scholarly journals Classification of Regular Parametrized One-relation Operads

2017 ◽  
Vol 69 (5) ◽  
pp. 992-1035 ◽  
Author(s):  
Murray Bremner ◽  
Vladimir Dotsenko
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Algebra ◽  
Free Algebra ◽  
Regular Representation ◽  
Polynomial Rings ◽  
Closed Field ◽  
Tensor Algebra ◽  
Determinantal Ideals

AbstractJean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: . Such an operad is called a parametrized one-relation operad. For a particular choice of parameters {xσ}, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space V is, as a graded vector space, isomorphic to the tensor algebra of V. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following ûve operads: the left-nilpotent operad defined by the relation ((a1a2)a3) = 0, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computationalmethods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

scholarly journals NILPOTENT MODULES OVER POLYNOMIAL RINGS

2020 ◽  
pp. 20-25
Author(s):  
А. Petravchuk ◽  
Ie. Chapovskyi ◽  
I. Klimenko ◽  
M. Sidorov
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Polynomial Rings ◽  
Closed Field ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Groups Of Automorphisms

Let K be an algebra ically closed field of characteristic zero, K[X ] the polynomial ring in n variables. The vector space Tn = K[X] is a K[X ] -module with the action i = xi 'x  v v for vTn . Every finite dimensional submodule V of Tn is nilpotent, i.e. every f  K[X ] acts nilpotently (by multiplication) on V . We prove that every nilpotent K[X ] -module V of finite dimension over K with one-dimensional socle can be isomorphically embedded in the module Tn . The groups of automorphisms of the module Tn and its finite dimensional monomial submodules are found. Similar results are obtained for (non-nilpotent) finite dimensional K[X ] -modules with one dimensional socle.

scholarly journals Linear Algebra and Differential Calculus in Pseudo-Intervals Vector Space

2016 ◽  
Vol 17 (3) ◽  
pp. 283
Author(s):  
Abdelouahab Kenoufi
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Free Algebra ◽  
Interval Arithmetic ◽  
Differential Calculus ◽  
Interval Matrix ◽  
Semi Group ◽  
New Approach ◽  
Group Completion ◽  
And Function

In this paper one proposes to use a new approach of interval arithmetic, the so-called pseudo- intervals [1, 5, 13]. It uses a construction which is more canonical and based on the semi-group completion into the group, and it allows to build a Banach vector space. This is achieved by embedding the vector space into free algebra of dimensions higher than 4. It permits to perform linear algebra and differential calculus with pseudo-intervals. Some numerical applications for interval matrix eigenmode calculation, inversion and function minimization are exhibited for simple examples. 

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

scholarly journals Probabilist Set Inversion using Pseudo-Intervals Arithmetic

2014 ◽  
Vol 15 (1) ◽  
pp. 097
Author(s):  
Abdelouahab KENOUFI
Keyword(s):  
Vector Space ◽  
Free Algebra ◽  
Interval Arithmetic ◽  
Semi Group ◽  
Numerical Examples ◽  
Inclusion Function ◽  
Python Programming

<pre><!--StartFragment-->In this paper, we present how to use an interval arithmetic framework based on free algebra construction, in order to build better defined inclusion function for interval semi-group and for its associated vector space. One introduces the <span>psi</span>-algorithm, which performs set inversion of functions and exhibits some numerical examples developed with the python programming langage<!--EndFragment--></pre>.

Constitutive relations in classical optics in terms of geometric algebra

2015 ◽  
Vol 55 (2) ◽  
Author(s):  
Adolfas Dargys
Keyword(s):  
Wave Propagation ◽  
Electromagnetic Wave ◽  
Vector Space ◽  
Closed System ◽  
Geometric Algebra ◽  
Electromagnetic Wave Propagation ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Basis Vectors

To have a closed system, the Maxwell electromagnetic equations should be supplemented by constitutive relations which describe medium properties and connect primary fields (E, B) with secondary ones (D, H). J.W. Gibbs and O. Heaviside introduced the basis vectors {i, j, k} to represent the fields and constitutive relations in the three-dimensional vectorial space. In this paper the constitutive relations are presented in a form of Cl3,0 algebra which describes the vector space by three basis vectors {σ1, σ2, σ3} that satisfy Pauli commutation relations. It is shown that the classification of electromagnetic wave propagation phenomena with the help of constitutive relations in this case comes from the structure of Cl3,0 itself. Concrete expressions for classical constitutive relations are presented including electromagnetic wave propagation in a moving dielectric.

Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

Rational permutation groups containing a full cycle

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
Temha Erkoç ◽  
Utku Yilmaztürk
Keyword(s):  
Symmetric Group ◽  
Proper Subgroup ◽  
Symmetric Groups ◽  
Transitive Permutation Group ◽  
Rational Group ◽  
Prime Degree ◽  
A Finite Group ◽  
Rational Groups ◽  
Complex Characters

AbstractA finite group whose irreducible complex characters are rational valued is called a rational group. Thus, G is a rational group if and only if N G(〈x〉)/C G(〈x〉) ≌ Aut(〈x〉) for every x ∈ G. For example, all symmetric groups and their Sylow 2-subgroups are rational groups. Structure of rational groups have been studied extensively, but the general classification of rational groups has not been able to be done up to now. In this paper, we show that a full symmetric group of prime degree does not have any rational transitive proper subgroup and that a rational doubly transitive permutation group containing a full cycle is the full symmetric group. We also obtain several results related to the study of rational groups.

Pencils of real binary cubics

1983 ◽  
Vol 93 (3) ◽  
pp. 477-484 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Real Case ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
The Real

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

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