JORDAN BIMODULES OVER THE SUPERALGEBRAS P(n) AND Q(n)

We extend the Jacobson's Coordinatization theorem to Jordan superalgebras. Using it we classify Jordan bimodules over superalgebras of types Q(n) and JP(n), n >= 3. Then we use the Tits-Kantor-Koecher construction and representation theory of Lie superalgebras to treat the remaining case Q(2).

Industrial apiculture in the Jordan valley during Biblical times with Anatolian honeybees

Although texts and wall paintings suggest that bees were kept in the Ancient Near East for the production of precious wax and honey, archaeological evidence for beekeeping has never been found. The Biblical term ""honey"" commonly was interpreted as the sweet product of fruits, such as dates and figs. The recent discovery of unfired clay cylinders similar to traditional hives still used in the Near East at the site of Tel Rehov in the Jordan valley in northern Israel suggests that a large-scale apiary was located inside the town, dating to the 10th-early 9th centuries B.C.E. This paper reports the discovery of remains of honeybee workers, drones, pupae, and larvae inside these hives. The exceptional preservation of these remains provides unequivocal identification of the clay cylinders as the most ancient beehives yet found. Morphometric analyses indicate that these bees differ from the local subspecies Apis mellifera syriaca and from all subspecies other than A. m. anatoliaca, which presently resides in parts of Turkey. This finding suggests either that the Western honeybee subspecies distribution has undergone rapid change during the last 3,000 years or that the ancient inhabitants of Tel Rehov imported bees superior to the local bees in terms of their milder temper and improved honey yield.

Jordan Gradings on Associative Algebras

In this paper we apply the method of functional identities to the study of group gradings by an abelian group G on simple Jordan algebras, under very mild restrictions on the grading group or the base field of coefficients.

Representation type of Jordan algebras

The problem of classification of Jordan bit-nodules 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. (c) 2010 Elsevier Inc. All rights reserved.

Noncommutative Jordan superalgebras of degree n > 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a ""nodal"" case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.

An algebraic theory for generalized Jordan chains and partial signatures in the Lagrangian Grassmannian

We discuss an algebraic theory for generalized Jordan chains and partial signatures, that are invariants associated to sequences of symmetric bilinear forms on a vector space. We introduce an intrinsic notion of partial signatures in the Lagrangian Grassmannian of a symplectic space that does not use local coordinates, and we give a formula for the Maslov index of arbitrary real analytic paths in terms of partial signatures.

Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees <= 6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees <= 6, and demonstrate the existence of further new identities in degree 7. our proofs depend on computer algebra: we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem. (C) 2009 Elsevier Inc. All rights reserved.