The Jordan normal form of Osserman algebraic curvature tensors

Let V be a vector space of signature (p,q). We construct two families of algebraic curvature tensors which generate the space of algebraic curvature tensors on V. We use these families to show that there exist Jordan Osserman algebraic curvature tensors with arbitrary Jordan normal form.     

The condition numbers of the matrix eigenvalue problem

Summary For ann ×n matrixA with distinct eigenvalues explicit expressions are obtained for certain condition numbers associated with the reduction ofA to its Jordan normal form. These condition numbers are also related by inequalities to (i) the departure from normality ofA, (ii) the discriminant of the eigenvalues ofA, (iii) the Gram determinant of the eigenvectors ofA.     

IDENTITIES FOR ALGEBRAS OBTAINED FROM THE CAYLEY-DICKSON PROCESS

The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 ⁿ for all n ≥ 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees ≤ 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree ≤ 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.     

The classification of N-dimensional non-associative Jordan algebras with (N − 3)-dimensional annihilator

The paper is devoted to give a complete classification of all n-dimensional non-associative Jordan algebras with (n?3)-dimensional annihilator over an algebraically closed field of characteristic ≠2. We also give a complete classification of all n-dimensional Jordan algebras with (n?1)- and (n?2)-dimensional annihilator.     

Admissible Orders of Jordan Loops

A commutative loop is Jordan if it satisfies the identity x²(yx) = (x²y)x. Using an amalgam construction and its generalizations, we prove that a nonassociative Jordan loop of order n exists if and only if n≧ 6 and n≠ 9. We also consider whether powers of elements in Jordan loops are well‐defined, and we construct an infinite family of finite simple nonassociative Jordan loops. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 103–118, 2009     

Osserman Conjecture in dimension <Emphasis Type="Italic">n</Emphasis> ≠ 8, 16

Let Mⁿ be a Riemannian manifold and R its curvature tensor. For a point p Mⁿ and a unit vector X T_pMⁿ, the Jacobi operator is defined by R_X=R(X,·)X. The manifold Mⁿ is called pointwise Osserman if, for every p Mⁿ, the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. Osserman conjectured that globally Osserman manifolds are two-point homogeneous. We prove the Osserman Conjecture for n8, 16, and its pointwise version for n2, 4, 8, 16. Partial result in the case n=16 is also given.Mathematics Subject Classification (2000): 53B20Work supported by MRDGS internal grant and by ARC grant S6005288.     

Osserman manifolds of dimension 8

For a Riemannian manifold Mⁿ with the curvature tensor R, the Jacobi operator R_X is defined by R_XY=R(X,Y)X. The manifold Mⁿ is called pointwise Osserman if, for every pMⁿ, the eigenvalues of the Jacobi operator R_X do not depend of a unit vector XT_pMⁿ, and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manifolds are flat or locally rank-one symmetric. This Conjecture is true for manifolds of dimension n8,16[14]. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.Mathematics Subject Classification (2000): 53B20     

Isotopes of prime (−1, 1)-and Jordan algebras

We deal with adjoint commutator and Jordan algebras of isotopes of prime strictly (−1, 1)-algebras. It is proved that a system of identities of the form [x ₁, x ₂, x ₂, x ₃,…, x _n ] for n = 2,.…, 5 is discernible on isotopes of prime (−1, 1)-algebras. Also it is shown that adjoint Jordan algebras for suitable isotopes of prime (−1, 1)-algebras may possess distinct sets of identities. In particular, isotopes of a prime Jordan monster have different sets of identities in general.     

Endoprimal implication algebras

An algebra A is endoprimal if, for all , the only maps which preserve the endomorphisms of A are the n-ary term functions of A. The theory of natural dualities has been a very effective tool for finding finite endoprimal algebras. We study endoprimality within the variety of implication algebras, which does not contain any non-trivial dualisable algebras. We show that there are no non-trivial finite endoprimal implication algebras. We also give some examples of infinite implication algebras which are endoprimal. Received July 28, 1998; accepted in final form January 18, 1999.     

