Dragomir Ž. Doković

Some infinite-dimensional simple Lie algebras in characteristic 0 related to those of Block

Abstract Given a nontrivial torsion-free abelian group (A, +, 0), a field F of characteristic 0, and a nondegenerate bi-additive skew-symmetric map ϑ:A × A → F, we study the Lie algebra L (A, ϑ) over F with basis {ex: x ϵ A⧹{0}} and multiplication [ex, ey] = ϑ(x, y)ex + y. We show that L (A, ϑ) is simple, determine its derivations, and show that the locally finite derivations D have the form D(ex) = μ(x)ex, μ ϵ Hom(A, F). We describe all isomorphisms between two such algebras. Finally, we compute H 2 ( L , F) .

Equivalence classes and representatives of Golay sequences

Abstract We introduce the notion of canonical form for Golay sequences such that every equivalence class contains exactly one member having the canonical form. Golay and Turyn have shown how to multiply Golay sequences of length m with Golay sequences of length n in order to construct Golay sequences of length mn . We say that Golay sequences of length n are constructible if they can be manufactured from Golay sequences of length n by using the multiplication operation and the elementary transformations. We list representatives of the equivalence classes of Golay sequences for all lengths ⩽40 and determine which classes are constructible.

Low-dimensional representations of Aut (F 2)

LetF2 be the free group of rank two, and Φ2 its automorphism group. We consider the problem of describing the representations of Φ2 of degreen for small values ofn. Our main result is the classification (up to equivalence) of all indecomposable representations ρ of Φ2 of degreen≤4 such that ρ(F2) ≠ 1. There are only finitely many such representations, and in all them ρ(F2) is solvable. This is no longer true in higher dimensions. Already forn=6 there exists a 1-parameter family of irreducible nonequivalent representations of Φ2 such that ρ(F2) contains a free non-abelian subgroup. We also obtain some new 4-dimensional representations of the braid groupB4 which are indecomposable and reducible at the same time. It would be interesting to find some applications of these representations.

Skew Hadamard matrices of order 4 x 37 and 4 x 43

Abstract In this paper we construct two skew Hadamard matrices of order 4 × 37 and one skew Hadamard matrix of size 4 × 43. No skew Hadamard matrices of these orders were known before. We also construct another Hadamard matrix (not of skew type) of size 4 × 37. All four of these matrices are of Goethals-Seidel type.

Pairs of involutions in the general linear group

In Theorem 1 we classify all finite-dimensional indecomposable representations of the infinite dihedral group over an arbitrary field k of char k ≠ 2. Of course, that result is equivalent to the problem of finding a “normal form” for a pair of involutions in GLn(k). In Theorem 2 we find necessary and sufficient conditions for a pair of conjugate involutions in GLn(k) to be conjugate via an involution, where k is assumed to be algebraically closed and char k ≠ 2.

Periodic Golay Pairs of Length 72

We construct supplementary difference sets (SDSs) with parameters (72; 36, 30; 30). These SDSs give periodic Golay pairs of length 72. No periodic Golay pair of length 72 was known previously. The smallest undecided order for periodic Golay pairs is now 90. The periodic Golay pairs constructed here are the first examples having length divisible by a prime congruent to 3 modulo 4. The main tool employed is a recently introduced compression method. We observe that Turyn’s multiplication of Golay pairs can be also used to multiply a Golay pair and a periodic Golay pair.

Williamson matrices of orders 4.29 and 4.31

Abstract It is established that there is only one, up to equivalence, Williamson matrix of order 4·29 and there are only two non-equivalent such matrices of order 4·31.

D-Optimal Matrices of Orders 118, 138, 150, 154 and 174

We construct supplementary difference sets (SDSs) with parameters (59; 28, 22; 21), (69; 31, 27; 24), (75; 36, 29; 28), (77; 34, 31; 27) and (87; 38, 36; 31). These SDSs give D-optimal designs (DO-designs) of two-circulant type of orders 118,138,150,154 and 174. Until now, no DO-designs of orders 138,154 and 174 were known. While a DO-design (not of two-circulant type) of order 150 was constructed previously by Holzmann and Kharaghani, no such design of two-circulant type was known. The smallest undecided order for DO-designs is now 198. We use a novel property of the compression map to speed up some computations.

Some new generating functions for weight multiplicities

Let g be a semisimple complex Lie algebra of rank l, and ω1,ω2,…,ωl its fundamental weights. Let Vω be a simple g-module with highest weight ω, and Vωλ its weight space of weight λ. We study the generating functions fλ=∑r1,…,r1≥0mλr1ωl+⋯+rlωltr11tr22⋯trll, where mωλ = dim Vωλ and obtain explicit formulae for fλ when g is simple of rank 2. All of these formulae seem to be new.

Quadratic cones invariant under some linear operators

A (solid) quadratic cone K in a finite-dimensional vector space V (over