2019, 26: 1-15. doi: 10.3934/era.2019.26.001

Cluster algebras with Grassmann variables

1. 

CNRS, Laboratoire de Mathématiques U.F.R. Sciences Exactes et Naturelles Moulin de la Housse - BP 1039 51687 REIMS cedex 2, France

2. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

We are grateful to Sophie Morier-Genoud, Gregg Musiker and Sergei Tabachnikov for a number of fruitful discussions

Received  September 06, 2018 Published  March 2019

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.

Citation: Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001
References:
[1]

H. S. M. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.  doi: 10.4064/aa-18-1-297-310.

[2]

J. A. Cruz Morales and S. Galkin, Upper bounds for mutations of potentials, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 005, 13 pp. doi: 10.3842/SIGMA.2013.005.

[3]

S. Fomin and A. Zelevinsky, Cluster algebras. Ⅰ. Foundations, J. Amer. Math. Soc., 15 (2002), 497-529.  doi: 10.1090/S0894-0347-01-00385-X.

[4]

S. Fomin and A. Zelevinsky, The Laurent phenomenon, Adv. in Appl. Math., 28 (2002), 119-144.  doi: 10.1006/aama.2001.0770.

[5]

A. Fordy and R. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 34 (2011), 19-66.  doi: 10.1007/s10801-010-0262-4.

[6]

S. Galkin and A. Usnich, Mutations of potentials, Preprint IPMU 10-0100, 2010.

[7]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J., 127 (2005), 291-311; and Correction to "Cluster algebras and Weil-Petersson forms", Duke Math. J., 139 (2007), 407-409. doi: 10.1215/S0012-7094-07-13925-5.

[8]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/surv/167.

[9]

M. GrossP. Hacking and S. Keel, Birational geometry of cluster algebras, Algebr. Geom., 2 (2015), 137-175.  doi: 10.14231/AG-2015-007.

[10]

I. IpR. Penner and A. Zeitlin, $N = 2$ super-Teichmüller theory, Adv. Math., 336 (2018), 409-454.  doi: 10.1016/j.aim.2018.08.001.

[11]

R. Marsh, Lecture Notes on Cluster Algebras, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.

[12]

L. Li, J. Mixco, B. Ransingh and A. Srivastava, An approach toward supersymmetric cluster algebras, arXiv: 1708.03851.

[13]

S. Morier-Genoud, Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc., 47 (2015), 895-938.  doi: 10.1112/blms/bdv070.

[14]

S. Morier-Genoud, V. Ovsienko, R. Schwartz and S. Tabachnikov, Linear difference equations, frieze patterns, and combinatorial Gale transform, Forum Math. Sigma, 2 (2014), e22, 45 pp. doi: 10.1017/fms.2014.20.

[15]

S. Morier-GenoudV. Ovsienko and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z., 281 (2015), 1061-1087.  doi: 10.1007/s00209-015-1520-x.

[16]

V. Ovsienko, A step towards cluster superalgebras, arXiv: 1503.01894.

[17]

V. Ovsienko and S. Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, Algebr. Represent. Theory, 21 (2018), 1119-1132. doi: 10.1007/s10468-018-9779-3.

[18]

R. Penner and A. Zeitlin, Decorated super-Teichmüller space, arXiv: 1509.06302.

[19]

L. Williams, Cluster algebras: An introduction, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 1-26.  doi: 10.1090/S0273-0979-2013-01417-4.

[20]

E. Witten, Notes On Super Riemann Surfaces And Their Moduli, arXiv: 1209.2459.

show all references

References:
[1]

H. S. M. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.  doi: 10.4064/aa-18-1-297-310.

[2]

J. A. Cruz Morales and S. Galkin, Upper bounds for mutations of potentials, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 005, 13 pp. doi: 10.3842/SIGMA.2013.005.

[3]

S. Fomin and A. Zelevinsky, Cluster algebras. Ⅰ. Foundations, J. Amer. Math. Soc., 15 (2002), 497-529.  doi: 10.1090/S0894-0347-01-00385-X.

[4]

S. Fomin and A. Zelevinsky, The Laurent phenomenon, Adv. in Appl. Math., 28 (2002), 119-144.  doi: 10.1006/aama.2001.0770.

[5]

A. Fordy and R. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 34 (2011), 19-66.  doi: 10.1007/s10801-010-0262-4.

[6]

S. Galkin and A. Usnich, Mutations of potentials, Preprint IPMU 10-0100, 2010.

[7]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J., 127 (2005), 291-311; and Correction to "Cluster algebras and Weil-Petersson forms", Duke Math. J., 139 (2007), 407-409. doi: 10.1215/S0012-7094-07-13925-5.

[8]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/surv/167.

[9]

M. GrossP. Hacking and S. Keel, Birational geometry of cluster algebras, Algebr. Geom., 2 (2015), 137-175.  doi: 10.14231/AG-2015-007.

[10]

I. IpR. Penner and A. Zeitlin, $N = 2$ super-Teichmüller theory, Adv. Math., 336 (2018), 409-454.  doi: 10.1016/j.aim.2018.08.001.

[11]

R. Marsh, Lecture Notes on Cluster Algebras, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.

[12]

L. Li, J. Mixco, B. Ransingh and A. Srivastava, An approach toward supersymmetric cluster algebras, arXiv: 1708.03851.

[13]

S. Morier-Genoud, Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc., 47 (2015), 895-938.  doi: 10.1112/blms/bdv070.

[14]

S. Morier-Genoud, V. Ovsienko, R. Schwartz and S. Tabachnikov, Linear difference equations, frieze patterns, and combinatorial Gale transform, Forum Math. Sigma, 2 (2014), e22, 45 pp. doi: 10.1017/fms.2014.20.

[15]

S. Morier-GenoudV. Ovsienko and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z., 281 (2015), 1061-1087.  doi: 10.1007/s00209-015-1520-x.

[16]

V. Ovsienko, A step towards cluster superalgebras, arXiv: 1503.01894.

[17]

V. Ovsienko and S. Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, Algebr. Represent. Theory, 21 (2018), 1119-1132. doi: 10.1007/s10468-018-9779-3.

[18]

R. Penner and A. Zeitlin, Decorated super-Teichmüller space, arXiv: 1509.06302.

[19]

L. Williams, Cluster algebras: An introduction, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 1-26.  doi: 10.1090/S0273-0979-2013-01417-4.

[20]

E. Witten, Notes On Super Riemann Surfaces And Their Moduli, arXiv: 1209.2459.

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