Search

Let be a composition algebra which is either the Hamilton quaternion algebra ℍ or the Cayley octonion algebra over ℝ. In a previous work, the nth symmetric power Sym n of is shown to be a direct sum of central simple algebras, corresponding to the partitions of n of length 2 , such that the component corresponding to the partition (m , n − m) is isomorphic to the component T 2 m − n of Sym 2 m − n corresponding to the partition (2 m − n , 0) of 2 m − n. In this work, we study the building blocks T n of these decompositions. We show that the "local" structure of , i.e. the complex-like subfields of , determine both the complement of T n in Sym n and the trace map of T n , induced from the trace map of . We also derive a recursive trace formula on the T n 's. We use the "local-global" results to define positive definite symmetric bilinear forms on the vector space ⊕ n = 0 ∞ T n , which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra T n is described. [ABSTRACT FROM AUTHOR]

We study systems of quadratic forms over fields and their isotropy over 2 -extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence we obtain that every central simple algebra of degree 1 6 is split by a 2 -extension of degree at most 2 1 6 . [ABSTRACT FROM AUTHOR]

The paper was motivated by Kovacs' paper (1973), Isaacs' paper (1980) and a recent paper, due to Bresar et al. (2018), concerning Skolem-Noether algebras. Let K be a unital commutative ring, not necessarily a field. Given a unital K-algebra S, where K is contained in the center of S, n ∈ ℕ, the goal of this paper is to study the question: when can a homomorphism ϕ: Mn(K) → Mn(S) be extended to an inner automorphism of Mn(S)? As an application of main results presented in the paper, it is proved that if S is a semilocal algebra with a central separable subalgebra R, then any homomorphism from R into S can be extended to an inner automorphism of S. [ABSTRACT FROM AUTHOR]

We give an example of infinite dimensional central simple (von Neumann) regular algebra with outer derivations. Given the algebra of all measurable operators affiliated with the type II hyperfinite factor we construct its -subalgebra which is dense in the measure topology. We prove that this algebra has a derivation which is discontinuous in the measure topology and hence is non-inner. We show that the algebra admits also a continuous (in the measure topology) derivation, implemented by a measurable operator, but which is still an outer derivation. [ABSTRACT FROM AUTHOR]

In a quaternion order of class number one, an element can be factored in multiple ways depending on the order of the factorization of its reduced norm. The fact that multiplication is not commutative causes an element to induce a permutation on the set of primes of a given reduced norm. We discuss this permutation and previously known results about the cycle structure, sign, and number of fixed points for quaternion orders. We generalize these results to other orders in central simple algebras over global fields. [ABSTRACT FROM AUTHOR]

We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra , endowed with an orthogonal involution. We construct examples of algebras isomorphic to a tensor product of quaternion algebras with k split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with k factors with involution. This extends an earlier result of Sivatski where the algebra considered is of degree 8 and index 4, and endowed with some orthogonal involution. [ABSTRACT FROM AUTHOR]

Using the theory of signatures of hermitian forms over algebras with involution, developed by us in earlier work, we introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to Hilbert's 17th problem, characterizing totally positive elements in terms of weighted sums of hermitian squares. As a consequence we obtain an earlier result of Procesi and Schacher and give a complete answer to their question about representation of elements as sums of hermitian squares. [ABSTRACT FROM AUTHOR]

Given a global field K and a positive integer n, we present a diophantine criterion for a polynomial in one variable of degree n over K not to have a root in K. This strengthens a result by Colliot-Thélène and Van Geel [Compositio Math. 151 (2015), 1965–1980] stating that the set of non-n th powers in a number field K is diophantine. We also deduce a diophantine criterion for a polynomial over K of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of Z in Q. [ABSTRACT FROM AUTHOR]

An algebra S is called a Skolem–Noether algebra (SN algebra for short) if for every central simple algebra R , every homomorphism R → R ⊗ S extends to an inner automorphism of R ⊗ S . One of the important properties of such an algebra is that each automorphism of a matrix algebra over S is the composition of an inner automorphism with an automorphism of S . The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem–Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra S is SN if and only if the power series algebra S [ [ ξ ] ] is SN. [ABSTRACT FROM AUTHOR]

Given a central simple algebra with involution over an arbitrary field, étale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and privileges a unified treatment for all characteristics whenever possible. As a consequence a conceptual proof of a theorem of Rowen is obtained, which asserts that every division algebra of exponent two and degree eight contains a maximal subfield that is a triquadratic extension of the centre. [ABSTRACT FROM AUTHOR]

In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra. [ABSTRACT FROM AUTHOR]

The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements whose squares lie in the square of the underlying field. [ABSTRACT FROM AUTHOR]

Let be a central simple algebra of degree over a number field , and be a strictly maximal subfield. We say that the ring of integers is selective if there exists an isomorphism class of maximal orders in no element of which contains . In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers by leveraging the theory of affine buildings for where is a local central division algebra. Then as an application, we use the local result and a local-global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in , and distinguish those which are guaranteed to contain . Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result, and give examples which clarify the interesting role of partial ramification in the algebra. [ABSTRACT FROM AUTHOR]

We study the conjugacy classes of elements of a finite-dimensional central simple associative algebra with respect to scalar extensions. We apply the results to investigate the products of two (skew-)symmetric elements in central simple algebras with involution. [ABSTRACT FROM AUTHOR]

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato's cohomology group, defined by cokernel of a general Artin-Schreier operator. Combining this with Tignol's result on the second trace form of simple algebras of degree , we show that over a field of characteristic . [ABSTRACT FROM AUTHOR]

LetAbe a central simple algebra over a fieldF. Letk1,…,krbe cyclic extensions ofFsuch thatk1 ⊗F… ⊗Fkris a field. We investigate conditions under whichAis a tensor product of symbol algebras where each fieldkilies in a symbolF-algebra factor of the same degree askioverF. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4. [ABSTRACT FROM PUBLISHER]

A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogonal type. [ABSTRACT FROM AUTHOR]

In this paper a further study is made of H -signatures of hermitian forms, introduced previously by the authors. It is shown that a tuple of reference forms H may be replaced by a single form and that the H -signature is invariant under Morita equivalence of algebras with involution. The “prime ideals” of the Witt group are studied, obtaining results that are analogues of the classification of prime ideals of the Witt ring by Harrison and Lorenz–Leicht. It follows that H -signatures canonically correspond to morphisms into the integers. [ABSTRACT FROM AUTHOR]

Positivity in *-algebras can be defined either algebraically, by quadratic modules, or analytically, by *-representations. By the induction procedure for *-representations, we can lift the analytical notion of positivity from a *-subalgebra to the entire *-algebra. The aim in this article is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the *-algebra is induced from its intersection with the *-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases. [ABSTRACT FROM AUTHOR]

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integer coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with its NC analogue. As an application, we compute the (NC) Chow motive of the Weil restriction of every smooth projective scheme whose category of perfect complexes admits a full exceptional collection. Finally, in the case of central simple algebras, we describe explicitly the NC analogue of the Weil restriction functor using solely the degree of the field extension. This leads to a “categorification” of the classical corestriction homomorphism between Brauer groups. [ABSTRACT FROM AUTHOR]