Noncommutative Jordan superalgebras of degree n ≥ 2

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

doi:10.1134/s1064562410020110

Cited by 4 publications

(13 citation statements)

References 6 publications

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“…All algebras and vector spaces in this paper are over a field F of characteristic not 2. As this work follows up on and uses the formulas from the papers [27] and [29], the operators in this paper act on the right.…”

Section: Preliminariesmentioning

confidence: 99%

“…On the other hand, a commutative noncommutative Jordan (super)algebra is Jordan -hence the name. In fact, the relation between noncommutative and commutative Jordan (super)algebras can be made precise using the notion of the symmetrized algebra: 27]). U is a noncommutative Jordan (super)algebra if and only if U is a flexible (super)algebra such that its symmetrized (super)algebra U (+) is a Jordan (super)algebra.…”

Section: Preliminariesmentioning

confidence: 99%

“…Here we recall some usual facts about Peirce decomposition, which is going to be our main tool during the paper. A more detailed exposition of Peirce decomposition can be found in [17], [27].…”

Section: Peirce Decompositionmentioning

confidence: 99%

“…Proof. From [27], Lemma 4.1, it follows that for i ∈ {0, 2} and x, y, z ∈ U 1 the following relation holds:…”

Section: Peirce Decompositionmentioning

confidence: 99%

“…Simple finite-dimensional central noncommutative Jordan superalgebras were described by Pozhidaev and Shestakov in [27,29,30]. Structure and derivations of low-dimensional simple noncommutative Jordan algebras were studied in [10].…”

Section: Introductionmentioning

confidence: 99%

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Representations of simple noncommutative Jordan superalgebras I

Popov

2020

Journal of Algebra

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In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree ≥ 3 we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras D t (α, β, γ) and K 3 (α, β, γ) and prove the Kronecker factorization theorem for superalgebras D t (α, β, γ). In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Peirce Decompositionmentioning

confidence: 99%

“…Proof. From [27], Lemma 4.1, it follows that for i ∈ {0, 2} and x, y, z ∈ U 1 the following relation holds:…”

Section: Peirce Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Representations of simple noncommutative Jordan superalgebras I

Popov

2020

Journal of Algebra

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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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Hom-Alternative, Hom-Malcev and Hom-Jordan Superalgebras

Abdaoui

Ammar

Makhlouf

2016

Bull. Malays. Math. Sci. Soc.

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Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras are Z 2 -graded generalizations of Hom-alternative, Hom-Malcev and Hom-Jordan algebras, which are Hom-type generalizations of alternative, Malcev and Jordan algebras. In this paper we prove that Hom-alternative superalgebras are Hom-Malcevadmissible and are also Hom-Jordan-admissible. Home-type generalizations of some well known identities in alternative superalgebras, including the Z 2 -graded Bruck-Kleinfled function are obtained.

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Noncommutative Jordan superalgebras of degree n ≥ 2

Cited by 4 publications

References 6 publications

Representations of simple noncommutative Jordan superalgebras I

Representations of simple noncommutative Jordan superalgebras I

The Structure of Simple Noncommutative Jordan Superalgebras

Hom-Alternative, Hom-Malcev and Hom-Jordan Superalgebras

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