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 \(SpO(2|1)\) 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 \(G(2|0; 4|1)\) of chiral conformal superspace (that is, \((2|0)\) planes inside the superspace \(\mathbb C^{4|1}\)) is a quotient of a cluster superalgebra.