Simple Jordan Superalgebras Research Articles

Differential algebras and simple Jordan superalgebras

In [14], a new example is constructed of a unital simple special Jordan superalgebra J over the field of reals. It turns out that J is a subsuperalgebra of a Jordan superalgebra of vector type but it cannot be isomorphic to a superalgebra of such a type. Moreover, the superalgebra of fractions of J is isomorphic to a Jordan superalgebra of vector type. In the present article, we find a similar example of a Jordan superalgebra. It is constructed over a field of characteristic 0 in which the equation t2 + 1 = 0 has no solutions.

δ-Superderivations of simple finite-dimensional Jordan and Lie superalgebras

We introduce the concept of a $\delta$-superderivation of a superalgebra. $\delta$-Derivations of Cartan-type Lie superalgebras are treated, as well as $\delta$-superderivations of simple finite-dimensional Lie superalgebras and Jordan superalgebras over an algebraically closed field of characteristic 0. We give a complete description of $\{1}{2}$-derivations for Cartan-type Lie superalgebras. It is proved that nontrivial $\delta$-(super)derivations are missing on the given classes of superalgebras, and as a consequence, $\delta$-superderivations are shown to be trivial on simple finite-dimensional noncommutative Jordan superalgebras of degree at least 2 over an algebraically closed field of characteristic 0. Also we consider $\delta$-derivations of unital flexible and semisimple finite-dimensional Jordan algebras over a field of characteristic not 2.

Maximal subalgebras of Jordan superalgebras

The maximal subalgebras of the finite-dimensional simple special Jordan superalgebras over an algebraically closed field of characteristic 0 are studied. This is a continuation of a previous paper by the same authors about maximal subalgebras of simple associative superalgebras, which is instrumental here.

Centroids of Quadratic Jordan Superalgebras

The centroid of a Jordan superalgebra consists of the natural “superscalar multiplications” on the superalgebra. A philosophical question is whether the natural concept of “scalar” in the category of superalgebras should be that of superscalars or ordinary scalars. Basic examples of Jordan superalgebras are the simple Jordan superalgebras with semisimple even part, which were classified over an algebraically closed field of characteristic ≠ 2 by Racine and Zelmanov. Here, we determine the centroids of the analog of these superalgebras over general rings of scalars and show that they have no odd centroid, suggesting that ordinary scalars are the proper concept.

δ-Derivations of simple finite-dimensional Jordan superalgebras

We describe non-trivial $\delta$-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero $\delta$-derivations are shown to be missing for $\delta\neq 0,{1}{2},1$, and we give a complete account of ${1}{2}$-derivations.

Classification of linearly compact simple Jordan and generalized Poisson superalgebras

We classify all linearly compact simple Jordan superalgebras over an algebraically closed field of characteristic zero. As a corollary, we deduce the classification of all linearly compact unital simple generalized Poisson superalgebras.

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.

REPRESENTATIONS OF EXCEPTIONAL SIMPLE JORDAN SUPERALGEBRAS OF CHARACTERISTIC 3#

ABSTRACT Representations of simple Jordan superalgebras of Hermitian 3 × 3 matrices over the exceptional simple alternative superalgebras B (1,2) and B (4,2) of characteristic 3 are studied. Every irreducible bimodule over these superalgebras up to isomorphism is either a regular bimodule or its opposite. As corollaries,some analogues of the Kronecker factorization theorem are proved for Jordan superalgebras that contain H3(B (1,2)) and H3(B(4,2)).

Simple Special Jordan Superalgebras with Associative Even Part

We describe the simple special unital Jordan superalgebras with associative even part A whose odd part M is an associative module over A. We prove that each of these superalgebras, not isomorphic to a superalgebra of nondegenerate bilinear superform, is isomorphically embedded into a twisted Jordan superalgebra of vector type. We exhibit a new example of a simple special Jordan superalgebra. We also describe the superalgebras such that M∩ [A,M]≠ 0.

Simple Jordan superalgebras with semisimple even part

Simple Jordan superalgebras with semisimple even part

Specializations of Jordan Superalgebras

AbstractIn this paper we study specializations and one-sided bimodules of simple Jordan superalgebras.

A construction of simple Jordan superalgebra of F type from a Jordan–Lie triple system

We study in this paper a construction of simple Jordan superalgebras from certain triple systems, in particular, we investigate F type from a Jordan–Lie triple system minutely.

