American Institute of Mathematical Sciences

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

Cluster algebras with Grassmann variables

Valentin Ovsienko 1, and  MichaeL Shapiro 2,

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.

Keywords: cluster algebra, quivers, superalgebra.
Mathematics Subject Classification: Primary: 13F60.
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

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

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Morier-Genoud, V. 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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Morier-Genoud, V. 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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

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