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Том 240, № 4 (2019)
- Год: 2019
- Статей: 9
- URL: https://bakhtiniada.ru/1072-3374/issue/view/15015
Article
On Stably Biserial Algebras and the Auslander–Reiten Conjecture for Special Biserial Algebras
Аннотация
According to a result claimed by Pogorza_ly, selfinjective special biserial algebras can be stably equivalent to stably biserial algebras only, and these two classes coincide. By an example of Ariki, Iijima, and Park, the classes of stably biserial and selfinjective special biserial algebras do not coincide. In these notes based on some ideas from the Pogorzały paper, a detailed proof is given for the fact that a selfinjective special biserial algebra can be stably equivalent to a stably biserial algebra only. The structure of symmetric stably biserial algebras is analyzed. It is shown that in characteristic other than 2, the classes of symmetric special biserial (Brauer graph) algebras and symmetric stably biserial algebras coincide. Also a proof of the Auslander–Reiten conjecture for special biserial algebras is given.
375-394
Hochschild Cohomology for Algebras of Semidihedral Type. VIII. The Family SD(2B)1
Аннотация
The Hochschild cohomology groups for algebras of semidihedral type, that are contained in the family SD(2B)1 (from the famous K. Erdmann’s classification), are computed. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used.
395-407
Hochschild Cohomology for Algebras of Dihedral Type. VII. The Family D(3R)
Аннотация
The Hochschild cohomology groups for algebras of dihedral type which are contained in the family D(3R) (from the famous K.Erdmann’s classification) are calculated. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used.
408-427
Double Cosets of Stabilizers of Totally Isotropic Subspaces in a Special Unitary Group II
Аннотация
In 2016, the authors considered the decomposition \( \mathrm{SU}\left(D,h\right)=\underset{i}{\cup }{P}_u{\gamma}_i{P}_{\upsilon } \), where SU(D, h) is a special unitary group over a division algebra D with involution, h is a symmetric or skew-symmetric nondegenerate Hermitian form, and Pu, Pυ are stabilizers of totally isotropic subspaces of the unitary space. Since Γ = SU(D, h) is a point group of a classical algebraic group \( \tilde{\Gamma} \), there is the “order of adherence” on the set of double cosets {PuγiPυ}, which is induced by the Zariski topology on \( \tilde{\Gamma} \). In the present paper, the adherence of such double cosets is described for the cases where \( \tilde{\Gamma} \) is an orthogonal or a symplectic group (that is, for groups of types Br, Cr, Dr).
428-446
Metacyclic 2-Extensions with Cyclic Kernel and Ultrasolvability Questions
Аннотация
Necessary and sufficient conditions for a metacyclic extension to be 2-local and ultrasolvable are established. These conditions are used to prove the ultrasolvability of an arbitrary group extension which has a local ultrasolvable associated subextension of the second type. The obtained reductions enables us to derive ultrasolvability results for a wide class of nonsplit 2-extensions with cyclic kernel.
447-458
459-473
474-480
When the Group Ring of a Finite Simple Group is Serial
Аннотация
A ring is said to be serial if its right and left regular modules are the direct sums of chain modules. The aim of the paper is to give an answer to the following question: for which finite simple groups, the group ring over a given field is serial.
481-493
The Existence of Root Subgroup Translated by a Given Element into its Opposite
Аннотация
Let Φ be a simply laced root system, K an algebraically closed field, and G = Gad(Φ,K) the adjoint group of type Φ over K. Then for every nontrivial element g ∈ G there exists a root element x of the Lie algebra of G such that x and gx are opposite.
494-502