# A step towards cluster superalgebras

@article{Ovsienko2015AST, title={A step towards cluster superalgebras}, author={Valentin Ovsienko}, journal={arXiv: Combinatorics}, year={2015} }

We introduce a class of commutative superalgebras generalizing cluster algebras. A cluster superalgebra is defined by a hypergraph called an "extended quiver", and transformations called mutations. We prove the super analog of the "Laurent phenomenon", i.e., that all elements of a given cluster superalgebra are Laurent polynomials in the initial variables, and find an invariant presymplectic form. Examples of cluster superalgebras are provided by superanalogs of Coxeter's frieze patterns. We…

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## 40 References

### Cluster algebras III: Upper bounds and double Bruhat cells

- Mathematics
- 2003

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of…

### Cluster algebras: an introduction

- Mathematics
- 2012

Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie…

### Grassmannians and Cluster Algebras

- Mathematics
- 2003

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian G(k, n) is a cluster algebra of geometric…

### Cluster algebras and Weil-Petersson forms

- Mathematics
- 2003

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of…

### Cluster algebras II: Finite type classification

- Mathematics
- 2002

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many…

### A family of linearizable recurrences with the Laurent property

- Mathematics
- 2014

We consider a family of non‐linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of…

### Introduction to Superanalysis

- Mathematics
- 1987

1. Grassmann Algebra.- 2. Superanalysis.- 3. Linear Algebra in Z2-Graded Spaces.- 4. Supermanifolds in General.- 5. Lie Superalgebras.- 1. Lie Superalgebras.- 2. Lie Supergroups.- 3. Laplace-Casimir…

### Cluster algebras IV: Coefficients

- MathematicsCompositio Mathematica
- 2007

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these…

### Introduction to the Theory of Supermanifolds

- Mathematics
- 1980

CONTENTSIntroduction Chapter I. Linear algebra in superspaces § 1. Linear superspaces § 2. Modules over superalgebras § 3. Matrix algebra § 4. Free modules § 5. Bilinear forms § 6. The supertrace §…

### Cluster Algebras and Poisson Geometry

- Mathematics
- 2002

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in…