Central Simple Algebras Of Degree Research Articles

Essential dimension of central simple algebras of degree 8 and exponent 2 in characteristic 2

The goal of this note is to reduce the existing upper bound for the essential dimension of central simple algebras of degree 8 and exponent 2 over fields of characteristic 2 from 10 to 9.

Cohomological invariants for central simple algebras of degree 8 and exponent 2

Cohomological invariants for central simple algebras of degree 8 and exponent 2

The metabolicity index of involutions in characteristic two

We define the notion of metabolicity index of involutions of the first kind in characteristic two. It is shown that this index is preserved over odd degree extensions of the base field. Also, its behavior over finite separable extensions is studied. As an application, it is shown that an orthogonal involution on a central simple algebra of degree a power of two which is either anisotropic or metabolic is totally decomposable if it is totally decomposable over some separable extension of the ground field. This result is then used to strengthen an earlier result of the author which proves a characteristic two counterpart of a conjecture concerning Pfister involutions formulated by Bayer-Fluckiger et al.

Splitting quaternion algebras defined over a finite field extension

We study systems of quadratic forms over fields and their isotropy over [Formula: see text]-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 [Formula: see text] is split by a [Formula: see text]-extension of degree at most [Formula: see text].

Essential Dimension, Symbol Length and -rank

AbstractWe prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

Decomposability of orthogonal involutions in degree 12

A theorem of Pfister asserts that every $12$-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from $2$ decomposes as a tensor product of a binary quadratic form and a $6$-dimensional quadratic form with trivial discriminant. The main result of the paper extends Pfister's result to orthogonal involutions: every central simple algebra of degree $12$ with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree $6$ with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree $12$, and to calculate the $f_3$-invariant of the involution if the algebra has index $2$.

Positive trace polynomials and the universal Procesi-Schacher conjecture

Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellens\"atze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schm\"udgen- and Putinar-type Positivstellens\"atze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree $n$: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this paper a negative answer for $n=3$ is presented. Consequently, including traces of hermitian squares as weights in the Positivstellens\"atze is indispensable.

Local selectivity of orders in central simple algebras

Let [Formula: see text] be a central simple algebra of degree [Formula: see text] over a number field [Formula: see text], and [Formula: see text] be a strictly maximal subfield. We say that the ring of integers [Formula: see text] is selective if there exists an isomorphism class of maximal orders in [Formula: see text] no element of which contains [Formula: see text]. 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 [Formula: see text] by leveraging the theory of affine buildings for [Formula: see text] where [Formula: see text] 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 [Formula: see text], and distinguish those which are guaranteed to contain [Formula: see text]. 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.

Square-central elements and standard generators for biquaternion algebras

Analyzing square-central elements in central simple algebras of degree 4, we show that every two elementary abelian Galois maximal subfields are connected by a chain of nontrivially-intersecting pairs. Similar results are proved for non-central quaternion subalgebras, and for central quaternion subalgebras when they exist. Along these lines we classify the maximal square-central subspaces.

Polynomial identity rings as rings of functions, II

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 PGLn-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 n=1 the definitions and results reduce to those of classical affine algebraic geometry.

Cohomological invariants of central simple algebras of degree 4

In this paper, we prove a result of Rost, which describes the cohomological invariants of central simple algebras of degree 4 with values in μ2 when the base field contains a square root of −1.

Essential dimension of simple algebras with involutions

Let 1⩽m⩽n be integers with m|n and Algn, m the class of central simple algebras of degree n and exponent dividing m. In this paper, we find new, improved upper bounds for the essential dimension and two-dimension of Algn, 2. In particular, we show that ed2(Alg16, 2)=24 over a field F of characteristic different from 2.

Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3

The maximal isotropic subspaces in split Cayley algebras were classified by van der Blij and Springer in Nieuw Archief voor Wiskunde VIII(3):158–169, 1960. Here we translate this classification to arbitrary composition algebras. We study intersection properties of such spaces in a symmetric composition algebra, and prove two triality results: one for two-D isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. We bound the distance between isotropic spaces of various dimensions, and study the strong orthogonality relation on isotropic vectors, with its own bound on the distance. The results are used to classify maximal p-central subspaces in central simple algebras of degree p = 3. We prove various linkage properties of maximal p-central spaces and p-central elements. Analogous results are obtained for symmetric p-central elements with respect to an involution of the second kind inverting a third root of unity.

Cohomological invariants for orthogonal involutions on degree 8 algebras

AbstractUsing triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.

Essential dimension of simple algebras in positive characteristic

Let p be a prime integer. For any integers 1 ⩽ s ⩽ r , ▪ p r , p s denotes the class of central simple algebras of degree p r and exponent dividing p s . For any s < r , we find a lower bound for the essential p-dimension of ▪ p r , p s . Furthermore, we compute an upper bound for ▪ 8 , 2 over a field of characteristic 2. As a result, we show ed 2 ( ▪ 4 , 2 ) = ed ( ▪ 4 , 2 ) = 3 and 3 ⩽ ed ( ▪ 8 , 2 ) ⩽ 10 over a field of characteristic 2.

Jordan Algebras Over Algebraic Varieties

General results on the module structure of Jordan algebras over locally ringed spaces are obtained. Albert algebras over a Brauer–Severi variety with associated central simple algebra of degree 3 are constructed using generalizations of the Tits process and the first Tits construction.

On ternary cubic forms that determine central simple algebras of degree 3

Fixing a field F of characteristic different from 2 and 3, we consider pairs ( A , V ) consisting of a degree 3 central simple F-algebra A and a 3-dimensional vector subspace V of the reduced trace zero elements of A which is totally isotropic for the trace quadratic form. Each such pair gives rise to a cubic form mapping an element of V to its cube; therefore we call it a cubic pair over F. Using the Okubo product in the case where F contains a primitive cube root of unity, and extending scalars otherwise, we give an explicit description of all isomorphism classes of such pairs over F. We deduce that a cubic form associated with an algebra in this manner determines the algebra up to (anti-)isomorphism.

Invariants of simple algebras

We determine the group of invariants with values in Galois cohomology with coefficients \({\mathbb{Z}/2\mathbb{Z}}\) of central simple algebras of degree at most 8 and exponent dividing 2.

Trialitarian automorphisms of lie algebras

Over an algebraically closed field of characteristic zero simple Lie algebras admit outer automorphisms of order 3 if and only if they are of type D4. Moreover, thereare two conjugacy classes of such automorphisms. Among orthogonal Lie algebras over arbitrary fields of characteristic zero, only orthogonal Lie algebras relative to quadratic norm forms of Cayley algebras admit outer automorphisms of order 3. We give a complete list of conjugacy classes of outer automorphisms of order 3 for orthogonal Lie algebras over arbitrary fields of characteristic zero. For the norm form of a given Cayley algebra, one class is associated with the Cayley algebra and the others with central simple algebras of degree 3 with involution of the second kind such that the cohomological invariant of the involution is the norm form.

Discriminant of Symplectic Involutions

We define an invariant of torsors under adjoint linear algebraic groups of type C_n-equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that takes values in H^3(F, mu_2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple algebra of degree 8 with symplectic involution can be decomposed as a tensor product of quaternion algebras with involution.