Noncommutative Jordan Algebras Research Articles

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.

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.

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.

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.

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.

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.

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.

Okubo algebras in characteristic 3 and their automorphisms

Associated to any eight-dimensional non-unital composition algebra with associative norm, there are outer automorphisms of order 3 of the corresponding spin group, such tiat the fixed subgroup is the automorphism group of the composition algebra. Over fields of characteristic ≠ 3 these are simple algebraic groups of types G 2 or A 2, related respectively to the para-octonion and the Okubo algebras A connection between the Okubo algebras over fields of characteristic 3 with some simple noncommutative Jordan algebras will be used to compute explicitly the automorphism groups and Lie algebras of derivations of these algebras. In contrast to the other characteristics, ths groups will no longer be of type A 2 and will either be trivial or contain a large unipotent radical.

Algèbres génétiques dimensionnellement nilpotentes

The structure of dimensionally nilpotent Lie algebras was studied by G. F. Leger and P. L. Manley for characteristic zero and by J. M. Osborn for characteristic p > 5. Moreover, J. M. Osborn studied dimensionally nilpotent commutative and noncommutative Jordan algebras. In this paper we give some results concerning Bernstein and genetic algebras which are dimensionally nilpotent in connection with Osborn's results about Jordan algebras.

Noncommutative matrix jordan algebras, cayley-dickson algebras, and schafer's theorem1

We provide a construction of noncommutative Jordan algebras of degree two. The construction can be iterated, and we show that after the first few iterations no new derivations arise. The relationship between this iterative process and the Cayley-Dickson process is studied, and the result on deriva¬tions is used to obtain a generalization of Schafer's classical theorem on the derivation algebras of Cayley-Dickson algebras.

Dimensionally Nilpotent Jordan Algebras

An algebra $A$ of dimension $n$ is called dimensionally nilpotent if it has a nilpotent derivation $\partial$ with the property that ${\partial ^{n - 1}} \ne 0$. Here we show that a dimensionally nilpotent Jordan algebra $A$ over a perfect field of characteristic not 2 or 3 is either (i) nilpotent, or (ii) one-dimensional modulo its maximal nilpotent ideal. This result is also extended to noncommutative Jordan algebras.

Dimensionally nilpotent Jordan algebras

An algebra A A of dimension n n is called dimensionally nilpotent if it has a nilpotent derivation ∂ \partial with the property that ∂ n − 1 ≠ 0 {\partial ^{n - 1}} \ne 0 . Here we show that a dimensionally nilpotent Jordan algebra A A over a perfect field of characteristic not 2 or 3 is either (i) nilpotent, or (ii) one-dimensional modulo its maximal nilpotent ideal. This result is also extended to noncommutative Jordan algebras.

Continuity of derivations on Mal'cev H*-algebras

A long time ago the concept of H*-algebra was introduced by Ambrose in [1] where the structure of complex associative H*-algebras was given. Since then this theory was extended to such classical types of non-associative algebras as alternative algebras (in [6]), Jordan algebras (in [5, 13, 14]), non-commutative Jordan algebras (in [5]), Lie algebras (in [3, 9, 10]) and Mal'cev algebras (in [2]).

Jordan radicals of u-algebras

In this paper we show that strong noncommutative Jordan algebras R over an arbitrary ring of scalars having the alternator mappings y,y,-1 as Jordon derivations are U-algebras, algebras such that Uablpar;crpar; lies in the Jordan ideal generated by a. For any U-algebra R we relate the radical theories of R and R+. Our main result is that any radical property p′ of U-algebras such that P′-radR⊂ p-radR+. If p is nondegenerate the P′ is nondegenerate and P′-radR=p-radR+. This applies in particular to the McCrimmon, locally nilpotent, nil, Jacobson and Brown-McCoy radicals of Jordan algebras

On the socle of a noncommutative Jordan algebra

In this paper we study some questions related to the socle of a nondegenerate noncommutative Jordan algebra. First we show that elements of finite rank belong to the socle, and that every element in the socle is von Neumann regular and has finite spectrum. Next we show that for Jordan Banach algebras the socle coincides with the maximal von Neumann regular ideal. For a nondegenerate noncommutative Jordan algebra, the annihilator of its socle can be regarded as a radical which is, generally, larger than Jacobson radical. Moreover, a nondegenerate noncommutative Jordan algebra whose socle has zero annihilator is isomorphic to a subdirect sum of primitive algebras having nonzero socle (which were described in [4]). Finally, these results are specialized to the particular case of an alternative algebra.

