Orthogonal Involution Research Articles

Orthogonal groups containing a given maximal torus

Let k be a field of characteristic different from 2 and let T be a fixed k-torus of dimension n. In this paper we study faithful k-representations ρ : T → SO (A,σ) , where ( A, σ) is a central simple algebra of degree 2 n with orthogonal involution σ. Note that in this case ρ( T ) is a maximal torus in SO (A,σ) . We are interested in describing the pairs ( A, σ) for which there is such a representation. We compute invariants for these algebras (discriminant and Clifford algebra), which are sufficient to determine their isomorphism class when I 3( k)=0 by a theorem of Lewis and Tignol. The first part of the paper is devoted to the case where A is split over k and an application to a theorem of Feit on orthogonal groups over Q is given.

The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain

We consider the Dirichlet problem Δu+λu+|u|2*−2u = 0 in Ω, u = 0 on ∂Ω where Ω is a bounded smooth domain in ℝN, N⩾4, and 2* = 2N/(N−2) is the critical Sobolev exponent. We show that if Ω is invariant under an orthogonal involution then, for λ>0 sufficiently small, there is an effect of the equivariant topology of Ω on the number of solutions which change sign exactly once.

Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential

We study the nonlinear Schr\"{o}dinger equation \[ -\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},\hbox{ \ }u\in \mathbb{R}^{N}, \] with critical exponent $2^{\ast }=2N/(N-2),$ $N\geq 4,$ where $a\geq 0$ has a potential well and is invariant under an orthogonal involution of $\mathbb{R} ^{N}$. Using variational methods we establish existence and multiplicity of solutions which change sign exactly once. These solutions localize near the potential well for $\mu $ small and $\lambda $ large.

Clifford algebras of hyperbolic involutions

For $\As$ a central simple algebra of even degree with hyperbolic orthogonal involution, we describe the canonically induced involution ${\underline \sigma}$ on the even Clifford algebra $(C_0(A,\sigma),{\underline \sigma})$ of $(A,\sigma)$ . When $\deg A \equiv 0 \mod{8}$ , $A \cong M_2(B)$ and the interesting part of ${\underline \sigma} $ is isomorphic to the canonical involution on an exterior power algebra of B. As a corollary, we get some properties of the involution on the exterior power algebra.

Hyperbolicité de certaines involutions sur le corps des fonctions d'une quadrique

Let F be a field of characteristic not 2. The aim of this paper is to investigate the hyperbolicity of an anisotropic orthogonal involution over the function field of a projective quadric. More precisely, we treat the case of a division algebra of degree less or equal to 8. Some results in the split case are included.

Twisted flag varieties of trialitarjan groups

For (A, σ) a central simple algebra of even degree with orthogonal involution, we present a method for constructing isotropic right ideals in the even Clifford algebra (C 0(A, σ)σ) from isotropic right ideals in (A, σ). We then use this construction to fully describe the twisted flag varieties associated to algebraic groups of type D 4 (including the trialitarian groups).

Zero cycles on some involution varieties

The group of zero cycles on involution varieties corresponding to central simple algebras of index at most two with an orthogonal involution is computed. This computation generalizes the Karpenko-Swan theorem on zero cycles on projective nonsingular quadrics. Bibliography:10 titles.

Galois cohomology of special orthogonal groups

If (A,s) is a central simple algebra of even degree with orthogonal involution, then for the map of Galois cohomology sets from H^1(F, SO(A,s)) to the 2-torsion in the Brauer group of F, we describe fully the image of a given element of H^1(F, SO(A,s)) and prove that this description is correct in two different ways. As an easy consequence, we derive a result of Bartels.

The Generalized Even Clifford Algebra

For a central simple algebra A with an orthogonal involution ∗ over a field F of characteristic not equal to two, we associate to it the generalized even Clifford algebraC0(A, ∗). We will present results on the structure of this algebra C0(A, ∗), some of which are due to Jacobson in (J. Algebra1 (1964), 288-300). Using a generic method, we then give a formula for computing C0(A, ∗) when (A, ∗) is decomposable as algebras with involution. Using these results, a necessary condition for decomposability of an algebra with orthogonal involution compatible with the involution is then obtained.

