Outer inheritance in jordan algebras

In the general structure theory of prime, simple, and division Jordan algebras developed by E.I. Zel’manov and applied in his solution of the Burnside problem, the Jordan classification in characteristic 2 required outer ideals of classical algebras. In this paper we show directly that over any ring of scalars the properties of nondegeneracy, strong primeness, unital simplicity, or divisibility are inherited by any ample outer ideal. This applies in particular to ample subspaces H ₀(A,*) of hermitian elements in associative algebras with involution.     

Ternary derivations of separable associative and jordan algebras

We describe the ternary derivations of finite-dimensional separable associative and Jordan algebras.     

Lie ideals of graded associative algebras

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Der _gr (A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Der _gr (A)).     

Zur konstruktion von Lie-Algebren aus Jordan-Tripelsystemen

In this paper we study certain Lie algebras which are constructed from the (-1)-eigenspaees of an involution of a Jordan algebra. The construction is a generalisation of the Koecher-Tits-construction. We give necessary conditions in terms of the Jordan algebras for the Lie algebras being simple. If the (-1)-spaces are Peirce-1/2-components then we obtain a close relation between the Lie algebras under consideration and the structure algebras of Jordan algebras. We finally give a list of those types of simple Lie algebras which can be formed by this construction; among them are Lie algebras of type E₆ and E₇.Of fundamental importance for our considerations is a close connection between the constructed Lie algebras and the standard imbeddings of Lie triple systems.     

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.     

Local and Subquotient Inheritance of Simplicity in Jordan Systems

In this paper we prove that the local algebras of a simple Jordan pair are simple. Jordan pairs all of which local algebras are simple are also studied, showing that they have a nonzero simple heart, which is described in terms of powers of the original pair. Similar results are given for Jordan triple systems and algebras. Finally, we characterize the inner ideals of a simple pair which determine simple subquotients, answering the question posed by O. Loos and E. Neher (1994, J. Algebra166, 255–295).     

Irreducible Lie-Yamaguti algebras of generic type

Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their inner derivation algebras are the algebraic counterparts of the isotropy irreducible homogeneous spaces.These systems splits into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types were classified in a previous paper through a generalized Tits Construction of Lie algebras. In this paper, the Lie-Yamaguti algebras of generic type are classified by relating them to several other nonassociative algebraic systems: Lie and Jordan algebras and triple systems, Jordan pairs or Freudenthal triple systems.     

Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a?J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a?P(a)²J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a?J with generalized inverse b the subtriple generated by {a, b} is strongly associative     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.     

Dialgebras and related triple systems

We consider some algebraical systems that lead to various nearly associative triple systems. We deal with a class of algebras which contains Leibniz-Poisson algebras, dialgebras, conformal algebras, and some triple systems. We describe all homogeneous structures of ternary Leibniz algebras on a dialgebra. For this purpose, in particular, we use the Leibniz-Poisson structure on a dialgebra. We then find a corollary describing the structure of a Lie triple system on an arbitrary dialgebra, a conformal associative algebra and a classical associative triple system. We also describe all homogeneous structures of an (ε, δ)-Freudenthal-Kantor triple system on a dialgebra.     

The Jordan socle and finitary Lie algebras

In this paper we introduce the notion of Jordan socle for nondegenerate Lie algebras, which extends the definition of socle given in [A. Fernández López et al., 3-Graded Lie algebras with Jordan finiteness conditions, Comm. Algebra, in press] for 3-graded Lie algebras. Any nondegenerate Lie algebra with essential Jordan socle is an essential subdirect product of strongly prime ones having nonzero Jordan socle. These last algebras are described, up to exceptional cases, in terms of simple Lie algebras of finite rank operators and their algebras of derivations. When working with Lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0, the exceptions disappear and the algebras of derivations are computed.     

Die struktur endlicher schwach abgeschlossener Jordanalgebren Teil II. Diskrete Jordanalgebren

In the preceding note [6] we reduced the study of continuous finite weakly closed Jordan algebras to real associative W^*-algebras of type II₁. Here we treat the remaining case of discrete finite weakly closed Jordan algebras and describe them completely by finite dimensional simple formally real Jordan algebras and by simple formally real Jordan algebras of quadratic forms of real Hilbert spaces. Jacobsons theory of Jordan algebras with minimum condition combined with W^*-algebra techniques constitutes an essential tool in the proof.     

Reelle Jordanalgebren mit endlicher Spur

We study real Jordan algebras of arbitrary dimension which admit an associative, positive definite trace form and have suitable continuity properties. We construct certain completions until we arrive at a class of Jordan algebras corresponding to associative W^*-algebras of finite type. For these Jordan algebras we derive some basic properties concerning positive elements and idempotents. In contrast to other related investigations exceptional Jordan algebras are not excluded.     

Additive Jordan derivations of reflexive algebras

Additive Jordan derivations of certain reflexive algebras are investigated. In particular, additive Jordan derivations of nest algebras on Banach spaces are shown to be additive derivations.     

Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse theusual process of constructing the Lie algebras of types G ₂and F ₄ as algebras of derivations of the splitoctonions or the exceptional Jordan algebra and instead beginwith their Dynkin diagrams and then construct the algebras togetherwith an action of the Lie algebras and associated Chevalley groups.This is shown to be a variation on a general construction ofall standard modules for simple Lie algebras and it is well suitedfor use in computational algebra systems. All the structure constantswhich occur are integral and hence the construction specialisesto all fields, without restriction on the characteristic, avoidingthe usual problems with characteristics 2 and 3.     

Prime Quotients of Jordan Systems and Lie Algebras

We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate.     

Representations of nongraded Lie algebras of block type

From his classification of quadratic conformal algebras corresponding to certain Hamiltonian pairs in integrable systems, Xu found a family of simple Lie algebras related to pairs of locally-finite derivations on certain commutative associative algebras. In this paper, we construct a large family of irreducible modules with four parameters for Xu's two-devivation algebras via the corresponding algebras of Weyl type. When the derivations are graded operators, we obtain a large family of uniformly-bounded irreducible weight modules for the Block algebras.     

Jordan operator algebras: basic theory

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.     

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.