Noncommutative Jordan superalgebras of degree n &gt; 2

Pozhidaev, A. P.; Shestakov, I. P.

doi:10.1007/s10469-010-9077-6

Cited by 14 publications

(27 citation statements)

References 21 publications

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“…The problem of classification of the simple finite-dimensional noncommutative Jordan superalgebras was posted in [6, Problem 3.100 a)] (see [26] as well). The present article continues our previous articles [23]- [24], where we classified the central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0. Now we consider the remaining modular case.…”

Section: Introductionmentioning

confidence: 54%

“…The theorem below was proved in [23,Theorem 2.4] for superalgebras over a field of characteristic zero, but the same proof is valid for the case of characteristic p = 2; one has only to use the mentioned above modular modification of the Cheng-Kac Theorem instead of the original theorem.…”

Section: An Analog Of Oehmke's Theoremmentioning

confidence: 98%

“…In [23]- [24] we called these brackets generalized Poisson brackets, but this term was used also by V. Kac in [4] for another algebra.…”

Section: Lemma 4 the Mappingmentioning

confidence: 99%

“…In the next sections we will determine the structure of a simple N J-superalgebra U according to the type of the simple Jordan superalgebra J = U (+) . Some of the cases were already investigated in [23] and [24] when char F = 0. We will refer to these cases when they are the same for the modular case.…”

Section: An Analog Of Oehmke's Theoremmentioning

confidence: 99%

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Simple finite-dimensional modular noncommutative Jordan superalgebras

Pozhidaev

Shestakov

2019

Journal of Pure and Applied Algebra

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We classify the central simple finite-dimensional noncommutative Jordan superalgebras of degree > 1 over an algebraically closed field of characteristic p > 2. The case of characteristic 0 was considered by the authors in the previous paper [24]. In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra B(m, n).

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Section: Introductionmentioning

confidence: 54%

Section: An Analog Of Oehmke's Theoremmentioning

confidence: 98%

“…In [23]- [24] we called these brackets generalized Poisson brackets, but this term was used also by V. Kac in [4] for another algebra.…”

Section: Lemma 4 the Mappingmentioning

confidence: 99%

Section: An Analog Of Oehmke's Theoremmentioning

confidence: 99%

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Simple finite-dimensional modular noncommutative Jordan superalgebras

Pozhidaev

Shestakov

2019

Journal of Pure and Applied Algebra

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“…В. Н. Желябин и И. П. Шестаков [8] получили описание йордановых супералгебр абелева типа без каких-либо ограничений для размерности. А. П. Пожидаев и И. П. Шестаков [9], [10] описали простые конечномерные некоммутативные йордановы супералгебры над полем характеристики нуль. В настоящей работе дается описание простых конечномерных правоальтернативных супералгебр абелева типа над полем характеристики нуль.…”

Section: *unclassified

Simple finite-dimensional right-alternative superalgebras of Abelian type of characteristic zero

Pchelintsev¹,

Shashkov²

2015

Известия Российской академии наук. Серия математическая

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Простые конечномерные правоальтернативные супералгебры абелева типа характеристики нуль Классифицированы простые конечномерные правоальтернативные супералгебры A = A0 ⊕ A1 над полем характеристики нуль, в которых четная часть A0 ассоциативна и коммутативна, а A1-ассоциативный A0-бимодуль. Доказано, что всякая такая супералгебра A = A0 ⊕ A1 получается удвоением полупростой четной части A0, а умножение в A определяется с помощью подходящего автоморфизма и линейного оператора, действующих на четной части A0. Библиография: 20 наименований.

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The Structure of Simple Noncommutative Jordan Superalgebras

2018

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In this paper we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras K 3 (α, β, γ) and D t (α, β, γ); and compute the derivations of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras. INTRODUCTIONThe class of noncommutative Jordan algebras is vast: for example, it includes alternative algebras, Jordan algebras, quasiassociative algebras, quadratic flexible algebras and anticommutative algebras. Schafer proved that a simple noncommutative Jordan algebra is either a simple Jordan algebra, or a simple quasiassociative algebra, or a simple flexible algebra of degree 2 [29]. Oehmke proved the analog of Schafer's classification for simple flexible algebras with strictly associative powers of and of characteristic = 2, 3 [20], McCrimmon classified simple noncommutative Jordan algebras of degree > 2 and characteristic = 2 [18, 19], and Smith described such algebras of degree 2 [28]. The case of nodal simple algebras of positive characteristic was considered in the papers of Kokoris [15,16].The case of simple finite-dimensional Jordan superalgebras over algebraically closed fields of characteristic 0 was studied by Kac [13] and Kantor [14]. Racine and Zelmanov [25] classified the finite-dimensional Jordan superalgebras of characteristic = 2 with semisimple even part, the case when even part is not semisimple was considered by Martinez and Zelmanov in [17] and Cantarini and Kac described all linearly compact simple Jordan superalgebras [2]. Simple noncommutative Jordan superalgebras were described by Pozhidaev and Shestakov in [23,24]. Representations of simple noncommutative superalgebras were described by Yu. Popov [22].Nowadays, a great interest is shown in the study of nonassociative algebras and superalgebras with derivations; in particular, Jordan algebras and superalgebras. For example, A. Popov determined the structure of differentiably simple Jordan algebras [21]; Kaygorodov and Yu. Popov determined the structure of Jordan algebras admitting invertible Leibniz derivations [9, 10]; Kaygorodov, Lopatin and Yu. Popov determined the structure of Jordan algebras admitting derivations with invertible values [8]; Barreiro, Elduque and Martínez described the derivations of Cheng-Kac Jordan superalgebra [1]; Kaygorodov, Shestakov and Zhelyabin studied generalized derivations of Jordan algebras and superalgebras [5-7, 27, 30]. Another interesting problem in the study of Jordan algebras and superalgebras is a description of maximal subalgebras and automorphisms [3,4].In this paper we are studying simple noncommutative Jordan superalgebras constructed in some papers of Pozhidaev and Shestakov. We describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras K 3 (α, β, γ) and D t (α, β, γ); and compute the derivations of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras.

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Noncommutative Jordan superalgebras of degree n > 2

Cited by 14 publications

References 21 publications

Simple finite-dimensional modular noncommutative Jordan superalgebras

Simple finite-dimensional modular noncommutative Jordan superalgebras

Simple finite-dimensional right-alternative superalgebras of Abelian type of characteristic zero

The Structure of Simple Noncommutative Jordan Superalgebras

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