Diagonalization and Jordan normal form – motivation through Maple ®

Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package Maple®.     

Noncommutative Jordan superalgebras of degree <Emphasis Type="Italic">n</Emphasis> > 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a “nodal” case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.     

Isospectral deformations of metrics on spheres

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R ⁿ for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric. Oblatum 19-VI-2000 & 21-II-2001?Published online: 4 May 2001     

Homotopy of ends and boundaries of CAT(0) groups

For n ≥ 0, we exhibit CAT(0) groups that are n-connected at infinity, and have boundary which is (n − 1)-connected, but this boundary has non-trivial nth-homotopy group. In particular, we construct 1-ended CAT(0) groups that are simply connected at infinity, but have a boundary with non-trivial fundamental group. Our base examples are 1-ended CAT(0) groups that have non-path connected boundaries. In particular, we show all parabolic semidirect products of the free group of rank 2 and have a non-path connected boundary.     

Decomposition of Jordan Automorphisms of Strictly Upper Triangular Matrix Algebra Over Commutative Rings

Over a 2-torsionfree commutative ring R with identity, the algebra of all strictly upper triangular n + 1 by n + 1 matrices is denoted by n ₁. In this article, we prove that any Jordan automorphism of n ₁ can be uniquely decomposed as a product of a graph automorphism, a diagonal automorphism, a central automorphism and an inner automorphism for n ≥ 3. In the cases n = 1, 2, we also give a decomposition for any Jordan automorphism of n ₁.     

On the notion of essential dimension for algebraic groups

We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our groupG isS _n , these objects are field extensions; ifG=O _n , they are quadratic forms; ifG=PGL _n , they are division algebras (all of degreen); ifG=G ₂, they are octonion algebras; ifG=F ₄, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, relating essential dimension to Galois cohomology.Partially supported by NSA grant MDA904-9610022 and NSF grant DMS-9801675     

The Tits–Kantor–Koecher Construction and Birepresentations of the Jordan Superpair SH(1, n)

Abstract

We introduce the concept of bimodule over a Jordan superpair and the Tits– Kantor–Koecher construction for bimodules. Using the construction we obtain the classification of irreducible bimodules over the Jordan superpair SH(1, n). We also prove semisimplicity for a class of finite dimensional SH(1, n)-bimodules for n ≥ 3.     

Canonical forms of some special matrices useful in statistics

In experimental situations wheren two or three level factors are involved andn observations are taken, then theD-optimal first order saturated design is ann ×n matrix with elements ±1 or 0, ±1 with the maximum determinant. Canonical forms are useful for the specification of the non-isomorphicD-optimal designs. In this paper, we study canonical forms such as the Smith normal form, the first, second and the Jordan canonical form ofD-optimal designs. Numerical algorithms for the computation of these forms are described and some numerical examples are also given.     

Multiplikatorsysteme und Charaktere Hermitescher Modulgruppen

We consider the hermitian modular group U(2n, o),n2, over an arbitrary imaginary-quadratic number field of discriminantd. By K-theoretic methods it is known that SU(2n, o) is generated by elementary matrices. Therefore, it is easy to determine the factor commutator group and all abelian characters of U(2n, o). Apart from characters which come via the determinant from non-trivial units, there is only one other non-trivial character of U(2n, o) in those cases wheren=2,d0 mod 4. This character arises in the same way as for Siegel's modular group Sp(4, ) by an action of U(4, o) on odd theta characteristics.Multiplier systems of 2n are characters, since they must have integral weight. Therefore as a corollary we obtain all multiplier systems of 2n.

     

On certain rational quadratic forms

We give safety neighbourhoods for the necessary conditions in the change of the Jordan canonical form of a matrix under small perturbations. We also obtain the minimum distance from an n × n complex matrix which has less than k nonconstant invariant factors (2≤ k≤ n) to the set of matrices which have more or equal to k. When k= 2, we get in particular the distance from a nonderogatory matrix to the set of derogatory matrices.