Noncommutative Jordan Algebras Research Articles

Nodal noncommutative Jordan algebras and simple Lie algebras of characteristic 𝑝

Nodal noncommutative Jordan algebras and simple Lie algebras of characteristic 𝑝

A Generalization of Noncommutative Jordan Algebras

and consider algebras satisfying (1) and (3). In ?1 we show that such algebras of characteristic not 2 or 3 are strictly power-associative, and in ?2 we establish several properties of the submodules A e(X) of these algebras. In the last section we use these results and follow the arguments in [6] to show the existence of an identity element in the finite dimensional semi-simple case. Then the results of [6] and [7] allow us to obtain the following theorems.

A generalization of non-commutative Jordan algebras

A generalization of non-commutative Jordan algebras

Restricted Noncommutative Jordan Algebras of Characteristic p

Restricted Noncommutative Jordan Algebras of Characteristic p

Restricted noncommutative Jordan algebras of characteristic 𝑝

In [7]2 we obtained a satisfactory structure theory for these algebras of finite dimension over F of characteristic 0 by proving that they are trace-admissible. Recent examples by L. A. Kokoris [4] show that the algebras satisfying (1) and (2) are not in general traceadmissible if F is of characteristic p > 2. It is natural to seek a generalization of (commutative) Jordan algebras of characteristic p > 2 in which the algebras are trace-admissible. We find this generalization in the algebras satisfying (1), (2), and

Noncommutative Jordan Algebras of Characteristic 0

Noncommutative Jordan Algebras of Characteristic 0

Noncommutative Jordan algebras of characteristic 0

That is, A is flexible (a weaker condition than commutativity). If a unity element is adjoined to A in the usual fashion, then a necessary and sufficient condition that (2) be satisfied in the extended algebra is that both (2) and (3) be satisfied in A. We define a noncommutative Jordan algebra A over an arbitrary field F to be an algebra satisfying (1) and (4). These algebras include the best-known nonassociative algebras (Jordan, alternative, quasiassociative, and-trivially-Lie algebras). In 1948 they were studied briefly by A. A. Albert in [1, pp. 574-575],2 but the assumptions (1) and (4) seemed to him inadequate to yield a satisfactory theory, and he restricted his attention to a less general class of algebras which he called standard. In this paper, using Albert's method of traceadmissibility3 and his results for trace-admissible algebras, we give a