Birational maps between generalized Severi-Brauer varieties

A conjecture of Amitsur states that two Severi-Brauer varieties V(A) and V(B) are birationally isomorphic if and only if the underlying algebras A and B are the same degree and generate the same cyclic subgroup of the Brauer group. We examine the question of finding birational isomorphisms between generalized Severi-Brauer varieties. As a first step, we exhibit a birational isomorphism between the generalized Severi-Brauer variety of an algebra and its opposite. We also extend a theorem of P. Roquette to generalized Severi-Brauer varieties and use this to show that one may often reduce the problem of finding birational isomorphisms to the case where each of the separable subfields of the corresponding algebras are maximal, and therefore to the case where the algebras have prime power degree. We observe that this fact allows us to verify Amitsur’s conjecture for many particular cases.     

On the Hopf–Schur group of a field

Let k be any field. We consider the Hopf–Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf–Schur group can be much larger than the Schur group of k.     

Nonassociative cyclic extensions of fields and central simple algebras

We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree m (resp., qs). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.     

Algebras whose right nucleus is a central simple algebra

We generalize Amitsur's construction of central simple algebras over a field F which are split by field extensions possessing a derivation with field of constants F to nonassociative algebras: for every central division algebra D over a field F of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is D and whose left and middle nucleus are a field extension K of F splitting D, where F is algebraically closed in K.We then give a short direct proof that every p-algebra of degree m, which has a purely inseparable splitting field K of degree m and exponent one, is a differential extension of K and cyclic. We obtain finite-dimensional division algebras over a field F of characteristic $p>0$ whose right nucleus is a division p-algebra.     

Covering points in permutation algebras

In this note we extend the main result of [8] concerning covering blocks of group algebras to the context of p-permutation algebras over arbitrary ground fields. We also establish several connections between the Green correspondence for points and the correspondences that arise using the Brauer homomorphism.     

Conjugation in Brauer algebras and applications to character theory

The Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define Brauer characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.     

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.     

Cellularity of twisted semigroup algebras

In this paper, the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of cellular twisted semigroup algebras of type JH. Partition algebras, Brauer algebras and Temperley-Lieb algebras all are examples of cellular twisted semigroup algebras of type JH. Our main result shows that the twisted semigroup algebra of a regular semigroup is cellular of type JH with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular algebras. Here we do not assume that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Moreover, for a twisted semigroup algebra, we do not require that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units in the ground ring. Note that trivially twisted semigroup algebras are the usual semigroup algebras. So, our results extend not only a recent result of East, but also some results of Wilcox.     

Character tables and normal left coideal subalgebras

We continue studying properties of semisimple Hopf algebras H over algebraically closed fields of characteristic 0 resulting from their generalized character tables. We show that the generalized character table of H reflects normal left coideal subalgebras of H. These are the Hopf analogues of normal subgroups in the sense that they arise from Hopf quotients. We apply these ideas to prove Hopf analogues of known results in group theory. Among the rest we prove that columns of the character table are orthogonal and that all entries are algebraic integers. We analyze ‘semi-kernels’ and their relations to the character table. We prove a full analogue of the Burnside–Brauer theorem for almost cocommutative H. We also prove the Hopf algebras analogue of the following (Burnside) theorem: If G is a non-abelian simple group then {1} is the only conjugacy class of G which has prime power order.     

Algebraic prime subalgebras in simple Artinian algebras

We prove a Wedderburn-Artin type theorem for algebraic prime subalgebras in simple Artinian algebras, giving a generalized version of Yahaghi’s theorem [B.R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl. 418 (2006) 599-613]. We also show that every semiprime left algebraic subring in a semiprime right Goldie ring must be a semiprime Artinian ring.     

Special forms of two symbols are division algebras

 If F is a valued field of characteristic p certain tensor products of cyclic F-algebras are called special forms. It is known that if F is maximally complete then every Brauer class of exponent p is represented by a special form. It is shown here that special forms of two symbols are division algebras, but that special forms of three symbols need not be, and can represent indecomposible division algebras of index $p^2$. Received: 16 October 2001     

Automorphisms of Green Orders and Their Derived Categories

In an earlier paper, Raphaël Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.     

The strong Brauer group of a cocommutative coalgebra

Using strong equivalences for coalgebras we define the strong Brauer group of a cocommutative coalgebra C, which is a subgroup of the Brauer group of C. In general there is not a good relation between the Brauer group of a coalgebra and the Brauer group of the dual algebra C∗, the former is not even a torsion group. We find that this subgroups embeds in the Brauer group of C∗. A key tool in this result is the use of techniques from torsion theory. Some cases where both subgroups coincide are shown, for example, C being coreflexive.     

Unramified division algebras do not always contain Azumaya maximal orders

We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups.     

Discriminants of orthogonal involutions on central simple algebras with tame gauges

In this article we generalize the results of Chacron et al. (1995) [4] concerning the computation of the discriminants of involutions of the first kind on central division algebras over Henselian valued fields of residue characteristic different from 2. We prove analogous results for orthogonal involutions on tame central simple algebras with simple residue algebras over a wide class of Henselian valued fields of arbitrary residue characteristic (see Corollary 2.18).     

Nonassociative cyclic algebras

Nonassociative quaternion algebras were first discovered over the real numbers independently by Dickson and Albert and provided some of the first examples of nonassociative division algebras. They were later classified completely by Waterhouse. Cyclic algebras can be seen as a natural generalisation of the classical quaternion algebras. With this in mind we generalise nonassociative quaternion algebras and introduce nonassociative cyclic algebras. These provide new examples of nonassociative central division algebras with Nucleus a separable field extension of degree n.     

On group rings of linear groups

Let H be a finitely generated group of matrices over a field F of characteristic zero. We consider the group ring KH of H over an arbitrary field K whose characteristic is either zero or greater than some number $N=N(H)$. We prove that KH is isomorphic to a subring of a ring S which is a crossed product of a division ring Δ with a finite group. Hence KH is isomorphic to a subring of a matrix ring over a skew field.     

n-Torsion of Brauer groups as relative Brauer groups of abelian extensions

It is now known [H. Kisilevsky, J. Sonn, Abelian extensions of global fields with constant local degrees, Math. Res. Lett. 13 (4) (2006) 599-607; C.D. Popescu, Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields, J. Number Theory 115 (2005) 27-44] that if F is a global field, then the n-torsion subgroup of its Brauer group Br(F) equals the relative Brauer group Br(Ln/F) of an abelian extension Ln/F, for all n∈Z?1. We conjecture that this property characterizes the global fields within the class of infinite fields which are finitely generated over their prime fields. In the first part of this paper, we make a first step towards proving this conjecture. Namely, we show that if F is a non-global infinite field, which is finitely generated over its prime field and ?≠char(F) is a prime number such that μ?²⊆F^×, then there does not exist an abelian extension L/F such that . The second and third parts of this paper are concerned with a close analysis of the link between the hypothesis μ?²⊆F^× and the existence of an abelian extension L/F such that , in the case where F is a Henselian valued field.