Involutorial Antiautomorphism Research Articles

Commutative Nilpotent Closed Algebras and Weil Representations

Let 𝔽 be the field GF(q 2) of q 2 elements, q odd, and let V be an 𝔽-vector space endowed with a nonsingular Hermitian form ϕ. Let σ be the adjoint involutory antiautomorphism of End𝔽 V associated to the form, and let U(ϕ) be the corresponding unitary group. We ask whether the restrictions of the Weil representation of U(ϕ) to certain subgroups are multiplicity-free. These subgroups consist of the members of U(ϕ) in subalgebras of the form 𝔽I + N, where N is a σ-stable commutative nilpotent subalgebra of End𝔽 V with the further property that N contains its annihilator. We give a necessary condition for multiplicity-freeness that depends on the dimensions of N and that annihilator. Moreover, the case that N is conjugate to its regular representation is completely settled. Several other classes of subalgebra are discussed in detail.

Mackey’s theory of $${\tau}$$ τ -conjugate representations for finite groups

The aim of the present paper is to expose two contributions of Mackey, together with a more recent result of Kawanaka and Matsuyama, generalized by Bump and Ginzburg, on the representation theory of a finite group equipped with an involutory anti-automorphism (e.g. the anti-automorphism \({g \mapsto g^{-1}}\)). Mackey’s first contribution is a detailed version of the so-called Gelfand criterion for weakly symmetric Gelfand pairs. Mackey’s second contribution is a characterization of simply reducible groups (a notion introduced by Wigner). The other result is a twisted version of the Frobenius–Schur theorem, where “twisted” refers to the above-mentioned involutory anti-automorphism.

Hamiltonian square roots of skew Hamiltonian quaternionic matrices

Criteria for existence of Hamiltonian quaternionic matrices that are square roots of a given skewHamiltonian quaternionic matrix are developed. The criteria are formulated in terms of respective canonical forms of skewHamiltonian quaternionic matrices. The Hamiltonian property is understood with respect to either the quaternionic conjugation, or an involutory antiautomorphism of the quaternions which is different from the quaternionic conjugation. Many results are stated and proved in a more general framework of symmetric and skewsymmetric matrices with respect to an invertible matrix which is skewsymmetric relative to an involutory antiautomorphism.

Canonical forms for mixed symmetric–skewsymmetric quaternion matrix pencils

Canonical forms are described for pairs of quaternionic matrices, or equivalently matrix pencils, where one matrix is symmetric and the other matrix is skewsymmetric, under strict equivalence and symmetry respecting congruence. The symmetry is understood in the sense of a fixed involutory antiautomorphism of the skew field of the real quaternions; the involutory antiautomorphism is assumed to be nonstandard, i.e., other than the quaternionic conjugation. Some applications are developed, such as canonical forms for quaternionic matrices under symmetry respecting congruence, and canonical forms for matrices that are skewsymmetric with respect to a nondegenerate symmetric or skewsymmetric quaternion valued inner product.

On the Lattice of Varieties of Involution Semigroups

In this paper, we investigate the lattice \smallbf L( \cal S ^\ast) of varieties of involution semigroups (semigroups endowed with an involutorial antiautomorphism ^\ast as a fundamental operation). There are two kinds of atoms in \smallbf L( \cal S ^\ast) : four nongroup ones and two countably infinite families of varieties of groups with involution. We exhibit the sublattices of \smallbf L( \cal S ^\ast) generated by both of these two collections of its atoms.

Minimal varieties of semirings with involution

We give a complete list of minimal varieties (varieties having no proper subvarieties) of semirings equipped with an involutorial antiautomorphism as a unary fundamental operation.

All Varieties of Normal Bands with Involution

In the present paper we completely determine the lattice L(NB*) of all varieties of normal bands equipped with an involutorial antiautomorphism as a fundamental operation.

Remarks on varieties of involution bands

In the present paper, we study varieties consisting of bands (idempotent semigroups) endowed with an involutorial antiautomorphism * as a fundamental operation. Our principal aim is here to provide an insight to some classes of these varieties from the structural point of view, especially in terms of semilattice decompositions, subdirect products and ideal extensions. In the course of such considerations, we shall extend the result of C. L. Adair [1], who described the lattice of all varieties of bands with a regular involution (i.e. with the identity x = xx * x) We depict a broader lattice of involution band varieties, which incorporates Adair’s lattice.

On the non-trace-valued forms

Let { + f be an involutory antiautomorphism of the divison ring k and E a k-vector space. A hermitian form @ on E is said to be trace-valued if, for all x E E, @(x, x) is a trace, i.e ., of the form < + f for some <E k [ 11. If the characteristic is not 2 or if the center of k is not fixed under the involution then all @ are trace-valued. Non-trace-valued forms are known as rather unmanageable-even in finite dimensions: nondegenerate isotropic planes may fail to be hyperbolic, the cancellation theorem is not valid and so Witt’s theorem does not hold either. This last fact seems to entail a particularly unfavourable verdict on the nontrace-valued forms as the classic theory of quadratic forms pivots on this theorem (see also the contention supported in [6, p. 2491). It is the purpose of this paper to demonstrate that matters may be looked at differentently and that the classification problem on subspaces in the nontrace-valued situation may be successfully attacked by what might be called the lattice method. The ground for this method has been laid in [2]. In order to give perspicuousness to the constituent parts we have stripped the problem of all unnecessary complications. In particular, we have made the following assumptions throughout the paper:

A Finitely Generated Modular Ortholattice

By an ortholattice we mean a lattice with 0 and 1 and a complementation operation which is an involutorial antiautomorphism. The free modular ortholattice on two generators has 96 elements—cf. J. Kotas [8].

Posibilidades de generalización de las lógicas cuánticas

Resumen

Bemerkung zu Einem Satz von Baer

Let α be an involutorial antiautomorphism of the division ring K with char K ≠ 2 and E := {r ∈ K | rrα = 1}. Refining a result of R. Baer we prove: (card E−1)2 < card K ⇒ α=id. This is done by computing card E and the cardinal numbers of the sets consisting of the α-symmetric respectively the α-skew-symmetric elements.

SIMPLE ALGEBRAS WITH INVOLUTION, AND UNITARY GROUPS

Let A be a central simple algebra on which an involutory antiautomorphism S is given whose restriction to the center K of A is not the identity. Let ?(A*) be the subgroup of the multiplicative group A* of A generated by the elements x A* such that xS = x, let NrdA/K : ?A?K be the reduced norm mapping of A into K, and let ? '(A*) be the subgroup of A* generated by the elements x A* whose reduced norm is invariant with respect to S. This paper considers the problem of when the groups ? '(A*) and ?(A*) coincide. Bibliography: 15 titles.