doi: 10.3934/era.2021068
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

The algebraic classification of nilpotent commutative algebras

1. 

National University of Uzbekistan, Tashkent, Uzbekistan, Institute of Mathematics named after V.I. Romanovsky Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

2. 

CMCC, Universidade Federal do ABC, Santo André, Brazil, Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia, Saint Petersburg University, Russia

3. 

National University of Uzbekistan, Tashkent, Uzbekistan, Institute of Mathematics named after V.I. Romanovsky, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

* Corresponding author: Ivan Kaygorodov

Received  February 2021 Revised  July 2021 Early access September 2021

Fund Project: The work is supported by the Russian Science Foundation under grant 19-71-10016

This paper is devoted to the complete algebraic classification of complex $ 5 $-dimensional nilpotent commutative algebras. Our method of classification is based on the standard method of classification of central extensions of smaller nilpotent commutative algebras and the recently obtained classification of complex $ 5 $-dimensional nilpotent commutative $ \mathfrak{CD} $-algebras.

Citation: Doston Jumaniyozov, Ivan Kaygorodov, Abror Khudoyberdiyev. The algebraic classification of nilpotent commutative algebras. Electronic Research Archive, doi: 10.3934/era.2021068
References:
[1]

Yo. Cabrera CasadoM. Siles Molina and M. V. Velasco, Classification of three-dimensional evolution algebras, Linear Algebra Appl., 524 (2017), 68-108.  doi: 10.1016/j.laa.2017.02.015.  Google Scholar

[2]

E. M. Cañete and A. Kh. Khudoyberdiyev, The classification of $4$-dimensional Leibniz algebras, Linear Algebra Appl., 439 (2013), 273-288.  doi: 10.1016/j.laa.2013.02.035.  Google Scholar

[3]

S. CicalòW. A. de Graaf and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl., 436 (2012), 163-189.  doi: 10.1016/j.laa.2011.06.037.  Google Scholar

[4]

I. Darijani and H. Usefi, The classification of 5-dimensional $p$-nilpotent restricted Lie algebras over perfect fields, I., J. Algebra, 464 (2016), 97-140.  doi: 10.1016/j.jalgebra.2016.06.011.  Google Scholar

[5]

E. Darpö and A. Rochdi, Classification of the four-dimensional power-commutative real division algebras, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1207-1223.  doi: 10.1017/S0308210510000259.  Google Scholar

[6]

W. A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not $2$, J. Algebra, 309 (2007), 640-653.  doi: 10.1016/j.jalgebra.2006.08.006.  Google Scholar

[7]

E. Dieterich and J. Öhman, On the classification of $4$-dimensional quadratic division algebras over square-ordered fields, J. London Math. Soc. (2), 65 (2002), 285-302.  doi: 10.1112/S0024610701003064.  Google Scholar

[8]

A. Fernández Ouaridi, I. Kaygorodov, M. Khrypchenko and Yu. Volkov, Degenerations of nilpotent algebras, J. Pure Appl. Algebra, 226 (2022), Paper No. 106850, 21 pp. doi: 10.1016/j.jpaa.2021.106850.  Google Scholar

[9]

A. S. Hegazi and H. Abdelwahab, Classification of five-dimensional nilpotent Jordan algebras, Linear Algebra Appl., 494 (2016), 165-218.  doi: 10.1016/j.laa.2016.01.015.  Google Scholar

[10]

A. S. HegaziH. Abdelwahab and A. J. Calderón Martín, The classification of $n$-dimensional non-Lie Malcev algebras with $(n-4)$-dimensional annihilator, Linear Algebra Appl., 505 (2016), 32-56.  doi: 10.1016/j.laa.2016.04.029.  Google Scholar

[11]

D. Jumaniyozov, I. Kaygorodov and A. Khudoyberdiyev, The algebraic classification of nilpotent commutative $\mathfrak {CD}$-algebras, Communications in Algebra, 49 (2021), 1464–1494. doi: 10.1080/00927872.2020.1837852.  Google Scholar

[12]

I. Kaygorodov, M. Khrypchenko and S. A. Lopes, The algebraic and geometric classification of nilpotent anticommutative algebras, Journal of Pure and Applied Algebra, 224 (2020), 106337, 32 pp. doi: 10.1016/j.jpaa.2020.106337.  Google Scholar

