Orthogonal Involution Research Articles

Trialitarian triples

Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical “trialitarian” isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.

Generating hyperbolic isometry groups by elementary matrices

We consider three families of groups: the Bianchi groups where is the ring of integers of an imaginary, quadratic field; the groups where is a -order of a definite, rational quaternion algebra with an orthogonal involution; and the groups where is an order of a definite, rational quaternion algebra. We show that such groups are generated by elementary matrices if and only if is semi-Euclidean (or -semi-Euclidean), which is a generalization of the usual notion of a Euclidean ring. The proofs are surprisingly simple and proceed by considering fundamental domains of Kleinian groups.

Degree 3 relative invariant for unitary involutions

Using the Rost invariant for non split simply connected groups, we define a relative degree $3$ cohomological invariant for pairs of orthogonal or unitary involutions having isomorphic Clifford or discriminant algebras. The main purpose of this paper is to study general properties of this invariant in the unitary case, that is for torsors under groups of outer type ${\sf A}$. If the underlying algebra is split, it can be reinterpreted in terms of the Arason invariant of quadratic forms, using the trace form of a hermitian form. When the algebra with unitary involution has a symplectic or orthogonal descent, or a symplectic or orthogonal quadratic extension, we provide comparison theorems between the corresponding invariants of unitary and orthogonal or symplectic types. We also prove the relative invariant is classifying in degree $4$, at least up to conjugation by the non-trivial automorphism of the underlying quadratic extension. In general, choosing a particular base point, the relative invariant also produces absolute Arason invariants, under some additional condition on the underlying algebra. Notably, if the algebra has even co-index, so that it admits a hyperbolic involution, which is unique up to isomorphism, we get a so-called {\em hyperbolic} Arason invariant. Assuming in addition the algebra has degree $8$, we may also define a {\em decomposable} Arason invariant. It generally does not coincide with the hyperbolic Arason invariant, as the hyperbolic involution need not be totally decomposable.

Totally positive field extensions and the pythagorean index

For a formally real field [Formula: see text], we study totally positive field extensions [Formula: see text] of [Formula: see text]. We prove that, if [Formula: see text] is Galois and totally positive then so is the corresponding extension of their pythagorean closures [Formula: see text]. We also study the behavior of weak isotropy and weak hyperbolicity of central simple algebras with an orthogonal involution over totally positive field extensions. We use some of these results and show that a conjecture due to Becher holds in some new cases.

Some generalizations of Springer’s theorem in characteristic two

It is shown that an anisotropic [Formula: see text]-fold system of hermitian forms over a quaternion algebra in characteristic two remains anisotropic over all odd degree extensions of the base field. It is also shown that an anisotropic orthogonal involution on a central simple algebra of co-index 3 in characteristic two is anisotropic over any odd degree extension of the ground field.

Supercuspidal representations of GLn(F) distinguished by an orthogonal involution

Let F be a non-archimedean locally compact field of residue characteristic p≠2, let G=GLn(F) and let H be an orthogonal subgroup of G. For π a complex smooth supercuspidal representation of G, we give a full characterization for the distinguished space HomH(π,1) being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of π in the space of smooth functions on the set of symmetric matrices in G, as a complex vector space, is non-zero and of dimension four, if and only if the central character of π evaluating at −1 is 1.

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.

The twisted Euclidean algorithm: Applications to number theory and geometry

We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution that admit such an algorithm. We give two applications: first, any order that admits such an algorithm has class number 1; second, we show how the existence of such an algorithm relates to the problem of constructing explicit Dirichlet domains for Kleinian subgroups of the isometry group of hyperbolic 4-space.

On the non-neutral component of outer forms of the orthogonal group

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.

On the possible decompositions of totally decomposable orthogonal involutions in characteristic two

In characteristic two, we determine all possible total decompositions of a totally decomposable algebra with orthogonal involution into tensor products of quaternion algebras with involution.

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 $2$.

Involutions on tensor products of quaternion algebras

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.

Azumaya algebras without involution

Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra A over a ring represents a 2-torsion class in the Brauer group if and only if there is an algebra A’ in the Brauer class of A admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose A’ such that deg A’ = 2 deg A . We show that 2 deg A is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra A of degree 4 and period 2 such that the degree of any algebra A’ in the Brauer class of A admitting an involution is divisible by 8. Separately, we provide examples of split and nonsplit Azumaya algebras of degree 2 admitting symplectic involutions, but no orthogonal involutions. These stand in contrast to the case of central simple algebras of even degree over fields, where the presence of a symplectic involution implies the existence of an orthogonal involution and vice versa.

Isotropy of orthogonal involutions in characteristic two

A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.

Separable descent of totally decomposable orthogonal involutions in characteristic two

ABSTRACTSome necessary and sufficient conditions are obtained for a totally decomposable algebra with orthogonal involution in characteristic two to have a separable descent.

On totally singular generalized quadratic forms

The algebra of similitudes of totally singular generalized quadratic forms in characteristic two is investigated. It is shown that this algebra satisfies certain functorial properties. An application of this study to central simple algebras with orthogonal involutions is also given.

Orthogonal involutions on central simple algebras and function fields of Severi–Brauer varieties

An orthogonal involution σ on a central simple algebra A, after scalar extension to the function field F(A) of the Severi–Brauer variety of A, is adjoint to a quadratic form qσ over F(A), which is uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution σ if and only if they hold for qσ. As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over F(A), so that the associated form qσ is a Pfister form. We also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms, thus proving that the form qσ does not determine the isomorphism class of σ, even when the underlying algebra has index 2. As a consequence, we show that the e3 invariant for orthogonal involutions is not classifying in degree 12, and does not detect totally decomposable involutions in degree 16, as opposed to what happens for quadratic forms.

On the Second Trace Form of Biquaternion Algebras with Involution

The second trace form of biquaternion algebras with involution of the first kind is investigated. It is shown that how this form is related to the pfaffians of these algebras with involution. As an application, a classification of biquaternion algebras with symplectic and decomposable orthogonal involution, in terms of their second trace form, is obtained.

On biquaternion algebras with orthogonal involution

We investigate the Pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their Pfaffians and some other related invariants is studied. Also, in arbitrary characteristic, a criterion is obtained for an orthogonal involution on a biquaternion algebra to be metabolic.

Existence of τ-antisymmetric solutions for a system in [formula omitted

In this paper we prove the existence of a nontrivial τ-antisymmetric solution for the following system{−Δu+u=|u|2p−2u+β(x)|v|p|u|p−2u,inRN−Δv+ω2v=|v|2p−2v+β(x)|u|p|v|p−2v,inRN where N≥3, 2<2p<2⁎, ω>0 is a positive constant and τ:RN→RN is a nontrivial orthogonal involution. Here β is a continuous function, invariant under τ, satisfying additional conditions, some of them related to ω. The basic tool employed here is the Concentration Compactness Principle.