Search

We extend, to the case of local fields of even characteristic, our previous computations for the set of maximal orders containing given quaternions as concrete sub-graphs of the Bruhat–Tits tree. Computing the relative position of such sub-graphs, the branches, is useful, for instance, to explicitly compute relative spinor images, thus solving the selectivity problem. They also play a role in the description of quotient graphs, which are useful to study matrix groups of arithmetical importance. As in earlier work, we concentrate on orders generated by two quaternions, but extending our method to larger generating sets is quite straightforward. In our previous work, the results were given mainly in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin–Scheier extensions, to take care of quaternions generating Galois field extensions. The latter extensions have no non-trivial elements of null trace. For this reason, we state our results in terms of a wider family of algebra presentations, in contrast to the aforementioned work, where we chose a pair of pure quaternions as generators of the order. [ABSTRACT FROM AUTHOR]

In this article, we present a decomposition theorem for unitary spaces over a central simple division algebra with involution in characteristic 2. This is a generalization of a decomposition result for quadratic forms in characteristic 2 from [4] and extends a generalization of the Witt decomposition theorem for nonsingular spaces in this setting to cover spaces that may be singular. [ABSTRACT FROM AUTHOR]

Let k be a field, K/k he a quadratic separable field extension, and A a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically dosed residue field of characteristic zero und the degree of A is odd, we prove that all K/k-involutions on A are cyclic. [ABSTRACT FROM AUTHOR]

We define a notion of Morita equivalence between algebras with antiautomorphisms such that two equivalent algebras have the same category of sesquilinear forms. This generalizes the Morita equivalence of algebras with involutions defined by Fröhlich and Mc Evett [5], and their categories of ϵ-hermitian forms. For two Morita equivalent algebras with involution, with an additional technical property (which is true for central simple algebras), we define a new algebra with antiautomorphism, called the orthogonal sum, which generalizes the usual notion of orthogonal sum of forms. We explore the invariants of this sum. [ABSTRACT FROM PUBLISHER]

We give a cohomological characterization of the set of conjugacy classes of finite subgroups of the projective multiplicative group of a finite dimensional algebra that become conjugate to a given group over some finite separable extension of the base field. We provide two applications. [ABSTRACT FROM AUTHOR]

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL1(A). For a field extension K of F, we study the first Galois Cohomology group H1(K,SL1(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series. [ABSTRACT FROM AUTHOR]

An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r# r where r ∈ R is nonzero. Suppose that R is a central simple algebra (i.e., R = Mn(D) for some integer n and central division algebra D) and # is an involution on R of the form r# = a-1 r* a, where * is some transpose involution on R and a is an invertible matrix such that a* = ±a. In Section 1 we characterize formal reality of # in terms of a and *|D. In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on D = (K/F, Φ) that extend to a formally real involution on the split algebra D ⊗F K ≅ Mn(K). Every such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x ↦ tr(x#x) is not positive semidefinite. [ABSTRACT FROM AUTHOR]

A crossed product representing an associative finite dimensional central simple algebra over a field is called regular if all values of the corresponding cocycle are roots of unity. Under a certain assumption such a crossed product is shown to allow the construction of Galois representations. The case of number fields is investigated more closely and several examples are discussed. [ABSTRACT FROM AUTHOR]

Let A be a central simple algebra over its center F. Define CK1A = Coker(K1F → K1A). We prove that if A and B are F-central simple algebras of coprime degrees, then CK1(A⊗FB) = CK1A × CK1B. [ABSTRACT FROM AUTHOR]