American Institute of Mathematical Sciences

Advanced
  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

[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

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

[1]

Peigen Cao, Fang Li, Siyang Liu, Jie Pan. A conjecture on cluster automorphisms of cluster algebras. Electronic Research Archive, 2019, 27: 1-6. doi: 10.3934/era.2019006
[2]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017
[3]

Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297
[4]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066
[5]

Neşet Deniz Turgay. On the mod p Steenrod algebra and the Leibniz-Hopf algebra. Electronic Research Archive, 2020, 28 (2) : 951-959. doi: 10.3934/era.2020050
[6]

Octav Cornea and Francois Lalonde. Cluster homology: An overview of the construction and results. Electronic Research Announcements, 2006, 12: 1-12.

[7]

Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523
[8]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[9]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420
[10]

Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023
[11]

Gerhard Keller, Carlangelo Liverani. Coupled map lattices without cluster expansion. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 325-335. doi: 10.3934/dcds.2004.11.325
[12]

Takashi Hara and Gordon Slade. The incipient infinite cluster in high-dimensional percolation. Electronic Research Announcements, 1998, 4: 48-55.

[13]

Shuping Li, Zhen Jin. Impacts of cluster on network topology structure and epidemic spreading. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3749-3770. doi: 10.3934/dcdsb.2017187
[14]

Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial & Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87
[15]

Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399
[16]

David E. Bernholdt, Mark R. Cianciosa, Clement Etienam, David L. Green, Kody J. H. Law, Jin M. Park. Corrigendum to "Cluster, classify, regress: A general method for learning discontinuous functions [1]". Foundations of Data Science, 2020, 2 (1) : 81-81. doi: 10.3934/fods.2020005
[17]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Convolutional codes with a matrix-algebra word-ambient. Advances in Mathematics of Communications, 2016, 10 (1) : 29-43. doi: 10.3934/amc.2016.10.29
[18]

H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50
[19]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239
[20]

Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (321)
  • HTML views (970)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences