A geometric construction of Moufang planes

An algebraic construction of degree 3 Jordan algebras (including the exceptional one) as trace 0 elements in a degree 4 Jordan algebra is translated to give a geometric construction of Barbilian planes coordinatized by composition algebras (including the Moufang plane) as skew polar line pairs and points on the quadratic surface determined by a polarity of projective 3-space over a smaller composition algebra.     

Universal envelopes of Malcev Algebras: Pointed Moufang bialgebras

Under study are the pointed unital coassociative cocommutative Moufang H-bialgebras. We prove an analog of the Cartier-Kostant-Milnor-Moore theorem for weakly associative Moufang H-bialgebras. If the primitive elements commute with group-like elements then these Moufang H-bialgebras are isomorphic to the tensor product of a universal enveloping algebra of a Malcev algebra and a loop algebra constructed by a Moufang loop.     

Solvable Jordan algebras of compact operators

It is proved that a Jordan algebra of compact operators which is closed is either an Engel Jordan algebra, or contains a nonzero finite rank operator; Moreover, it is showed that any solvable Jordan algebra of compact operators on an infinite dimensional Banach space is triangularizable.     

Distributions vectorielles homogènes sur une algèbre de Jordan

We study distributions on a Euclidean Jordan algebra V with values in a finite dimensional representation space for the identity component G of the structure group of V and homogeneous equivariance condition. We show that such distributions exist if and only if the representation is spherical, and that then the dimension of the space of these distributions is r+1 (where r is the rank of V). We give also construction of these distributions and of those that are invariant under the semi-simple part of G.     

Trace and spectrum preserving linear mappings in Jordan-Banach algebras

In the first section we define the trace on the socle of a Jordan-Banach algebra in a purely spectral way and we prove that it satisfies several identities. In particular this trace defines the Faulkner bilinear form. In the second section, using analytic tools and the properties of the trace, we prove that a spectrum preserving linear mapping fromJ ontoJ ¹, whereJ andJ ¹ are semisimple Jordan-Banach algebras, is not far from being a Jordan isomorphism. It is in particular a Jordan isomorphism ifJ ¹ is primitive with non-zero socle.     

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.     

On algebras satisfying $x^2x^2=N(x)x$

The commutative algebras satisfying the “adjoint identity”: , where N is a cubic form, are shown to be related to a class of generically algebraic Jordan algebras of degree at most 4 and to the pseudo-composition algebras. They are classified under a nondegeneracy condition. As byproducts, the associativity of the norm of any pseudo-composition algebra is proven and the unital commutative and power-associative algebras of degree are shown to be Jordan algebras. Received January 26, 1999; in final form August 26, 1999 / Published online July 3, 2000     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

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.     

Martindale quotients of Jordan algebras

In this paper we introduce Martindale quotients of Jordan algebras over arbitrary rings of scalars with respect to denominator filters of ideals. For any denominatored algebra, we show the existence of maximal Martindale quotients naturally containing all Martindale quotients of the algebra with respect to the given denominator filter.     

Scorza varieties and Jordan algebras

In this paper, I give two very direct proves of the correspondance between a geometric object (Scorza varieties) and an algebraic one (Jordan algebras). I also give a short proof of the homogeneity of Scorza varieties, and a new and very simple proof of properties of the automorphism group of a Jordan algebra.     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

Polynomial identities of bicommutative algebras,Lie and Jordan elements

ABSTRACT

An algebra with identities a(bc)?=?b(ac), (ab)c?=?(ac)b is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan.     

A note on the Ostrowski-Schneider type inertia theorem in Euclidean Jordan algebras

In a recent paper [7], Gowda et al. extended Ostrowski-Schneider type inertia results to certain linear transformations on Euclidean Jordan algebras. In particular, they showed that In(a)=In(x) whenever a°x>0 by the min-max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined by

In(x):=(π(x),ν(x),δ(x)),     

Springer forms and the first Tits construction of exceptional Jordan division algebras

In this paper, a certain quadratic form, originally due to Springer [15], which may be associated with any separable cubic subfield living inside an exceptional simple Jordan algebra is related to the coordinate algebra of an appropriate scalar extension. We use this relation to show that, in the presence of the third roots of unity, exceptional Jordan division algebras arising from the first Tits construction are precisely those where reducing fields and splitting fields agree, and that all isotopes of a first construction exceptional division algebra are isomorphic.The research of the second author was supported in part by an NSERC grant and by a von Humboldt Fellowship at the University of Münster     

Poisson identities of enveloping algebras

Let L be a restricted Lie algebra. The symmetric algebra S_p(L) of the restricted enveloping algebra u(L) has the structure of a Poisson algebra. We give necessary and sufficient conditions on L in order for the symmetric algebra S_p(L) to satisfy a multilinear Poisson identity. We also settle the same problem for the symmetric algebra S(L) of a Lie algebra L over an arbitrary field. The first author was partially supported by MIUR of Italy. The second author was partially supported by Grant RFBR-04-01- 00739. Received: 31 October 2005     

Oligomorphic clones

A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if its set of permutations forms an oligomorphic permutation group. There is a close relationship to ω-categorical structures, i.e., countably infinite structures with a first-order theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an ω-categorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an ω-categorical structure that contains all first-order definable relations. There is a similar Galois connection between locally closed oligomorphic clones and ω-categorical structures containing all primitive positive definable relations. In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define minimal oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones. Presented by A. Szendrei. Received July 12, 2005; accepted in final form August 29, 2006.     

The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras

We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements [a, b] and [a, b]³, where a and b are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic F-module by two derivations. We show that the Jordan commutator s-identities follow from the Glennie-Shestakov s-identity.