[13]

I. Kaygorodov, M. Khrypchenko and S. Lopes, The algebraic classification of nilpotent algebras, arXiv: 2012.00525. Google Scholar

[14]

I. Kaygorodov, M. Khrypchenko and Yu. Popov, The algebraic and geometric classification of nilpotent terminal algebras, J. Pure Appl. Algebra, 225 (2021), 106625, 41 pp. doi: 10.1016/j.jpaa.2020.106625.  Google Scholar

[15]

I. Kaygorodov, I. Rakhimov and Sh. K. Said Husain, The algebraic classification of nilpotent associative commutative algebras, J. Algebra Appl., 19 (2020), 2050220, 14 pp. doi: 10.1142/S0219498820502205.  Google Scholar

[16]

I. Kaygorodov and Yu. Volkov, The variety of $2$-dimensional algebras over an algebraically closed field, Canad. J. Math., 71 (2019), 819-842.  doi: 10.4153/S0008414X18000056.  Google Scholar

[17]

Yu. KobayashiK. ShirayanagiS.-Ei. Takahasi and M. Tsukada, Classification of three-dimensional zeropotent algebras over an algebraically closed field, Comm. Algebra, 45 (2017), 5037-5052.  doi: 10.1080/00927872.2017.1313426.  Google Scholar

[18]

G. Mazzola, Generic finite schemes and Hochschild cocycles, Comment. Math. Helv., 55 (1980), 267-293.  doi: 10.1007/BF02566686.  Google Scholar

[19]

H. P. Petersson, The classification of two-dimensional nonassociative algebras, Results Math., 37 (2000), 120-154.  doi: 10.1007/BF03322518.  Google Scholar

[20]

T. Skjelbred and T. Sund, Sur la classification des algebres de Lie nilpotentes, C. R. Acad. Sci. Paris Sér. A-B, 286 (1978), A241–A242.  Google Scholar

show all references

References:
[1]

Yo. Cabrera CasadoM. Siles Molina and M. V. Velasco, Classification of three-dimensional evolution algebras, Linear Algebra Appl., 524 (2017), 68-108.  doi: 10.1016/j.laa.2017.02.015.  Google Scholar

[2]

E. M. Cañete and A. Kh. Khudoyberdiyev, The classification of $4$-dimensional Leibniz algebras, Linear Algebra Appl., 439 (2013), 273-288.  doi: 10.1016/j.laa.2013.02.035.  Google Scholar

[3]

S. CicalòW. A. de Graaf and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl., 436 (2012), 163-189.  doi: 10.1016/j.laa.2011.06.037.  Google Scholar

[4]

I. Darijani and H. Usefi, The classification of 5-dimensional $p$-nilpotent restricted Lie algebras over perfect fields, I., J. Algebra, 464 (2016), 97-140.  doi: 10.1016/j.jalgebra.2016.06.011.  Google Scholar

[5]

E. Darpö and A. Rochdi, Classification of the four-dimensional power-commutative real division algebras, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1207-1223.  doi: 10.1017/S0308210510000259.  Google Scholar

[6]

W. A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not $2$, J. Algebra, 309 (2007), 640-653.  doi: 10.1016/j.jalgebra.2006.08.006.  Google Scholar

[7]

E. Dieterich and J. Öhman, On the classification of $4$-dimensional quadratic division algebras over square-ordered fields, J. London Math. Soc. (2), 65 (2002), 285-302.  doi: 10.1112/S0024610701003064.  Google Scholar

[8]

A. Fernández Ouaridi, I. Kaygorodov, M. Khrypchenko and Yu. Volkov, Degenerations of nilpotent algebras, J. Pure Appl. Algebra, 226 (2022), Paper No. 106850, 21 pp. doi: 10.1016/j.jpaa.2021.106850.  Google Scholar

[9]

A. S. Hegazi and H. Abdelwahab, Classification of five-dimensional nilpotent Jordan algebras, Linear Algebra Appl., 494 (2016), 165-218.  doi: 10.1016/j.laa.2016.01.015.  Google Scholar

