Volume 243, Nº 4 (2019)

Decomposition of unipotents gives short polynomial expressions of the conjugates of elementary generators as products of elementaries. It turns out that with some minor twist the decomposition of unipotents can be read backwards to give very short polynomial expressions of the elementary generators themselves in terms of elementary conjugates of an arbitrary matrix and its inverse. For absolute elementary subgroups of classical groups this was recently observed by Raimund Preusser. I discuss various generalizations of these results for exceptional groups, specifically those of types E₆ and E₇, and also mention further possible generalizations and applications.

The Hochschild cohomology algebra for the algebras of dihedral type in the subfamily of the family D(2ℬ), for which the parameter c is equal to 0, are described. The calculation of multiplication in this cohomology algebra, uses the minimal bimodule projective resolution for algebras under consideration, that was constructed in the previous paper of the authors. The obtained results allow to describe the Hochschild cohomology algebra also for algebras with c = 0 in the family D(2\( \mathcal{A} \)).

A set of additive subgroups σ = (σ_ij), 1 ≤ i, j ≤ n, of a field (or ring) K is called a net of order n over K if σ_irσ_rj ⊆ σ_ij for all values of the indices i, r, j. The same system, but without diagonal, is called an elementary net. A full or elementary net σ = (σ_ij) is said to be irreducible if all the additive subgroups σ_ij are different from zero. An elementary net σ is closed if the subgroup E(σ) does not contain new elementary transvections. The present paper is related to a question posed by Y. N. Nuzhin in connection with V. M. Levchuk’s question No. 15.46 from the Kourovka notebook about the admissibility (closure) of elementary net (carpet) σ = (σ_ij) over a field K. Let J be an arbitrary subset of {1, . . . , n}, n ≥ 3, and the cardinality m of J satisfies the condition 2 ≤ m ≤ n − 1. Let R be a commutative integral domain (non-field) with identity, and let K be the quotient field of R. An example of a net σ = (σ_ij) of order n over K, for which the group E(σ) ∩ 〈t_ij(K) : i, j ∈ J〉 is not contained in the group 〈t_ij(σ_ij) : i, j ∈ J〉, is constructed.

Several explicit systems of equations defining the exterior square of the general linear group ⋀² GL_n as affine group scheme are presented. Algebraic ingredients of the equations, so called exterior numbers, are translated to the language of weight diagrams corresponding to a Lie group of type A_n−1 in the representation with the highest weight ϖ₂.

In 1997, B. I. Plotkin introduced a concept of geometric equivalence of algebraic structures and posed a question: is it true that every nilpotent torsion-free group is geometrically equivalent to its Mal’cev’s closure? A negative answer in the form of three counterexamples was given by V. V. Bludov and B. V. Gusev in 2007. In the present paper, an infinite series of counterexamples of unbounded Hirsch rank and nilpotency degree is constructed.

An S-ring (a Schur ring) is said to be separable with respect to a class of groups if every its algebraic isomorphism to an S-ring over a group from is induced by a combinatorial isomorphism. It is proved that every Schur ring over an Abelian group G of order 4p, where p is a prime, is separable with respect to the class of Abelian groups. This implies that the Weisfeiler-Lehman dimension of the class of Cayley graphs over G is at most 3.