On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

An explicit example of a noncrossed product division algebra

The paper presents an explicit example of a noncrossed product division algebra of index and exponent 8 over the field ?(s)(t). It is an iterated twisted function field in two variables D(x, σ)(y, τ ) over a quaternion division algebra D which is defined over the number field ?(√3,√?7). The automorphisms σ and τ are computed by solving relative norm equations in extensions of number fields. The example is explicit in the sense that its structure constants are known. Moreover, it is pointed out that the same arguments also yield another example, this time over the field ?((s))((t)), given by an iterated twisted Laurent series ring D((x, σ))((y, τ )) over the same quaternion division algebra D. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectra of quantized hyperalgebras

We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflo's Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztig's tensor product theorem.

     

Finite groups embeddable in division rings

In a tour de force in 1955, S. A. Amitsur classified all finite groups that are embeddable in division rings. In particular, he disproved a conjecture of Herstein which stated that odd-order emdeddable groups were cyclic. The smallest counterexample turned out to be a group of order 63. In this note, we prove a non-embedding result for a class of metacyclic groups, and present an alternative approach to a part of Amitsur's results, with an eye to ``de-mystifying" the order 63 counterexample.

     

Gelfand–Mazur Theorems in normed algebras: A survey

The Gelfand–Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$ or the quaternions $\mathbb{H}$; (Complex version) Every complex normed division algebra is isometrically isomorphic to $\mathbb{C}$. This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues.     

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 .     

A twisted Laurent series ring that is a noncrossed product

The striking results on noncrossed products were their existence (Amitsur [1]) and the determination of ℚ(t) and ℚ((t)) as their smallest possible centres (Brussel [3]). This paper gives the first fully explicit noncrossed product example over ℚ((t)). As a consequence, the use of deep number theoretic theorems (local-global principles such as the Hasse norm theorem and density theorems) in order to prove existence is eliminated. Instead, the example can be verified by direct calculations. The noncrossed product proof is short and elementary. Supported in part by the DAAD (Kennziffer D/02/00701).     

A counterexample concerning the 3-error linear complexity of 2 n -periodic binary sequences

In this article, we present a counterexample to Theorem 4.2 and Theorem 5.2 by Kavuluru (Des Codes Cryptogr 53:75–97, 2009). We conclude that the counting functions for the number of 2 ⁿ -periodic binary sequences with fixed 3-error linear complexity by Kavuluru are not correct.     

Identities in algebras with involution

One of the main features of the theory of polynomial identities is the existence (for anyn) of a division algebra of degreen, formed by adjoining quotients of central elements of the algebra of genericn×n matrices; this division algebra is extremely interesting and has been used by Amitsur (forn divisible by either 8 or the square of an odd prime) as an example of a non-crossed product central division algebra. The main object of this paper is to obtain, in a parallel method, division algebras with involution of the first kind, knowledge of which would answer some long-standing questions in the theory of division algebras with involution. One such question is, “Is every division algebra with involution of the first kind a tensor product of quaternion division algebras?” In the process, a theory of (polynomial) identities in algebras with involution is developed with emphasis on prime PI-algebras with involution.     

An affine PI Hopf algebra not finite over a normal commutative Hopf subalgebra

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra , provides a noetherian prime counterexample.

     

Infinite Rings with Planar Zero-Divisor Graphs

In this article, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of ?₂-graded algebras. Essentially, the results are the same as in the classical case, except that the notion of a ?₂-graded division algebra needs to be modified. We classify all finite dimensional ?₂-graded division algebras over ? and ?.     

A noncommutative Brooks-Jewett Theorem

In classical measure theory the Brooks-Jewett Theorem provides a finitely-additive-analogue to the Vitali-Hahn-Saks Theorem. In this paper, it is studied whether the Brooks-Jewett Theorem allows for a noncommutative extension. It will be seen that, in general, a bona-fide extension is not valid. Indeed, it will be shown that a C^*-algebra A satisfies the Brooks-Jewett property if, and only if, it is Grothendieck, and every irreducible representation of A is finite-dimensional; and a von Neumann algebra satisfies the Brooks-Jewett property if, and only if, it is topologically equivalent to an abelian algebra.     

Remarks on Ginzburg's bivariant Chern classes

The convolution product is an important tool in the geometric representation theory. Ginzburg constructed the bivariant Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we give some remarks on the Ginzburg bivariant Chern classes.

     

On Stone's theorem and the Axiom of Choice

It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.

     

A spectral mapping theorem for representations of one-parameter groups

In this paper we present some generalization (at the same time a new and a short proof in the Banach algebra context) of the Weak Spectral Mapping Theorem (WSMT) for non-quasianalytic representations of one-parameter groups.

     

A note on a theorem of Raubenheimer and Rode

We prove the converse of Raubenheimer and Rode's Banach algebra version of the Perron-Frobenius Theorem.

     

On torsion subgroups of whitehead groups of division algebras

We completely determine all torsion abelian groups that can occur as the torsion subgroup of the Whitehead group of a division algebra of prime index. More precisely, we prove that if D is a division algebra of prime index, then the torsion subgroup of K ₁(D) is locally cyclic. Conversely, if A is a torsion locally cyclic group, then there exists a division algebra D of prime index such that the torsion subgroup of K ₁(D) is isomorphic to A. Our result can be considered as a non-commutative version of May’s Theorem.     

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.     

Analytic study on a state observer synchronizing a class of linear fractional differential systems

This paper is concerned with Theorem 2 in Matignon and d’André-Novel (1997) [1], which was sufficient and necessary criterion on a state observer for a class of linear fractional differential systems. Based on the stability theory, the dual principle and the pole assignment theory of the fractional differential system, we have proved the validity of sufficiency of Theorem 2 in details. A counterexample is provided to show that the condition of Theorem 2 is not necessary.