The Structure of Simple Noncommutative Jordan Superalgebras

Abstract

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.

Non-associative algebraic structures: classification and structure

“…The present part is based on the papers written together with Antonio Jesús Calderón, Artem Lopatin, Bakhrom Omirov, Gulkhayo Solijanova, Luisa Camacho, Pasha Zusmanovich and Yury Popov [44,59,159,180].…”

“…In our paper with Lopatin and Popov, we study simple noncommutative Jordan superalgebras constructed in some papers by Pozhidaev and Shestakov [225]. We describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras K 3 (α, β, γ) and D t (α, β, γ) [159]. We also compute the derivations of the nontrivial simple finitedimensional noncommutative Jordan superalgebras [159, Theorems 9 and 10]].…”

Representations of simple noncommutative Jordan superalgebras I

“…In the work [20] the universal envelopes for one-sided representations of simple Jordan superalgebras were constructed, and also irreducible one-sided bimodules over the superalgebras D t were described. Shtern [36] classified irreducible bimodules over the exceptional Kac superalgebra K 10 .…”

“…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]. Generalizations of derivations of simple noncommutative Jordan superalgebras were studied in [11].…”

The algebraic and geometric classification of nilpotent noncommutative Jordan algebras

“…However, the structure theory of this class is far from being nice. Nevertheless, a certain progress was made in the study of structure theory of noncommutative Jordan algebras and superalgebras (see, for example [5,26,[34][35][36][37]). In this paper, our goal is to obtain a complete algebraic and geometric description of the variety of all 4dimensional nilpotent noncommutative Jordan algebras over the complex field.…”