[10]

A. S. HegaziH. Abdelwahab and A. J. Calderón Martín, The classification of $n$-dimensional non-Lie Malcev algebras with $(n-4)$-dimensional annihilator, Linear Algebra Appl., 505 (2016), 32-56.  doi: 10.1016/j.laa.2016.04.029.  Google Scholar

[11]

D. Jumaniyozov, I. Kaygorodov and A. Khudoyberdiyev, The algebraic classification of nilpotent commutative $\mathfrak {CD}$-algebras, Communications in Algebra, 49 (2021), 1464–1494. doi: 10.1080/00927872.2020.1837852.  Google Scholar

[12]

I. Kaygorodov, M. Khrypchenko and S. A. Lopes, The algebraic and geometric classification of nilpotent anticommutative algebras, Journal of Pure and Applied Algebra, 224 (2020), 106337, 32 pp. doi: 10.1016/j.jpaa.2020.106337.  Google Scholar

[13]

I. Kaygorodov, M. Khrypchenko and S. Lopes, The algebraic classification of nilpotent algebras, arXiv: 2012.00525. Google Scholar

[14]

I. Kaygorodov, M. Khrypchenko and Yu. Popov, The algebraic and geometric classification of nilpotent terminal algebras, J. Pure Appl. Algebra, 225 (2021), 106625, 41 pp. doi: 10.1016/j.jpaa.2020.106625.  Google Scholar

[15]

I. Kaygorodov, I. Rakhimov and Sh. K. Said Husain, The algebraic classification of nilpotent associative commutative algebras, J. Algebra Appl., 19 (2020), 2050220, 14 pp. doi: 10.1142/S0219498820502205.  Google Scholar

[16]

I. Kaygorodov and Yu. Volkov, The variety of $2$-dimensional algebras over an algebraically closed field, Canad. J. Math., 71 (2019), 819-842.  doi: 10.4153/S0008414X18000056.  Google Scholar

[17]

Yu. KobayashiK. ShirayanagiS.-Ei. Takahasi and M. Tsukada, Classification of three-dimensional zeropotent algebras over an algebraically closed field, Comm. Algebra, 45 (2017), 5037-5052.  doi: 10.1080/00927872.2017.1313426.  Google Scholar

[18]

G. Mazzola, Generic finite schemes and Hochschild cocycles, Comment. Math. Helv., 55 (1980), 267-293.  doi: 10.1007/BF02566686.  Google Scholar

[19]

H. P. Petersson, The classification of two-dimensional nonassociative algebras, Results Math., 37 (2000), 120-154.  doi: 10.1007/BF03322518.  Google Scholar

[20]

T. Skjelbred and T. Sund, Sur la classification des algebres de Lie nilpotentes, C. R. Acad. Sci. Paris Sér. A-B, 286 (1978), A241–A242.  Google Scholar

[1]

Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121

[2]

Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469

[3]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[4]

Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39

[5]

A. A. Kirillov. Family algebras. Electronic Research Announcements, 2000, 6: 7-20.

[6]

Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167

[7]

Steffen Konig and Changchang Xi. Cellular algebras and quasi-hereditary algebras: a comparison. Electronic Research Announcements, 1999, 5: 71-75.

[8]

Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62.

[9]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39

[10]

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

[11]

L. S. Grinblat. Theorems on sets not belonging to algebras. Electronic Research Announcements, 2004, 10: 51-57.

[12]

Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009

[13]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124

[14]

Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, 2021, 29 (4) : 2619-2636. doi: 10.3934/era.2021004

[15]

Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $ n $-slice algebras and related algebras. Electronic Research Archive, 2021, 29 (4) : 2687-2718. doi: 10.3934/era.2021009

[16]

Randall Dougherty and Thomas Jech. Left-distributive embedding algebras. Electronic Research Announcements, 1997, 3: 28-37.

[17]

G. Mashevitzky, B. Plotkin and E. Plotkin. Automorphisms of categories of free algebras of varieties. Electronic Research Announcements, 2002, 8: 1-10.

[18]

María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637

[19]

A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

[20]

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

2020 Impact Factor: 1.833

Article outline

Figures and Tables

[Back to Top]