Noncommutative Jordan Research Articles

Representations of simple noncommutative Jordan superalgebras II

In this article we continue the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. We prove that an irreducible unital bimodule over a simple noncommutative Jordan superalgebra U of degree ≥2 is either an irreducible bimodule over its symmetrized superalgebra U ( + ) or is equal to one of its Peirce components. Applying this result, we describe irreducible finite-dimensional representations of simple noncommutative Jordan superalgebras U ( V , f , ⋆ ) and K ( Γ n , A ) , completing the description of simple finite-dimensional unital bimodules over simple noncommutative Jordan superalgebras over an algebraically closed field of characteristic 0.

Multiplicative semiprimeness of strongly semiprime non-commutative Jordan algebras

In this paper, we prove that the multiplication algebra of a nondegenerate non-commutative Jordan algebra is semiprime as a consequence of the multiplicative primeness of strongly prime non-commutative Jordan algebras, obtained previously by the two first named authors. For that we prove first the coincidence of the McCrimmon radicals for a non-commutative Jordan algebra and its symmetrization, which allows us to describe the nondegenerate ones as a subdirect product of a family of strongly prime algebras. Indeed, we obtain that the minimal nondegenerate ideals of the symmetrization of any flexible algebra are ideals of the original algebra. This is based on its invariance under the derivations of the algebra, which requires working on vector spaces over a field of 0 characteristic.

Degenerations of Jordan Algebras and “Marginal” Algebras

We describe all degenerations of the variety [Formula: see text] of Jordan algebras of dimension three over [Formula: see text]. In particular, we describe all irreducible components in [Formula: see text]. For every [Formula: see text] we define an [Formula: see text]-dimensional rigid “marginal” Jordan algebra of level one. Moreover, we discuss marginal algebras in associative, alternative, left alternative, non-commutative Jordan, Leibniz and anticommutative cases.

The algebraic and geometric classification of nilpotent noncommutative Jordan algebras

We give algebraic and geometric classifications of complex four-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only [Formula: see text] non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of three rigid algebras and two one-parametric families of algebras.

Representations of simple noncommutative Jordan superalgebras I

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 Dt(α,β,γ) and K3(α,β,γ) and prove the Kronecker factorization theorem for superalgebras Dt(α,β,γ). In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.

Multiplicative primeness of strongly prime non-commutative Jordan algebras

We prove that the multiplication algebra of any strongly prime non-commutative Jordan algebra is a prime algebra.

Simple finite-dimensional modular noncommutative Jordan superalgebras

We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic p>2. The case of characteristic 0 was considered by the authors in the previous paper [21]. 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).

The Structure of Simple Noncommutative Jordan Superalgebras

In this paper we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras $K_3(\alpha,\beta,\gamma)$ and $D_t(\alpha,\beta,\gamma)$ and compute the derivations of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras.

A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations

Moens proved that a finite-dimensional Lie algebra over a field of characteristic zero is nilpotent if and only if it has an invertible Leibniz-derivation. In this article we prove the analogous results for finite-dimensional Malcev, Jordan, (−1,1)-, right alternative, Zinbiel and Malcev-admissible noncommutative Jordan algebras over a field of characteristic zero. Also, we describe all Leibniz-derivations of semisimple Jordan, right alternative and Malcev algebras.

Simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0

We classify the central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0. As a corollary, we describe the Poisson brackets on the simple finite-dimensional Jordan superalgebras of characteristic 0.

On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

Let be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .

Algebras of Hamieh and Abbas Used in the Dirac Equation

Hamieh and Abbas [1] propose using a 3-dimensional real algebra in a solution of the Dirac equation. We show that this algebra, denoted by , belongs to a large class of quadratic Jordan algebras with subalgebras isomorphic to the complex numbers and that the spinor matrices associated with the solution of the Dirac equation generate a six-dimensional real noncommutative Jordan algebra.

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.

Noncommutative Jordan superalgebras of degree n ≥ 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a “nodal” case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.

Noncommutative Jordan superalgebras of degree n > 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a “nodal” case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.

Nonassociative unitary Banach algebras

We generalize the theory of associative unitary normed algebras to the setting of noncommutative Jordan algebras. Special attention is devoted to the case of alternative algebras.

Multialternating Jordan polynomials and codimension growth of matrix algebras

Let R be the Jordan algebra of k × k matrices over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables of order k 2 and we prove that f is not a polynomial identity of R. We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f, we are able to prove that the exponential rate of growth of the sequence of Jordan codimensions of R in precisely k 2.

Formula omitted]-Balanced Freudenthal Kantor triple systems and noncommutative Jordan algebras

A noncommutative Jordan algebra of a specific type is attached to any ( − 1 , − 1 ) -balanced Freudenthal Kantor triple system, in such a way that the triple product in this system is determined by the binary product in the algebra. Over fields of characteristic zero, the simple noncommutative Jordan algebras of this type are classified.

IDENTITIES FOR ALGEBRAS OBTAINED FROM THE CAYLEY-DICKSON PROCESS

The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 n for all n ≥ 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees ≤ 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree ≤ 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.

On the Variety Determined by Symmetric Quadratic Algebras

We consider some polynomial identities of degree ≤5 which are satisfied by all symmetric quadratic algebras. We call rings satisfying these identities generalized quadratic rings, or GQ-rings. We show that when the ring is not flexible, these identities are enough to make the ring quadratic over its center. Therefore, simple nonflexible GQ-rings are symmetric quadratic algebras over their center, which is a field. For prime GQ-rings, the center has no nonzero zero divisors. Prime GQ-rings, which are not flexible, are subrings of the quadratic algebra formed by extending the center to its field of quotients. Flexible GQ-rings are noncommutative Jordan rings which satisfy a quadratic condition over their commutative center. We show that any semi-prime GQ-ring is a subdirect sum of a noncommutative Jordan ring (which satisfies a quadratic condition over its commutative center) and a nonflexible ring (which is symmetric quadratic over its center).