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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]

In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree. [ABSTRACT FROM AUTHOR]

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup lvi of GLn(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLND,(K*) = M, K* ≏ M, K/F is Galois with Gal(K/F) ≅ M/K*, and F[M] = Mn(D). In particular, when F is global or local, it is proved that if ([D : F], Char(F)) = 1, then every non- abelian maximal subgroup of GL1 (D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5. [ABSTRACT FROM AUTHOR]

Abstract: We solve Kaplansky's problem concerning the structure of linear preservers of invertibility in the special case of maps on central simple algebras. [Copyright &y& Elsevier]

Abstract: Let F be a non-Archimedean local field, and let A be a central simple F-algebra. In the present paper, based on the Bushnell–Kutzko theory of types for , which has recently been developed by Sécherre and Stevens, we derive an explicit formula for the formal degree of an irreducible discrete series representation of the multiplicative group . [Copyright &y& Elsevier]

Abstract: In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a -action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In the present paper, much of this is extended to prime characteristic. In addition, a mistake in the earlier paper is corrected. One of the results is that the finitely generated prime PI-algebras of degree n are precisely the rings that arise as “coordinate rings” of “n-varieties” in this setting. For the definitions and results reduce to those of classical affine algebraic geometry. [Copyright &y& Elsevier]

Let A be a finite dimensional central simple algebra. By the Skolem–Noether theorem, every automorphism of A is inner. We will give a short proof of a somewhat more general result. The concept behind this proof is the fact that every linear map on A belongs to the multiplication algebra of A. As an application we will describe linear maps α, β : A → A such that α(x)β(y) = 0 whenever xy = 0. [ABSTRACT FROM PUBLISHER]

Given an F-central simple algebra A = Mn(D), denote by A′ the derived group of its unit group A*. Here, the Frattini subgroup Φ(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′), otherwise Φ(A*) = ⋂p∤deg(A) F*p. Furthermore, it is also shown that Φ(A*) = k* whenever F is either a field of rational functions over a divisible field k or a finitely generated extension of an algebraically closed field k. [ABSTRACT FROM AUTHOR]

Abstract: Let be a fixed algebraic number field and let be an associative algebra over given by structure constants such that holds for some positive integer n. Suppose that n is bounded. Then an isomorphism can be constructed by a polynomial time ff-algorithm. An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields. As a consequence, we obtain a polynomial time ff-algorithm to compute isomorphisms of central simple algebras of bounded degree over . [Copyright &y& Elsevier]

Abstract: We prove a new upper bound on the essential p-dimension of the projective linear group . [Copyright &y& Elsevier]

This is a variation on a theme of Bayer-Fluckiger, Shapiro, and Tignol related to hyperbolic involutions. More precisely, criteria for the hyperbolicity of involutions of quadratic extensions of simple algebras and involutions of the form σ ⊗ τ and σ ⊗ ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A, and ρ is an involution of a quaternion algebra are obtained. [ABSTRACT FROM AUTHOR]

Abstract: In 1976 Procesi and Schacher developed an Artin–Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size. [Copyright &y& Elsevier]

We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative central simple algebra, the center k of which is not absolutely algebraic, and consider units a,b of R such that {a,ab} freely generate a free group. It is shown that such b can be chosen from suitable Zariski dense open subsets of R, while the a can be chosen from a set of cardinality |k| (which need not be open). [ABSTRACT FROM AUTHOR]

Abstract: We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions. [Copyright &y& Elsevier]

Abstract: We give an algorithm for finding an isomorphism of a given central simple algebra of degree 4 over the rationals and the full matrix algebra, provided it exists. It reduces the task to classical problems in number theory, for which there are already known algorithms. [Copyright &y& Elsevier]

Abstract: We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a -action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For our definitions and results reduce to those of classical affine algebraic geometry. [Copyright &y& Elsevier]

Abstract: We study actions of linear algebraic groups on central simple algebras using algebro-geometric techniques. Suppose an algebraic group G acts on a central simple algebra A of degree n. We are interested in questions of the following type: (a) Do the G-fixed elements form a central simple subalgebra of A of degree n? (b) Does A have a G-invariant maximal subfield? (c) Does A have a splitting field with a G-action, extending the G-action on the center of A? Somewhat surprisingly, we find that under mild assumptions on A and the actions, one can answer these questions by using techniques from birational invariant theory (i.e., the study of group actions on algebraic varieties, up to equivariant birational isomorphisms). In fact, group actions on central simple algebras turn out to be related to some of the central problems in birational invariant theory, such as the existence of sections, stabilizers in general position, affine models, etc. In this paper we explain these connections and explore them to give partial answers to questions (a)–(c). [Copyright &y& Elsevier]

Abstract: Let be the algebra of all matrices over a commutative unital ring , and let be a -module. Various characterizations of bilinear maps with the property that whenever x any y commute are given. As the main application of this result we obtain the definitive solution of the problem of describing (not necessarily bijective) commutativity preserving linear maps from into for the case where is an arbitrary field; moreover, this description is valid in every finite-dimensional central simple algebra. [Copyright &y& Elsevier]

Abstract: In this paper we present a cohomological description of the equivariant Brauer group relative to a Galois finite extension of fields endowed with the action of a group of operators. This description is a natural generalization of the classic Brauer–Hasse–Noether''s theorem, and it is established by means of a three-term exact sequence linking the relative equivariant Brauer group, the 2nd cohomology group of the semidirect product of the Galois group of the extension by the group of operators and the 2nd cohomology group of the group of operators. [Copyright &y& Elsevier]

LetD be a finite-dimensionalF-central division algebra. A criterion is given forD to be a supersoluble (nilpotent) crossed product division algebra in terms of subgroups of the multiplicative groupD* ofD. More precisely, it is shown thatD is a supersoluble (nilpotent) crossed product if and only ifD* contains an abelian-by-supersoluble (abelian-by-nilpotent) generating subgroup. [ABSTRACT FROM AUTHOR]

We discuss standard pairs of generators of cyclic divisionp-algebras of degreep, and prove forp=3 that any two Artin-Schreier elements are connected by a chain of standard pairs. This result has immediate applications to the presentations of such algebras. [ABSTRACT FROM AUTHOR]

A field extension L / F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L / F is called universally excellent if L ⋅ E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras. [ABSTRACT FROM AUTHOR]

Let (A , σ) be a central simple algebra with an orthogonal involution. It is well-known that O (A , σ) contains elements of reduced norm −1 if and only if the Brauer class of A is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the "if" part fails if one allows the base ring to be arbitrary. [ABSTRACT FROM AUTHOR]