Quadratic descent of involutions in degree 2 and 4

If K/F is a quadratic extension, we give necessary and sufficient conditions in terms of the discriminant (resp. the Clifford algebra) for a quadratic form of dimension 2 (resp. 4) over K to be similar to a form over F. We give similar criteria for an orthogonal involution over a central simple algebra A of degree 2 (resp. 4) over K to be such that A = A ′ ⊗ F K A = A’ { \otimes _F}K , where A ′ A’ is invariant under the involution. This leads us to an example of a quadratic form over K which is not similar to a form over F but such that the corresponding involution comes from an involution defined over F.

A Variety Associated to an Algebra with Involution

A variety IV( A, *) is canonically associated to a central simple algebra A with orthogonal involution of the first kind *. The properties of this variety are the main topic of this paper. The paper is divided into two parts. We first define this variety to be the subvariety of the Grassmannian of the n-dimensional subspaces of the n 2-dimensional vector space A consisting of all the n-dimensional right ideals I of A, such that I* I = 0. The geometric properties of IV( A, *) are investigated. In particular the generic property of this variety is identified. Further, the kernel of the scalar extension map Br( F) → Br( K), where K is the function field of IV( A, *), is also computed. Lastly, a criterion on the embeddability of the field K into the function field of the Brauer-Severi variety of A is given. In the second part, the Quillen K-theory of IV( A, *) is computed. The K-theory result then leads to an index reduction formula for the function field K.

Products of two involutions in classical groups of characteristic 2

In a corresponding result for fields not of characteristic 2, Wonenburger showed in [6] that every orthogonal transformation is the product of two orthogonal involutions, whereas every symplectic transformation is the product of two skew-symplectic involutions (a skew-symplectic transformation T satisfies B(Tu, Tu) = -(u, u) for u, v in v). It follows from our theorem that if k is a finite field of even characteristic, each element of the symplectic group or of the two nonisomorphic orthogonal groups is conjugate to its inverse, and thus the complex characters of these three families of finite groups are all real-valued. 583 0021.8693/81/080583-09$02.00/0

A Decomposition of Orthogonal Transformations

The purpose of the present note is to give a partial answer to a question raised by Professor Coxeter, namely, if an orthogonal transformation is expressed as a product of orthogonal involutions, how many involutions do we need? Our answer is partial because we are going to consider only non-degenerate symmetric bilinear forms of index 0 and fields of characteristic ≠2. Under these conditions we prove that any orthogonal transformation is the product of at most two orthogonal involutions, which implies that we can write any orthogonal transformation as the product of two involutions.

Fixed Point Free Involutions on the 3-Sphere

In [1], P. A. Smith has proved that the set of fixed points of a periodic homeomorphism T of the 3-sphere S3 onto itself is an i-sphere imbedded in S3, i = -1, 0, 1, or 2, where the -1-sphere denotes the empty set. In trying to prove that an involution (map of period 2) on S3 is conjugate, in the group of all homeomorphisms of S3 onto itself, to an orthogonal involution, it is natural to divide the problem into four cases, as the dimension, n, of the fixed point sphere is -1, 0, 1, or 2, respectively. R. H. Bing, [2], has settled the case n=2 in the negative, unless one assumes that the 2-sphere of fixed points is tamely imbedded. Regarding the case n= 1, Montgomery and Samelson, [3], have shown that, if T is semi-linear, the circle of fixed points cannot be a knotted torus knot of type (p, 2), and moreover, that if the circle of fixed points is not knotted, then the involution, T, is conjugate to an orthogonal involution. This latter result suggests an approach for the case n = -1. An invariant set will mean a set mapped onto itself by the involution. In the absence of fixed points, the invariant circles become important. In fact, we have: