Biquaternion division algebras and fourth power-central elements

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 ×2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.

Let K/k be a separable field extension of degree 2, D be a finite-dimensional central division algebra over K with a K/k-involution τ, h be an Hermitian anisotropic form on a right D-vector space with respect to τ, and let U(h) be the unitary group of h. Then the reduced Whitehead group of its special linear subgroup is defined as follows: $$ \mathrm{SUK}_1^{\mathrm{an}}(h)={{{\mathrm{SU}(h)}} \left/ {{\left[ {U(h),U(h)} \right]}} \right.} $$ , where [U(h), U(h)] is the commutator subgroup of U(h). The first main result establishes a link between the above group and its analog SUK1(h) for the case of isotropic h (with respect to the same τ). Theorem. There exists a surjective homomorphism from $$ \mathrm{SUK}_1^{\mathrm{an}}(h) $$ to SUK1(h). Furthermore, we also give a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups. Bibliography: 32 titles.

Canonical dimension of a smooth complete connected variety is the minimal dimension of image of its rational endomorphism. The $i$-th canonical dimension of a non-degenerate quadratic form is the canonical dimension of its $i$-th orthogonal grassmannian. The maximum of a canonical dimension for quadratic forms of a fixed dimension is known to be equal to the dimension of the corresponding grassmannian. This article is about the minima of the canonical dimensions of an anisotropic quadratic form. We conjecture that they equal the canonical dimensions of an excellent anisotropic quadratic form of the same dimension and we prove it in a wide range of cases.

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of these results to the algebraic equations and De Moivre’s formula in generalized quaternion algebras and in generalized octonion division algebras.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

Quadratic forms over fields with four quaternion algebras

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer N N , first we give the classification of positive definite ternary quadratic forms of level 4 N 4N explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4 N 4N , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level 4 N 4N . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight 3 / 2 3/2 and level 4 N 4N , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

Explicit equivalence of quadratic forms over [formula omitted

Introduction. Let G1 and G be connected semisimple algebraic groups defined over a field K of characteristic zero and assume that there is an isomorphism f of G1 onto G which is defined over R, the algebraic closure of K. If p: G -? GL(V) is an absolutely irreducible (finite-dimensional) representation of G defined over K, then p of is an absolutely irreducible representation of G1 defined over K. Satake [7, p. 230] has shown that there is a field K1 which is a finite extension of K, a (unique) central simple division algebra K# defined over K1, a finite-dimensional right vector space V1 over K#, and a K1-homomorphism pi: G1 -GL(V1/K#) (the group of all nonsingular K#-linear endomorphisms of V1) such that (p of)(g) = 01(p1(g)) for all g E G1 where 01 is a unique absolutely irreducible representation of End (V1/K#) (the algebra of all K#-linear endomorphisms of V1) onto End (V). In this paper we are interested in the case where K= K1 and where there are invariant forms on V and V1. More precisely, we state the following two problems. PROBLEM 1. Assume that K#=K and that there are invariant bilinear forms B on V and B1 on V1 which are defined over K. What is the relationship between these two forms over K? Of course, if B is alternating, so is B1 and both are determined by dim V=dim V1. Hence, we shall always take B and B1 to be symmetric. PROBLEM 2. Assume that K# is a nontrivial division algebra over K (i.e., K# AK) and that there is an invariant bilinear form B on V and an invariant ?hermitian form F (c = + 1 or 1) on V1 both of which are defined over K. What is the relationship between these two forms over K? We are especially interested in the case K= Q,, a p-adic field. (In a future paper, we shall discuss the case K=R.) Here, some simplifications are immediately available. In Problem 2, it can be shown [7, p. 232] that K# has an involution of the first kind; but over Q,, it is known that the only such division algebra is the quaternion division algebra. Furthermore, it is known that a hermitian form on a finite-dimensional vector space over a quaternion division algebra defined over Q, is determined only by the dimension of the vector space. Therefore, in Problem 2 we shall always take F to be skew-hermitian; in the case where K# is a quaterion division algebra, this means that the form B is symmetric [7, p. 233]. If W is a vector space defined over K and if S is a symmetric form on W which is also defined over K, then three invariants can be associated with the pair (W, S),

Nonstandard quadratic forms over rational function fields

We describe some constructions of orthonormal lattices in totally real subfields of cyclotomic fields, obtained by endowing their ring of integers with a trace form. We also describe constructions of quaternion division algebras over such fields. Orthonormal lattices and quaternion division algebras over totally real fields find use in wireless networks in ultra wideband communication, and we describe the application.

In this paper we prove local–global principles for the existence of an embedding (E, σ) ↪ (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local–global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S -arithmetic groups in all absolutely simple algebraic groups of type different from D_4 which was initiated in our paper [23]. More precisely, we prove that in a group of type D_n , n even > 4 , two weakly commensurable Zariski-dense S -arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n + 7 , with n ≥ 1 . The appendix contains a Galois-cohomological interpretation of our embedding theorems.

Our purpose in this chapter is to introduce three algebras of importance in the theory of quadratic forms, the Clifford algebra, the quaternion algebra, and the Hasse algebra. The Clifford algebra will be developed from first principles and its main use for us will be in the definition of an invariant called the spinor norm. The quaternion algebra and the Hasse algebra play an important role in the arithmetic theory of quadratic forms. The definition of the Hasse algebra depends on some of the structure theory of central simple algebras, in particular it needs Wedderburn’s theorem and the theory of similarity of algebras that is normally used in defining the Brauer group. We have therefore included a proof of Wedderburn’s theorem and some of its consequences. Also included as a convenience to the reader is a brief discussion of the tensor product of finite dimensional vector spaces1.

Theorem. If a tensor product of a division algebra D D with a quaternion algebra Q Q is not a division algebra, then either D D and Q Q possess a common quadratic subfield or D D contains a splitting field of Q Q not quadratic over the base field. The above theorem generalizes a recently published result of Albert’s. Theorem. If the tensor product of two division algebras over a local or a global field K K is not a division algebra, then they contain a common extension field of K K .

This chapter investigates the consequences of the assumption that one Moufang set is weakly isomorphic to another. It first introduces some well-known facts about involutions which are assembled in a few lemmas, including those dealing with an involutory set, a biquaternion division algebra, and a quaternion division algebra with a standard involution. It then presents a notation for a non-trivial anisotropic quadratic space and another for an involutory set are presented, along with assumptions for a pointed anisotropic quadratic space and the standard involution of a quaternion. It also makes a number of propositions regarding the standard involution of a quaternion and a biquaternion. Results about weak isomorphisms between Moufang sets arising from involutory sets are given.