Decomposability for division algebras of exponent two and associated forms

This article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated Witt ring is torsion free.      

The isometry classification of hermitian forms over division algebras

The isometry classification problem occupies a central role in the theory of quadratic and hermitian forms. This article is a survey of results on the problem for quadratic and hermitian forms over a field and also for hermitian and skew-hermitian forms over a noncommutative division algebra with involution. Rather than adopting a very abstract approach, the problems are stated in matrix or linear-algebraic terms. The known solutions depend crucially on the particular field considered, although there are some general results which are mentioned. While many of the results date back a long time, some recent results, especially those on skew-hermitian forms over a quaternion algebra over a number field, are included.     

Hermitian forms over division algebras over real function fields

We develop here the theory of (skew-) hermitian forms over division algebras over the real function field and its completions. In particular, a local and local-global classification for forms of all types are given and some Hasse principles are proved.     

All dihedral division algebras of degree five are cyclic

In 1982 Rowen and Saltman proved that every division algebra which is split by a dihedral extension of degree of the center, odd, is in fact cyclic. The proof requires roots of unity of order in the center. We show that for , this assumption can be removed. It then follows that , the -torsion part of the Brauer group, is generated by cyclic algebras, generalizing a result of Merkurjev (1983) on the and torsion parts.

     

Sklyanin algebras and their symbols

Three-dimensional Sklyanin algebras are a new class of possible counterexamples to the cyclicity problem. In this note, we collect a few observations concerning this problem.     

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     

On algebras of two dimensional singular integral operators with homogeneous discontinuities in symbols

We describe the Fredholm symbol algebra for theC ^*-algebra generated by two dimensional singular integral operators, acting onL ₂(²), and whose symbols admit homogeneous discontinuities. Locally these discontinuities are modeled by homogeneous functions having slowly oscillating (and, in particular, piecewise continuous) discontinuities on a system of rays outgoing from the origin.These results extend the well-known Plamenevsky results for the two dimensional case. We present here an alternative and much clearer approach to the problem.Rostov State University RussiaPartially supported by Russian Fund for Fundamental Investigations, RFFI-98-01-01-023, and by CONACYT project 32424-EPartially supported by CONACYT Project 27934-E, México.     

Cyclic division algebras

A general example of cyclic division algebra is given, based on a construction of Brauer, yielding examples of division algebras of arbitrary prime exponent without proper central subalgebras, and also noncrossed products of arbitrary exponent. This research was supported in part by the U.S.-Israel Binational Science Foundation. An erratum to this article is available at .     

Graded division algebras

Mathematische Zeitschrift -     

On central division algebras

For any integern such that 8|n or for which there exists an odd primeq such thatq ²|n, there is a central division algebra of dimensionn ² over its center which is not a crossed product. The algebra constructed in this paper is the algebraQ(X ₁,…,X)_m, the algebra generated over the rationalQ bym(≧2) generic matrices. To the memory of A. A. Albert This paper was originally presented in November, 1971 for publication elsewhere in a volume in honor of Prof. A. A. Albert on the occasion of his 65th birthday. The volume was never published due to the death of Prof. Albert in June 1972.     

Selectivity in division algebras

A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, maximal orders in the algebra. We completely characterize selective orders in central division algebras, of dimension 9 or greater, in terms of the characterization of selective orders given by Chinburg and Friedman in the quaternionic case.     

Semidirect product division algebras

SupposeD is a division algebra of degreep over its centerF, which contains a primitivep-root of 1. Also supposeD has a maximal separable subfield overF whose Galois group is the semidirect product of the cyclic groupsC _p C _q , whereq=2, 3, 4, or 6 and is relatively prime top (In particular this is the case whenp is prime ≤7 andD has a maximal separable subfield whose Galois group is solvable.) ThenD is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.     

On finite division algebras

A review of the known facts about division algebras of small dimensions over finite fields is given. The cases of dimensions three and four for the commutative algebras are shown to lead to interesting linear spaces of quadrics. This leads to a geometrical classification of the three-dimensional case and a spread of lines of PG(3, q) constructed from any four-dimensional commutative division algebra.