Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-18T18:47:52.761Z Has data issue: false hasContentIssue false
  • English
  • Français

POTENTIAL ALGEBRAS WITH FEW GENERATORS

Part of: General nonassociative rings

Published online by Cambridge University Press:  01 July 2020

NATALIA IYUDU  and
STANISLAV SHKARIN
STANISLAV SHKARIN
Affiliation:
e-mail: s.shkarin@qub.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a complete description of quadratic twisted potential algebras on three generators as well as cubic twisted potential algebras on two generators up to graded algebra isomorphisms under the assumption that the ground field is algebraically closed and has characteristic other than 2 or 3.

MSC classification

Secondary: 17A45: Quadratic algebras (but not quadratic Jordan algebras)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Special Issue S1: Workshop on Nonassociative algebras and their applications , December 2020 , pp. S28 - S76
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Artin, M. and Shelter, W., Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171–216.CrossRefGoogle Scholar
Artin, M., Tate, J. and Van den Bergh, M., Some algebras associated to automorphisms of elliptic curves, in The Grothendieck Festschrift I (Cartier, P., Illusie, L., Kaz, N., Laumon, G., Manin, Y. and Ribet, K., Editors), Progress in Mathematics, vol. 86 (Birkhäuser, Boston, 1990), 33–85.Google Scholar
Bocklandt, R., Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (2008), 14–32.Google Scholar
Bocklandt, R., Schedler, T. and Wemyss, M., Superpotentials and higher order derivations, J. Pure Appl. Algebra 214(9) (2010), 1501–1522.CrossRefGoogle Scholar
Donovan, W. and Wemyss, M., Noncommutative deformations and flops, Duke Math. J. 165 (2016), 1397–1474.CrossRefGoogle Scholar
Drinfeld, V., On quadratic quasi-commutational relations in quasi-classical limit, Selecta Math. Sovietica 11 (1992), 317–326.Google Scholar
Dubois-Violette, M., Multilinear forms and graded algebras, J. Algebra 317 (2007), 198–225.CrossRefGoogle Scholar
Dubois-Violette, M., Graded algebras and multilinear forms, C. R. Math. Acad. Sci. Paris. 341 (2005), 719–724.CrossRefGoogle Scholar
Ginzburg, V., Calabi Yau algebras, ArXiv:math/0612139v3 (2007).Google Scholar
Gurevich, G., Foundations of the theory of algebraic invariants, Noordhoff (1964).Google Scholar
Iyudu, N. and Shkarin, S., Sklyanin algebras and Gröbner bases, J. Algebra. 470 (2017), 379–419.CrossRefGoogle Scholar
Iyudu, N. and Shkarin, S., Sklyanin algebras and a cubic root of 1, MPIM Preprint. 49 (2017), 1–19.Google Scholar
Iyudu, N. and Smoktunowitz, A., Golod–Shafarevich type theorems and potential algebras, IMRN. 15(2019), 4822–4844.CrossRefGoogle Scholar
Kontsevich, M., Formal (non) commutative symplectic geometry, in The Gelfand Mathematical Seminars (Paris 1992) (Gelfand, I. M., Corwin, L. and Lepowsky, J., Editors), Progress in Mathematics, vol. 120 (Birkhäuser, Basel, 1994), 97–121.Google Scholar
Kraft, K., Geometrische Methoden in Invarianttheorie (Friedrich Vieweg & Sohn, Brauunschweig, 1985).CrossRefGoogle Scholar
Polishchuk, A. and Positselski, L., Quadratic algebras, University Lecture Series, vol. 37 (American Mathematical Society, Providence, RI, 2005).Google Scholar
Shearer, J., A graded algebra with non-rational Hilbert series, J. Algebra 62 (1980), 228–231.CrossRefGoogle Scholar
Toda, Y., Noncommutative width and Gopakumar–Vafa invariants, Manuscripta Mathematica 148 (2015), 521–533.CrossRefGoogle Scholar
Zelmanov, E., Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44 (2007), 1185–1195.CrossRefGoogle Scholar
Ufnarovskij, V., Combinatorial and asymptotic methods in algebra, Encyclopaedia of Mathematical Sciences, vol. 57 (Kostrikin, A. and Shafarevich, I., Editors) (Springer-Verlag, Berlin, Heidelberg, New York, 1995), 1–196.Google Scholar