The algebraic classification of nilpotent commutative algebras

Doston Jumaniyozov; Ivan Kaygorodov; Abror Khudoyberdiyev

doi:10.3934/era.2021068

The algebraic classification of nilpotent commutative algebras

Electronic Research Archive ◽

10.3934/era.2021068 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Doston Jumaniyozov ◽

Ivan Kaygorodov ◽

Abror Khudoyberdiyev

Keyword(s):

Standard Method ◽

Central Extensions ◽

Algebraic Classification ◽

Commutative Algebras

This paper is devoted to the complete algebraic classification of complex <inline-formula><tex-math id="M1">\begin{document}$ 5 $\end{document}</tex-math></inline-formula>-dimensional nilpotent commutative algebras. Our method of classification is based on the standard method of classification of central extensions of smaller nilpotent commutative algebras and the recently obtained classification of complex <inline-formula><tex-math id="M2">\begin{document}$ 5 $\end{document}</tex-math></inline-formula>-dimensional nilpotent commutative <inline-formula><tex-math id="M3">\begin{document}$ \mathfrak{CD} $\end{document}</tex-math></inline-formula>-algebras.

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The algebraic classification of nilpotent associative commutative algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498820502205 ◽

2019 ◽

Vol 19 (11) ◽

pp. 2050220 ◽

Cited By ~ 5

Author(s):

Ivan Kaygorodov ◽

Isamiddin Rakhimov ◽

Sh. K. Said Husain

Keyword(s):

Algebraic Classification ◽

Commutative Algebras

In this paper, we give a complete algebraic classification of [Formula: see text]-dimensional complex nilpotent associative commutative algebras.

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THE DYNAMICS OF NQ-SYSTEMS IN THE PLANE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409022786 ◽

2009 ◽

Vol 19 (01) ◽

pp. 117-133 ◽

Cited By ~ 4

Author(s):

MATEJ MENCINGER ◽

MILAN KUTNJAK

Keyword(s):

Algebraic Approach ◽

Dynamical Analysis ◽

Algebraic Classification ◽

Commutative Algebras ◽

Planar Maps ◽

Points Of View ◽

One To One

The dynamics of discrete homogeneous quadratic planar maps is considered via the algebraic approach. There is a one-to-one correspondence between these systems and 2D commutative algebras (c.f. [Markus, 1960]). In particular, we consider the systems corresponding to algebras which contain some nilpotents of rank two (i.e. NQ-systems). Markus algebraic classification is used to obtain the class representatives. The case-by-case dynamical analysis is presented. It is proven that there is no chaos in NQ-systems. Yet, some cases are really interesting from the dynamical and bifurcational points of view.

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Classification of central extensions of Lax operator algebras

10.1063/1.3043863 ◽

2008 ◽

Author(s):

Martin Schlichenmaier ◽

Piotr Kielanowski ◽

Anatol Odzijewicz ◽

Martin Schlichenmaier ◽

Theodore Voronov

Keyword(s):

Operator Algebras ◽

Central Extensions ◽

Lax Operator

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Algebraic Classification of Field Equations

The General Theory of Relativity ◽

10.1007/978-1-4614-3658-4_7 ◽

2012 ◽

pp. 465-536

Author(s):

Anadijiban Das ◽

Andrew DeBenedictis

Keyword(s):

Field Equations ◽

Algebraic Classification

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The Algebraic and Geometric Classification of Four Dimensional Superalgebras

10.26686/wgtn.16915645.v1 ◽

2021 ◽

Author(s):

◽

Aaron Armour

Keyword(s):

Classification Problem ◽

Associative Algebras ◽

New Variety ◽

Graded Algebras ◽

Finite Representation ◽

Algebraic Classification ◽

Irreducible Components ◽

Module Algebras ◽

Sweedler’S Hopf Algebra

The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12]. Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras. This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties. The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.

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