Dragomir Ž. Djoković

Distance-preserving subgraphs of hypercubes

Abstract We give a characterization of connected subgraphs G of hypercubes H such that the distance in G between any two vertices a, b ϵ G is the same as their distance in H. The hypercubes are graphs which generalize the ordinary cube graph.

Hermitian matrices over polynomial rings

Abstract Let R be a twisted polynomial ring F[X; S, D] where F is a division ring, S is an automorphism of F and D is an S-derivation of F. Thus Xα = αSX + αD holds for every α ϵ F. Let ∗ be an involution of R such that F ∗ = F . Then we show that every anisotropic hermitian matrix a over R is congruent to a matrix b such that the degree of every diagonal entry of b is strictly larger than the degree of any off-diagonal entry in the same row. It follows that if a is also unimodular then it must be congruent to a diagonal matrix. If F = CorH (complexes or quaternions) and S = 1, D = 0, X ∗ = X while the restriction of ∗ to F is the conjugation then every positive semidefinite matrix a over R admits a factorization a = b ∗ b with b a square matrix over R.

Classification of trivectors of an eight-dimensional real vector space

(1983). Classification of trivectors of an eight-dimensional real vector space. Linear and Multilinear Algebra: Vol. 13, No. 1, pp. 3-39.

On regular graphs, VI☆

This paper is a continuation of [1] and we shall use the same terminology. The main result of this part is the following: Suppose that the automorphism group of a connected graph of valency p + 1, p a prime, has a subgroup which acts as a regular permutation group on the set of s-arcs of the graph. Then s ≤ 7 and s ≠ 6. The case in which the graph is finite and p = 2 was studied by Tutte [3]. Note that we do not assume that graphs are finite.

Classification of some 2-graded lie algebras

We continue the study of the structure of 2-graded Lie algebras (see next section for the definition) which was begun by Hochschild [6] and Hochschild and the author [4]. AU semi-simple 2-graded Lie algebras over an algebraically closed field of characteristic 0 have been determined in [4]. In this paper we introduce the analog of the Killing form for 2-graded Lie algebras. AS in the case of ordinary Lie algebras, this form plays an important role in the structure theory. For semi-simple 2-graded Lie algebras this form is non-degenerate although the converse is not true. Semi-simplicity of 2-graded Lie algebras is preserved under both extension and restriction of scalars. Our main result is the classification of indecomposable 2-graded Lie algebras L over an algebraically closed field of characteristic 0 under the condition that the zero component Lo of L be simple (Theorem 8). There are seven infinite series of such algebras. This classification is accomplished in two steps. The first step is the classification of those 2-graded Lie algebras L which are simple and have Lo simple (Theorem 6). In the proof of this theorem we need the following fact: If‘ L, is a simple Lie algebra then the multiplicity 172 r) (resp. nzl ) of L, in the symmetric power S2(Lo) (resp. the exterior power A-(Lo)) is nzo = I if Lo =: rl ,t (112 2),

Classification of pairs consisting of a linear and a semilinear map

Abstract L. Kronecker has found normal forms for pairs ( A , B ) of m -by- n matrices over a field F when the admissible transformations are of the type ( A , B )→( SAT , SBT ), where S and T are invertible matrices over F . For the details about these normal forms we refer to Gantmachers book on matrices [5, Chapter XII]. See also Dicksons paper [3]. We treat here the following more general problem: Find the normal forms for pairs ( A , B ) of m -by- n matrices over a division ring D if the admissible transformations are of the type ( A , B )→( SAT , SBJ ( T )) where J is an automorphism of D . It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D = C , J =conjugation is essentially equivalent to the recent problem of Dlab and Ringel [4]. This is explained thoroughly in Sec. 6. We conclude with two open problems.

Orbits of Nilpotent Matrices

Abstract The action under conjugation of invertible lower triangular n × n matrices (over an infinite field) on lower triangular nilpotent matrices, N n , divides N n into orbits. We show that for n ⩾6 the number of orbits is infinite.

Products of reflections in the quaternionic unitary group

Let Sp(n) denote the unitary group of a positive definite hermitian form on an n-dimensional right quaternionic vector space. Thus Sp(n) is the compact real form of the complex symplectic group Sp(2n, C). For each A ϵ Sp(n), n ⩾ 2, we determine the smallest number m such that A can be written as a product of m reflections in Sp(n). In particular, we show that if n ⩾ 4 then every A ϵ Sp(n) is a product of n, n + 1, or n + 2 reflections. In the projective group PSp(n), n > 4, every element is a product of n reflections.

Irreducible connected Lie subgroups of Gl n (R) are closed

IfG is a connected real Lie group and π:G→Aut (V) a continuous irreducible finite-dimensional real representation then we show that π(G) is closed in Aut(V). A similar result is valid in the complex case.

Positive semi-definite matrices as sums of squares

Abstract If K is a field and char K ≠ 2, then an element α ϵ K is a sum of squares in K if and only if α ⩾ 0 for every ordering of K . This is the famous theorem of Artin and Landau. It has been generalized to symmetric matrices over K by D. Gondard and P. Ribenboim. They have also shown that Artins theorem on positive definite rational functions has a suitable extension to positive definite matrix functions. In this paper we attain two goals. First, we show that similar theorems are valid for Hermitian matrices instead of symmetric ones. Second, we simplify D. Gondard and P. Ribenboims proof of their second theorem and strengthen it.