Quaternionic Case Research Articles

The cut-off phenomenon for random reflections II: Complex and quaternionic cases

The cut-off phenomenon for random reflections II: Complex and quaternionic cases

Maximal Orders in Central Simple Algebras and Bruhat–Tits Buildings

We study the affine building forSLnover a local field and give a characterization of distance involving Hecke operators. Forn=3 we give an explicitly computable distance formula. We use this local information to show that the class number of a maximal order in a central simple algebra of dimensionn2over a number fieldKis equal to the number of orbits of a group of isometries (related to the unit group of the maximal order) acting on a Bruhat-Tits building forSLn(K). This generalizes results of Serre and Vignéras who considered the quaternion case in which the Bruhat-Tits building is a tree.

The local zeta function for the non-trivial characters associated with the singular Jordan algebras

This paper investigates the local integrals \[ Z m ( t , χ ) = ∫ H m ( O C ) χ ( det ( x ) ) | det ( x ) | s d x Z_m(t,\chi )=\int _{H_m(O_C)} \chi ( \det (x)) | \det (x) |^s dx \] where O C O_C represents the integers of a composition algebra over a non-archimedean local field K K and χ \chi is a non-trivial character on the units in the ring of integers of K K extended to K ∗ K^* by setting χ ( π ) = 1 \chi (\pi )=1 . The local zeta function for the trivial character is known for all composition algebras C C . In this paper, we show in the quaternion case that Z ( t , χ ) = 0 Z(t, \chi )=0 for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for χ \chi equal to the quadratic character. In this latter case, Z ( t , χ ) = 0 Z(t, \chi )=0 for any character of order greater than 2 2 .

On a Noether Normalization for a mod-2 Cohomology Ring

An almost extraspecial group with a nonsingular quadratic form is treated. There are three cases: real, complex, and quaternion. The mod-2 cohomology ring has three subrings: the subring generated by Stiefel-Whitney classes of a unique faithful irreducible real representation, an invariant subring of the orthogonal group, and that of its commutator subgroup. The object of this paper is to show that the three subrings are very close to one another. Particularly in the quaternion case, they coincide with each other module the nilradical of the cohomology ring. It is well known that the first is a Noether normalization for the cohomology ring, while it was shown by us that the third is a subring of universally stable elements defined by Evens and Priddy (with a few exceptions of small order). Thus it may be concluded that the last is nothing but a Noether normalization for the cohomology ring.

Determination of Grassmann Manifolds Which are Boundaries

AbstractLetFGn,kdenote the Grassmann manifold of allk-dimensional (left)F-vector subspace ofFnfor F = ℝ, the reals,C, the complex numbers, orHthe quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in [4]. Partial results were obtained in [4] for the caseF= ℝ, including a sufficient condition, due to A. Dold, on n and k for ℝGn,kto bound. Here, we show that Dold's condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well.

The loops onU(n)/O(n) andU(2n)/Sp(n)

In [6] and [9] the second author and Bill Richter showed that the natural ‘degree’ filtration on the homology of ΩSU(n) has a geometric realization, and that this filtration stably splits (as conjectured by M. Hopkins and M. Mahowald). The purpose of the present paper is to prove the real and quaternionic analogues of these results. To explain what this means, consider the following two ways of viewing the filtration and splitting for ΩSU(n). Whenn= ∞, ΩSU=BU. The filtration isBU(1)⊆BU(2)⊆… and the splittingBU≅ V1≤<∞is a theorem of Snaith[14]. The result for ΩSU(n) may then be viewed as a ‘restriction’ of the result forBU. On the other hand there is a well-known inclusion ℂPn−1. This extends to a map ΩΣℂPn−1→ΩSU(n), and the filtration (or splitting) may be viewed, at least algebraically, as a ‘quotient’ of the James filtration (or splitting) of ΩΣℂPn−1. It is now clear what is meant by the ‘real and quaternionic analogues’. In the quaternionic case, we replaceBUbyBSp=Ω(SU/SP), ΩSU(n) by Ω(SU(2n)/SP(n))and ℂPn−1by ℍPn−1. The integral homology of Ω(SU(2n)/SP(n)) is the symmetric algebra on the homology of ℍPn−1, and may be filtered by the various symmetric powers. We show that this filtration can be realized geometrically, and that the spaces of the filtration are certain (singular) real algebraic varieties (exactly as in the complex case). The strata of the filtration are vector bundles over filtrations of Ω(SU(2n−2)/SP(n−1)), and the filtration stably splits. See Theorems (1·7) and (2·1) for the precise statement. In the real case we replaceBUby Ω(SU/SO), Ω(SU(n)/SO(n)) and ℂPn−1by ℝPn−1. Here integral homology must be replaced by mod 2 homology, and splitting is only obtained after localization at 2. (Snaith's splitting ofBOin [14] can be refined [2, 8] so as to be exactly analogous to the splitting ofBU:BO≅V1≤<∞MO(k).)

