion of cluster superalgebras which is a supersymmetric version of the classical cluster algebras introduced by Fomin and Zelevinsky. We show that the symplectic-orthogonal supergroup $SpO(2\mathrm{\mid }1)$ admits a cluster superalgebra structure and as a consequence of this, we deduce that the supercommutative superalgebra generated by all the entries of a superfrieze is a subalgebra of a cluster superalgebra. We also show that the coordinate superalgebra of the super Grassmannian $G(2\mathrm{\mid }0;4\mathrm{\mid }1)$ of chiral conformal superspace (that is, $(2\mathrm{\mid }0)$ planes inside the superspace $\mathbb C^{4|1}$) is a quotient of a cluster superalgebra.