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

https://www.aimsciences.org/article/exportPdf?id=ac0bf832-2a83-4280-82e1-5f5261be0b48

Cluster algebras with Grassmann variables

Motivated by the definition of super Teichmuller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super Teichmuller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super $\lambda$-lengths associated to arcs in a bordered surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super $\lambda$-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's $T$-path formulas for type $A$ cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type $A_n$. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the $\mu$-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.

/pdf/an-expansion-formula-for-decorated-super-teichmuller-spaces-mndwn51bdy.pdf

An Expansion Formula for Decorated Super-Teichmüller Spaces

We define a super analog of the classical Plucker embedding of the Grassmannian into a projective space. The difficulty of the problem is rooted in the fact that super exterior powers $\Lambda^{r|s}(V)$ are not a simple generalization from the completely even case (this works only for $r|0$ when it is possible to use $\Lambda^r(V)$). To construct the embedding we need to non-trivially combine a super vector space $V$ and its parity-reversion $\Pi V$. Our "super Plucker map" takes the Grassmann supermanifold $G_{r|s}(V)$ to a "weighted projective space" $P\left(\Lambda^{r|s}(V)\oplus \Lambda^{s|r}(\Pi V)\right)$ with weights $+1,-1$. A simpler map $G_{r|0}(V)\to P(\Lambda^r(V))$ works for the case $s=0$. We construct a super analog of Plucker coordinates, prove that our map is an embedding, and obtain "super Plucker relations". It is interesting that another type of relations (due to Khudaverdian) is equivalent to the (super) Plucker relations in the case $r|s=2|0$. We discuss application to much sought-after super cluster algebras and construct a super cluster structure for $G_2(\mathbb{R}^{4|1})$ and $G_2(\mathbb{R}^{5|1})$.

On super Pl\"{u}cker embedding and cluster algebras

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super $\lambda$-lengths in a marked disk, generalizing Schiffler's $T$-path formula. In the present paper, we give an alternate combinatorial expression for these super $\lambda$-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.

Double Dimer Covers on Snake Graphs from Super Cluster Expansions

Matrix Formulae for Decorated Super Teichmüller Spaces

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 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.

An approach toward 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.

/pdf/an-introduction-to-supersymmetric-cluster-algebras-4xw5mcu89o.pdf

B. Ransingh

Papers

An approach toward supersymmetric cluster algebras

An Introduction to Supersymmetric Cluster Algebras

An Introduction to Supersymmetric Cluster Algebras