Generators and relations in the special multiplicative group of a weakly perfect ring

1991 ◽  
Vol 31 (3) ◽  
pp. 498-505
Author(s):  
Zh. S. Satarov
Keyword(s):  
Multiplicative Group ◽  
Perfect Ring ◽  
Generators And Relations

scholarly journals Quiver origami: discrete gauging and folding

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Antoine Bourget ◽  
Amihay Hanany ◽  
Dominik Miketa
Keyword(s):  
Gauge Theories ◽  
Generators And Relations

Abstract We study two types of discrete operations on Coulomb branches of 3d$$ \mathcal{N} $$ N = 4 quiver gauge theories using both abelianisation and the monopole formula. We generalise previous work on discrete quotients of Coulomb branches and introduce novel wreathed quiver theories. We further study quiver folding which produces Coulomb branches of non-simply laced quivers. Our methods explicitly describe Coulomb branches in terms of generators and relations including mass deformations.

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

scholarly journals REGULAR FUNCTIONS ON THE SHILOV BOUNDARY

2005 ◽  
Vol 04 (06) ◽  
pp. 613-629 ◽  
Author(s):  
OLGA BERSHTEIN
Keyword(s):  
Symmetric Domain ◽  
Principal Series ◽  
Bounded Symmetric Domain ◽  
Bounded Symmetric Domains ◽  
Shilov Boundary ◽  
Generators And Relations ◽  
Regular Functions ◽  
Degenerate Principal Series

In this paper a *-algebra of regular functions on the Shilov boundary S(𝔻) of bounded symmetric domain 𝔻 is constructed. The algebras of regular functions on S(𝔻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(𝔻) = Un is investigated.

On multiplicative systems defined by generators and relations

1953 ◽  
Vol 49 (4) ◽  
pp. 579-589 ◽  
Author(s):  
Trevor Evans
Keyword(s):  
Automorphism Group ◽  
Finite Number ◽  
Cyclic Group ◽  
Complete Information ◽  
Isomorphism Problem ◽  
Infinite Cyclic Group ◽  
Generators And Relations ◽  
Decision Method ◽  
Multiplicative Systems ◽  
Finitely Related

The techniques developed in (9) are used here to study the properties of multiplicative systems generated by one element (monogenie systems). The results are of two kinds. First, we obtain fairly complete information about the automorphisms and endo-morphisms of free and finitely related loops. The automorphism group of the free monogenie loop is the infinite cyclic group, each automorphism being obtained by mapping the generator on one of its repeated inverses. A monogenie loop with a finite, non-empty set of relations has only a finite number of endomorphisms. These are obtained by mapping the generator on some of the components, or their repeated inverses, occurring in the relations. We use the same methods to solve the isomorphism problem for monogenie loops, i.e. we give a method for determining whether two finitely related monogenie loops are isomorphic. The decision method consists essentially of constructing all homomorphisms between two given finitely related monogenie loops.

LORENTZ COVARIANCE FOR THE QUANTUM ALGEBRA OF OBSERVABLES: NAMBU–GOTO STRINGS IN 3 + 1 DIMENSIONS

2002 ◽  
Vol 17 (17) ◽  
pp. 2331-2349 ◽  
Author(s):  
GERRIT HANDRICH
Keyword(s):  
Rest Frame ◽  
Poisson Algebra ◽  
Quantum Algebra ◽  
Substantial Part ◽  
Quantum Case ◽  
Generators And Relations ◽  
Defining Relations ◽  
Algebra Of Observables ◽  
The One ◽  
The Relationship

To postulate correspondence for the observables only is a promising approach to a fully satisfying quantization of the Nambu–Goto string. The relationship between the Poisson algebra of observables and the corresponding quantum algebra is established in the language of generators and relations. A very valuable tool is the transformation to the string's rest frame, since a substantial part of the relations are solved. It is the aim of this paper to clarify the relationship between the fully covariant and the rest frame description. Both in the classical and in the quantum case, an efficient method for recovering the covariant algebra from the one in the rest frame is described. Restrictions on the quantum defining relations are obtained, which are not taken into account when one postulates correspondence for the rest frame algebra. For the part of the algebra studied up to now in explicit computations, these further restrictions alone determine the quantum algebra uniquely — in full consistency with the further restrictions found in the rest frame.

scholarly journals ON REGULAR GROUPS AND FIELDS

2014 ◽  
Vol 79 (3) ◽  
pp. 826-844 ◽  
Author(s):  
TOMASZ GOGACZ ◽  
KRZYSZTOF KRUPIŃSKI
Keyword(s):  
Conjugacy Class ◽  
Classical Group ◽  
Multiplicative Group ◽  
Finite Extension ◽  
Generic Type ◽  
Common Generalization ◽  
Counter Example ◽  
Minimal Groups ◽  
Unbounded Orbit

AbstractRegular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. LetKbe a regular field which is not generically stable and letpbe its global generic type. We observe that ifKhas a finite extensionLof degreen, thenP(n)has unbounded orbit under the action of the multiplicative group ofL.Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable.

scholarly journals Digital signature scheme on the 2 x 2 matrix algebra

2021 ◽  
Vol 17 (3) ◽  
pp. 254-261
Author(s):  
Nikolay A. Moldovyan ◽  
◽  
Alexandr A. Moldovyan ◽  
Keyword(s):  
Digital Signature ◽  
High Performance ◽  
Matrix Algebra ◽  
Multiplicative Group ◽  
Discrete Logarithm ◽  
Signature Scheme ◽  
The Public ◽  
Signature Generation ◽  
Commutative Subalgebras ◽  
Digital Signature Scheme

The article considers the structure of the 2x2 matrix algebra set over a ground finite field GF(p). It is shown that this algebra contains three types of commutative subalgebras of order p2, which differ in the value of the order of their multiplicative group. Formulas describing the number of subalgebras of every type are derived. A new post-quantum digital signature scheme is introduced based on a novel form of the hidden discrete logarithm problem. The scheme is characterized in using scalar multiplication as an additional operation masking the hidden cyclic group in which the basic exponentiation operation is performed when generating the public key. The advantage of the developed signature scheme is the comparatively high performance of the signature generation and verification algorithms as well as the possibility to implement a blind signature protocol on its base.

Sign in / Sign up

Export Citation Format

Share Document