Isotopes of prime (−1, 1)-and Jordan algebras

We deal with adjoint commutator and Jordan algebras of isotopes of prime strictly (−1, 1)-algebras. It is proved that a system of identities of the form [x ₁, x ₂, x ₂, x ₃,…, x _n ] for n = 2,.…, 5 is discernible on isotopes of prime (−1, 1)-algebras. Also it is shown that adjoint Jordan algebras for suitable isotopes of prime (−1, 1)-algebras may possess distinct sets of identities. In particular, isotopes of a prime Jordan monster have different sets of identities in general.     

Jordan centroids

Central simple triples are important for the classification of prime Jordan triples of Clifford type in arbitrary characterstics. For triples and pairs (or even for unital Jordan algebras of characteristic 2), there is no workable notion of center, and the concept of “central simple” system must be understood as “centroid-simple”. The centroid of a Jordan system (algebra, triple, or pair) consists of the “natural” scalars for that system: the largest unital, commutative ring Γ such that the system can be considered as a quadratic Jordan system over Γ. In this paper we will characterize the centroids of the basic simple Jordan algebras, triples, and pairs. (Consideration of the tangled ample outer ideals in Jordan algebras of quadratic forms will be left to a separate paper.) A powerful tool is the Eigenvalue Lemma, that a centroidal transformation on a prime system over φ which has an eigenvalue α in φ must actually be scalar multiplication by α. An important consequence is that a prime system over φ with reduced elements P_xJ = φx (or which grows reduced elements under controlled conditions) must already be central, Γ = φ.     

The Tits–Kantor–Koecher Construction and Birepresentations of the Jordan Superpair SH(1, n)

Abstract

We introduce the concept of bimodule over a Jordan superpair and the Tits– Kantor–Koecher construction for bimodules. Using the construction we obtain the classification of irreducible bimodules over the Jordan superpair SH(1, n). We also prove semisimplicity for a class of finite dimensional SH(1, n)-bimodules for n ≥ 3.     

Generalized koszul complexes and the extension functor

We show that Jordan triple homomorphisms and derivations between prime special quadral Jordan triple systems on which Zel’manov polynomials do not vanish extend to associative homomorphis and derivations of associative ?-envelopes (either associative triple systems or Z₂-graded associative algebras). This generalizes results of Zel'manov and McCrimmon for Jordan algebras (which in turn generalized results of Martindale).     

Hyperinvariant subspaces ofC 0-operators over a multiply connected region

In this paper we find a relation between the lattice of hyperinvariant subspaces of an operatorT of classC ₀ over a multiply connected region and that of its Jordan modelT. It is shown that, generally, the lattice corresponding toT can be identified with a retract of that corresponding toT. Thus the Jordan model has the smallest lattice of hyperinvariant subspaces in a given quasisimilarity class.     

Additive Derivations on Jordan Banach Pairs

Abstract

We show the invariance of “almost all” primitive ideals under additive derivations on a Jordan Banach pair and we extend the well known result of Johnson and Sinclair to the Jordan Banach pairs framework.     

Prime Algebras Connected With Monsters

We study the overalgebras and the ideals of the Jordan algebras possessing prime (?1, 1)-envelopings. If a Jordan algebra possesses a prime nonassociative (?1, 1)-enveloping then we prove that it is also prime; furthermore, its every ideal is a prime algebra. In particular, the overalgebras and metaideals of Jordan monsters are prime.     

NOETHERIAN UNIQUE FACTORIZATION SEMIGROUP ALGEBRAS

We investigate when semigroup algebras K[S] of submonoids S of torsion free polycyclic-by-finite groups G are Noetherian unique factorization rings in the sense of Chatters and Jordan, that is, every prime ideal contains a principal height one prime ideal. For the group algebra K[G] this problem was solved by Brown.     

INDECOMPOSABLE INJECTIVE MODULES AND A THEOREM OF KAPLANSKY

Abstract

Every tripotent e of a generalized Jordan triple system of second order uniquely defines a decomposition of the space of the triple into a direct sum of eight components. This decomposition is a generalization of the Peirce decomposition for the Jordan triple system. The relations between components are studied in the case when e is a left unit.     

Prime virtually semisimple modules and rings

Abstract

We show that the multiplication algebra of a nondegenerate Jordan algebra is a semiprime algebra.     

Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.     

Hessian Potentials with Parallel Derivatives

Let ${U \subset \mathbb A^n}$ be an open subset of real affine space. We consider functions ${F: U \to \mathbb R}$ with non-degenerate Hessian such that the first or the third derivative of F is parallel with respect to the Levi-Civita connection defined by the Hessian metric ${F{^\prime{^\prime}}}$ . In the former case the solutions are given precisely by the logarithmically homogeneous functions, while the latter case is closely linked to metrised Jordan algebras. Both conditions together are related to unital metrised Jordan algebras. Both conditions combined with convexity provide a local characterization of canonical barriers on symmetric cones.     

Decidability for modules over a group ring - III

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)).     

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.     

The orthogonality relation classifies finite-dimensional formally real simple Jordan algebras

Abstract

It is shown that a finite-dimensional formally real simple Jordan algebra is completely determined by the relation of Jordan-orthogonality.

Communicated by Prof. Alberto Elduque     

Prime Jordan algebras and the Kantor construction

Two new constructions of prime Jordan algebras containing nonzero trivial elements are presented. It is proved that a Jordan superalgebra of Poisson brackets is a homomorphic image of a special one.Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 301–316, May–June, 1994.     

A Kronecker factorization theorem for the exceptional Jordan superalgebra

We prove that a Jordan superalgebra J containing the 10-dimensional exceptional Kac superalgebra K₁₀ is isomorphic to (K₁₀⊗_FS)⊕J′, where S is an associative commutative algebra.     

Homotope polynomial identities in prime Jordan systems

We prove an analogue of the Posner-Rowen theorem for strongly prime Jordan pairs and triple systems: the central closure of a strongly prime Jordan system satisfying a homotope polynomial identity is simple with finite capacity. We also prove that if a Jordan system satisfies a homotope polynomial identity it also satisfies a strict homotope polynomial identity.     

On Composition series relative to a module

ABSTRACT

A commutative algebra with the identity (a * b) * (c * d) ? (a * d) * (c * b) = (a, b, c) * d ? (a, d, c) * b is called Novikov–Jordan. Example: K[x] under multiplication a * b = ?(ab) is Novikov–Jordan. A special identity for Novikov–Jordan algebras of degree 5 is constructed. Free Novikov–Jordan algebras with q generators are exceptional for any q ≥ 1.