Том 243, № 4 (2019)
- Жылы: 2019
- Мақалалар: 12
- URL: https://journals.rcsi.science/1072-3374/issue/view/15038
Article
On Certain Multiplicative Structures on Cubic Extensions
Аннотация
Multiplicative properties of a certain correspondence between the elements of a cyclic cubic extension of rational number field and elements in a suitable pure cubic extension are investigated. The case of Shanks cubic polynomial is considered to connect the multiplication of pure cubic irrational and the summation of points of an associated elliptic curve.
Towards the Reverse Decomposition of Unipotents
Аннотация
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.
Unrelativized Standard Commutator Formula
Аннотация
In the present note, which is a marginalia to the previous papers by Roozbeh Hazrat, Alexei Stepanov, Zuhong Zhang, and the author, I observe that for any ideals A,B≤R of a commutative ring R and all n ≥ 3 the birelative standard commutator formula also holds in the unrelativized form, as [E(n,A),GL(n,B)] = [E(n,A),E(n,B)] and discuss some obvious corollaries thereof.
Hochschild Cohomology of Algebras of Dihedral Type. VIII. Hochshild Cohomology Algebra for the Family D(2ℬ)(k, s, 0) in Characteristic 2
Аннотация
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\( \mathcal{A} \)).
Products of Commutators on a General Linear Group Over a Division Algebra
Аннотация
The word maps \( \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) \) and \( \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } \) for a word \( w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], \) where D is a division algebra over a field K, are considered. It is proved that if \( \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], \) then \( \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), \) where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then \( \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). \) The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.
On a Question About Generalized Congruence Subgroups. I
Аннотация
A set of additive subgroups σ = (σij), 1 ≤ i, j ≤ n, of a field (or ring) K is called a net of order n over K if σirσrj ⊆ σij 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 = (σij) is said to be irreducible if all the additive subgroups σij 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 = (σij) 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 = (σij) of order n over K, for which the group E(σ) ∩ 〈tij(K) : i, j ∈ J〉 is not contained in the group 〈tij(σij) : i, j ∈ J〉, is constructed.
Explicit Equations for Exterior Square of the General Linear Group
Аннотация
Several explicit systems of equations defining the exterior square of the general linear group ⋀2 GLn 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 An−1 in the representation with the highest weight ϖ2.
Plotkin’s Geometric Equivalence, Mal’cev’s Closure, and Incompressible Nilpotent Groups
Аннотация
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.
Eparability of Schur Rings Over an Abelian Group of Order 4p
Аннотация
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.