A generalization of the momentum mapping construction for quaternionic K�hler manifolds

We present a method of reduction of any quaternionic Kahler manifold with isometries to another quaternionic Kahler manifold in which the isometries are divided out. Our method is a generalization of the Marsden-Weinstein construction for symplectic manifolds to the non-symplectic geometry of the quaternionic Kahler case. We compare our results with the known construction for Kahler and hyperKahler manifolds. We also discuss the relevance of our results to the physics of supersymmetric non-linear σ-models and some applications of the method. In particular, we show that the Wolf spaces can be obtained as theU(1) andSU(2) quotients of quaternionic projective spaceHP(n). We also construct an interesting example of compact riemannianV-manifolds(orbifolds) whose metrics are quaternionic Kahler and not symmetric.

Groups Preserving a Class of Bilinear Functions

AbstractGiven a finite dimensional Euclidean vector space V, ( , ) and an involution τ of V, one can form the bilinear function ( , )τ defined by (x, y)τ = (τ(x), y), x,y ∊ V.Let O(τ) = {ϕ ∊ GL(V)|(ϕx, ϕy)τ = (x, y)τ}.If t is self-adjoint the structure of O(t) is well known. The purpose of this paper is to detemine the structure of O(t) in the general case. This structure is also determined in the complex and quaternionic case, as well as the case when the condition on t is replaced by τ2 = ∊ι, ∊ ∈ ℝ.

Statistical analysis based on quaternion normal random variables

The quaternion normal distribution is derived and a number of characteristics are highlighted. The maximum likelihood estimation procedure in the quaternion case is examined and the conclusion is reached that the estimation procedure is simplified if the unknown parameters of the associated real probability density function are estimated. The quaternion estimator is then obtained by regarding these estimators as the components of the quaternion estimator. By means of a example attention is given to a test criterium which can be used in the quaternion model.

Quantum mechanics of the quaternionic Hilbert spaces based upon the imprimitivity theorem

We survey the realization of quantum mechanics in quaternionic Hilbert spaces following the methods of Mackey, who examined the complex and real cases exploiting the imprimitivity theorem. We show that there exists a unique unitary skew-adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic imaginary unit. Difficulties in the definition of time reversal, however, arise because of the properties of the quaternionic field. The introduction of an extra imaginary unit, commuting with the others, is suggested in order to implement time reversal properly. In the Appendix we give the proof of the imprimitivity theorem, in the quaternionic case, that we use in the paper.

Symmetric systems with semi-simple structure algebra: The quaternionic case

The basic theory of linear systems over the quaternions is developed.

Quadratic Forms Permitting Triple Composition

In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, $Q\left (\{{xyz} \} \right ) = Q(x)Q(y)Q(z)$. In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples $\{xyz \} = (xy)z$ or $x(yz)$ determined by a composition algebra, with the quadratic form $Q$ the usual norm form. For any fixed $Q$ this leads to $1$ isotopy class in dimensions $1$ and $2$, $3$ classes in the dimension $4$ quaternion case, and $6$ classes in the dimension $8$ octonion case.

Some James numbers of Stiefel manifolds

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

Rational parametrisation of normalised stiefel manifolds, and explicit non-'t hooft solutions of the Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations for Sp(n)

It is proved that normalised Stiefel manifolds admit a rational parametrisation which generalises Cayley's parametrisation of the unitary groups. Applying (the quaternionic case of) this parametrisation to the Atiyah-Drinfeld-Hitchin-Manin (ADHM) instanton matrix equations, large families of new explicit rational solutions emerge. In particular, new explicit non-'t Hooft solutions are presented.

Finite quaternionic reflection groups

In this article the quaternionic reflection groups are classified. Such a group is defined so as to generalize the notion of reflection groups appearing in [4, 171, i.e., it is a group of linear transformations in a quaternionic vector space of dimension n < cc generated by elements that fix an (n 1)-dimensional subspace pointwise. Moreover, the notion of root system as given in 13, 41 is extended to the quaternionic case. These systems can be used to construct the groups involved and vice versa. Relevant definitions can be found in Section 1. In the following section, the imprimitive groups are determined in a relatively simple manner that resembles the complex case. The primitive groups are classified by means of their complexifications (i.e., their natural isomorphic images in the group of all invertible linear transformations in the 2n-dimensional complex vector space underlying the given quaternionic one). The primitive groups with imprimitive complexifications are dealt with in Section 3. In order to determine the remaining groups, extensive use is made of the work of’Huffman and Wales [12-15, 201 on the classification of primitive unimodular complex linear groups generated by elements that fix a subspace of codimension 2 pointwise. If the complexification of a quaternionic reflection group is primitive, it belongs to this family of groups. Section 4 is about these groups and their root systems. The main results are comprised in Theorems 2.6, 2.9, 3.6, and 4.2.

Geometric quantization and internal symmetry

As a part of an attempt to geometrize physics, internal symmetries in the covariant classification of matter by itsT μυ type are considered in relation to phase transformations generated by complex and quaternionic structures on space-time. The Rainich theory of electromagnetism and neutrinos is compared with the theory ofU(1) ×SO(1, 3) torsional gauge fields, and extended to the quaternionic case. It is shown by the Kostant technique of geometric quantization that complex and quaternionic phase transformations for an Einstein space are associated with one-dimensional and three-dimensional harmonic oscillators.

Dual models with a colour symmetry

It is shown that all dual models derived maively from a recently discovered class of infinite graded Lie algebras with O( N) symmmetry posses ghosts if N > 2 with the possible exception of the quaternion case.