A new construction of the Kac Jordan superalgebra

We present an elementary construction of the 10-dimensional simple Jordan superalgebra K 10 of Kac using the 3-dimensional tiny Kaplansky superalgebra. This new realization of K 10 enables us to derive many of its properties.

Simple Special Jordan Superalgebras with Associative Nil-Semisimple Even Part

We describe unital simple special Jordan superalgebras with associative nil-semisimple even part. In every such superalgebra J=A+M, either M is an associative and commutative A-module, or the associator space (A,A,M) coincides with M. In the former case, if JJ is not a superalgebra of the non-degenerate bilinear superform then its even part A is a differentiably simple algebra and its odd part M is a finitely generated projective A-module of rank 1. Multiplication in M is defined by fixed finite sets of derivations and elements of A. If, in addition, M is one-generated then the initial superalgebra is a twisted superalgebra of vector type. The condition of being one-generated for M is satisfied, for instance, if A is local or isomorphic to a polynomial algebra. We also give a description of superalgebras for which (A,A,M)≠0 and M⋂[A,M]≠0, where [,] is a commutator in the associative enveloping superalgebra of J. It is shown that such each infinite-dimensional superalgebra may be obtained from a simple Jordan superalgebra whose odd part is an associative module over the even.

Simple Finite-Dimensional Jordan Superalgebras of Prime Characteristic

Simple Finite-Dimensional Jordan Superalgebras of Prime Characteristic

Graded simple Jordan superalgebras of growth one

Introduction Structure of the even part Cartan type Even part is direct sum of two loop algebras $A$ is a loop algebra $J$ is a finite dimensional Jordan superalgebra or a Jordan superalgebra of a superform The main case Impossible cases Bibliography.

Primitive Superalgebras with Superinvolution

Our main purpose is to provide for primitive associative superalgebras a structure theory analogous to that for algebras [5,6,10] and to classify primitive superrings with superinvolution having a minimal one-sided superideal. We were led to this problem by our work on finite dimensional central simple Jordan superalgebras over fields of characteristic not 2 [9] (see also [7]). Of course, just as symmetric elements give rise to Jordan superalgebras, skewsymmetric elements give rise to Lie superalgebras [8,4]. The results and methods are closely related to those of structure theory of associative rings and central simple associative algebras with involution [5, Chap. I;6, Chaps. II, III;1, Chap. X;10, Chap. 2]. Some of the results have been announced in [13].

Prime alternative superalgebras of arbitrary characteristic

Simple nonassociative alternative superalgebras are classified. Any such superalgebra either is trivial (i.e., has zero odd part) or has characteristic 2 or 3 and is isomorphic over its center to a superalgebra of one of the following five types: in characteristic 3, these are two superalgebras of dimensions 3 and 6 and a “twisted superalgebra of vector type,” which either is infinite-dimensional or has dimension 2·3n; in characteristic 2, those are either a Cayley-Dixon algebra with a grading induced by the Cayley-Dixon process or a “double Cayley-Dixon algebra.” Under certain constraints on the structure of even parts, we also give a description of prime nonassociative alternative nontrivial superalgebras in terms of central orders of simple superalgebras. The simple superalgebras of dimensions 3 and 6 are then used to construct simple Jordan superalgebras of characteristic 3 and of dimensions 12 and 21, respectively.

Graded Polynomial Identities of the Jordan Superalgebra of a Bilinear Form

LetWsp,2q=W0(p)⊕W1(2q)be a direct sum of two vector spaces of dimensionpand 2q, respectively, over a fieldkof characteristic zero,p=2,3,…,∞;q=1,2,…,∞; and let 〈x,y〉 be a nondegenerate bilinear form onWsp,2qwhich is symmetric onW0(p)and skew-symmetric onW1(2q)and such that 〈W0(p),W1(2q)〉=〈W1(2q),W0(p)〉=0. Then the vector spaceBsp,2q=k⊕Wsp,2qequipped with the product (α+v)(β+u)=(αβ+〈v,u〉)+(αu+βv), α,β∈k,v,u∈Wsp,2q, is a simple Jordan superalgebra, withk+W0(p)as its even component andW1(2q)as the odd one. In this paper explicit finite bases for graded polynomial identities of the series of Jordan superalgebrasBsp,2q,p=2,3,…,∞;q=1,2,…,∞; are found.

Classification of simple z-graded lie superalgebras and simple jordan superalgebras

(1977). Classification of simple z-graded lie superalgebras and simple jordan superalgebras. Communications in Algebra: Vol. 5, No. 13, pp. 1375-1400.