On lie algebras associated with nodal noncommutative Jordan algebras

On lie algebras associated with nodal noncommutative Jordan algebras

Primitive Noncommutative Jordan Algebras with Nonzero Socle

Let $A$ be a nondegenerate noncommutative Jordan algebra over a field $K$ of characteristic $\ne 2$. Defining the socle $S(A)$ of $A$ to be the socle of the plus algebra ${A^ + }$, we prove that $S(A)$ is an ideal of $A$; then we prove that if $A$ has nonzero socle, $A$ is prime if and only if it is primitive, extending a result of Osborn and Racine [6] for the associative case. We also describe the prime noncommutative Jordan algebras with nonzero socle and in particular the simple noncommutative Jordan algebras containing a completely primitive idempotent. In fact we prove that a nondegenerate prime noncommutative Jordan algebra with nonzero socle is either (i) a noncommutative Jordan division algebra, (ii) a simple flexible quadratic algebra over an extension of the base field, (iii) a nondegenerate prime (commutative) Jordan algebra with nonzero socle, or (iv) a $K$-subalgebra of ${L_W}{(V)^{(\lambda )}}$ containing ${F_W}(V)$ or of $H{({L_V}(V), * )^{(\lambda )}}$ containing $H({F_V}(V), * )$ where in the first case $(V,W)$ is a pair of dual vector spaces over an associative division $K$-algebra $D$ and $\lambda \ne 1/2$ is a central element of $D$, and where in the second case $V$ is self-dual with respect to an hermitian inner product $(|),D$ has an involution $\alpha \to \bar \alpha$ and $\lambda \ne 1/2$ is a central element of $D$ with $\lambda + \bar \lambda = 1$.

Primitive noncommutative Jordan algebras with nonzero socle

Let A A be a nondegenerate noncommutative Jordan algebra over a field K K of characteristic ≠ 2 \ne 2 . Defining the socle S ( A ) S(A) of A A to be the socle of the plus algebra A + {A^ + } , we prove that S ( A ) S(A) is an ideal of A A ; then we prove that if A A has nonzero socle, A A is prime if and only if it is primitive, extending a result of Osborn and Racine [6] for the associative case. We also describe the prime noncommutative Jordan algebras with nonzero socle and in particular the simple noncommutative Jordan algebras containing a completely primitive idempotent. In fact we prove that a nondegenerate prime noncommutative Jordan algebra with nonzero socle is either (i) a noncommutative Jordan division algebra, (ii) a simple flexible quadratic algebra over an extension of the base field, (iii) a nondegenerate prime (commutative) Jordan algebra with nonzero socle, or (iv) a K K -subalgebra of L W ( V ) ( λ ) {L_W}{(V)^{(\lambda )}} containing F W ( V ) {F_W}(V) or of H ( L V ( V ) , ∗ ) ( λ ) H{({L_V}(V), * )^{(\lambda )}} containing H ( F V ( V ) , ∗ ) H({F_V}(V), * ) where in the first case ( V , W ) (V,W) is a pair of dual vector spaces over an associative division K K -algebra D D and λ ≠ 1 / 2 \lambda \ne 1/2 is a central element of D D , and where in the second case V V is self-dual with respect to an hermitian inner product ( | ) , D (|),D has an involution α → α ¯ \alpha \to \bar \alpha and λ ≠ 1 / 2 \lambda \ne 1/2 is a central element of D D with λ + λ ¯ = 1 \lambda + \bar \lambda = 1 .