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The Variety of Two-dimensional Algebras Over an Algebraically Closed Field

Part of: Algebraic groups

Published online by Cambridge University Press:  16 October 2018

Ivan Kaygorodov  and
Yury Volkov
Ivan Kaygorodov
Affiliation:
Universidade Federal do ABC, CMCC, Santo André, Brazil Email: [email protected]
Yury Volkov
Affiliation:
Saint Petersburg State University, Saint Petersburg, Russia Email: [email protected]
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Abstract

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The work is devoted to the variety of two-dimensional algebras over algebraically closed fields. First we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of two-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

Keywords

two-dimensional algebrasorbit closuredegenerationrigid algebra

MSC classification

Primary: 14L30: Group actions on varieties or schemes (quotients)
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 4 , August 2019 , pp. 819 - 842
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The work was supported by FAPESP 14/24519-8; RFBR 17-51-04004; the President’s Program “Support of Young Russian Scientists” (grant MK-1378.2017.1).

References

Ananin, A. and Mironov, A., The moduli space of two-dimensional algebras . Comm. Algebra 28(2000), 9, 4481–4488.Google Scholar
Bai, C. and Meng, D., The classification of Novikov algebras in low dimension . J. Phys. A 34(2001), 1581–1594. https://doi.org/10.1088/0305-4470/34/8/305.Google Scholar
Benes, T. and Burde, D., Degenerations of pre-Lie algebras . J. Math. Phys. 50(2009), no. 11, 112102, 9. https://doi.org/10.1063/1.3246608.Google Scholar
Benes, T. and Burde, D., Classification of orbit closures in the variety of three-dimensional Novikov algebras . J. Algebra Appl. 13(2014), no. 2, 1350081, 33 pp. https://doi.org/10.1142/S0219498813500813.Google Scholar
Burde, D., Degenerations of nilpotent Lie algebras . J. Lie Theory 9(1999), no. 1, 193–202.Google Scholar
Burde, D., Degenerations of 7-dimensional nilpotent Lie algebras . Comm. Algebra 33(2005), no. 4, 1259–1277. https://doi.org/10.1081/AGB-200053956.Google Scholar
Burde, D. and Steinhoff, C., Classification of orbit closures of 4-dimensional complex Lie algebras . J. Algebra 214(1999), no. 2, 729–739. https://doi.org/10.1006/jabr.1998.7714.Google Scholar
Calderón, A., Fernández Ouaridi, A., and Kaygorodov, I., Classification of bilinear maps with radical of codimension 2, arxiv:1806.07009.Google Scholar
Cañete, E. and Khudoyberdiyev, A., The classification of 4-dimensional Leibniz algebras . Linear Algebra Appl. 439(2013), no. 1, 273–288.Google Scholar
Dzhumadil’daev, A., Ismailov, N., and Tulenbaev, K., Free bicommutative algebras . Serdica Math. J. 37(2011), no. 1, 25–44.Google Scholar
Gorbatsevich, V., On contractions and degeneracy of finite-dimensional algebras . Soviet Math. (Iz. VUZ) 35(1991), no. 10, 17–24.Google Scholar
Gorbatsevich, V., Anticommutative finite-dimensional algebras of the first three levels of complexity . St. Petersburg Math. J. 5(1994), 505–521.Google Scholar
Gorbatsevich, V., On the level of some solvable Lie algebras . Sib. Math. J. 39(1998), 5, 872–883. https://doi.org/10.1007/BF02672909.Google Scholar
Goze, M. and Remm, E., 2-dimensional algebras . Afr. J. Math. Phys. 10(2011), no. 1, 81–91.Google Scholar
Grunewald, F. and O’Halloran, J., Varieties of nilpotent Lie algebras of dimension less than six . J. Algebra 112(1988), 315–325. https://doi.org/10.1016/0021-8693(88)90093-2.Google Scholar
Grunewald, F. and O’Halloran, J., Characterization of orbit closure and applications . J. Algebra 116(1988), 163–175. https://doi.org/10.1016/0021-8693(88)90199-8.Google Scholar
Grunewald, F. and O’Halloran, J., Deformations of Lie algebras . J. Algebra 162(1993), no. 1, 210–224. https://doi.org/10.1006/jabr.1993.1250.Google Scholar
Ismailov, N., Kaygorodov, I., and Volkov, Y., The geometric classification of Leibniz algebras . Internat. J. Math. 29(2018), no. 5, 1850035, 12. https://doi.org/10.1142/S0129167X18500350.Google Scholar
Kashuba, I. and Martin, M., Deformations of Jordan algebras of dimension four . J. Algebra 399(2014), 277–289. https://doi.org/10.1016/j.jalgebra.2013.09.040.Google Scholar
Kaygorodov, I., Popov, Y., and Volkov, Y., Degenerations of binary Lie and nilpotent Malcev algebras . Comm. Algebra 46(2018), no. 11, 4929–4941. https://doi.org/10.1080/00927872.2018.1459647.Google Scholar
Kaygorodov, I., Popov, Y., Pozhidaev, A., and Volkov, Y., Degenerations of Zinbiel and nilpotent Leibniz algebras . Linear Multilinear Algebra 66(2018), no. 4, 704–716. https://doi.org/10.1080/00927872.2018.1459647.Google Scholar
Kaygorodov, I. and Volkov, Y., Complete classification of algebras of level two . Moscow Math. J., to appear. arxiv:1710.08943.Google Scholar
Khudoyberdiyev, A., The classification of algebras of level two . J. Geom. Phys. 98(2015), 13–20. https://doi.org/10.1016/j.geomphys.2015.07.020.Google Scholar
Khudoyberdiyev, A. and Omirov, B., The classification of algebras of level one . Linear Algebra Appl. 439(2013), no. 11, 3460–3463. https://doi.org/10.1016/j.laa.2013.09.020.Google Scholar
Lauret, J., Degenerations of Lie algebras and geometry of Lie groups . Differ. Geom. Appl. 18(2003), 2, 177–194. https://doi.org/10.1016/S0926-2245(02)00146-8.Google Scholar
Mazzola, G., The algebraic and geometric classification of associative algebras of dimension five . Manuscripta Math. 27(1979), 81–101. https://doi.org/10.1007/BF01297739.Google Scholar
Mazzola, G., Generic finite schemes and Hochschild cocycles . Comment. Math. Helv. 55(1980), 267–293. https://doi.org/10.1007/BF02566686.Google Scholar
Petersson, H., The classification of two-dimensional nonassociative algebras . Results Math. 37(2000), no. 1–2, 120–154. https://doi.org/10.1007/BF03322518.Google Scholar
Seeley, C., Degenerations of 6-dimensional nilpotent Lie algebras over ℂ . Comm. Algebra 18(1990), 3493–3505. https://doi.org/10.1080/00927879008824088.Google Scholar
Snobl, L. and Winternitz, P., Classification and identification of Lie algebras (CRM Monograph Series) , American Mathematical Society, Providence, RI, 2014.Google Scholar