scholarly journals Infinite dimensional representations of

1990 ◽  
Vol 32 (1) ◽  
pp. 25-33 ◽  
Author(s):  
A. Dean ◽  
F. Zorzitto
Keyword(s):  
Dynkin Diagram ◽  
Finite Dimension ◽  
Vector Spaces ◽  
Closed Field ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Image Position ◽  
Subspace Problem ◽  
Extended Dynkin Diagram

By a representation of the extended Dynkin diagram we shall mean a list of 5 vector spaces P, E1, E2, E3, E4 over an algebraically closed field K, and 4 linear maps a1, a2, a3, a4 as shown.The spaces need not be of finite dimension.In their solution of the 4-subspace problem [6], Gelfand and Ponomarev have classified such representations when the spaces are finite dimensional. A representation like (1) can also be viewed as a module over the K-algebra R4 consisting of all 5 × 5 matrices having zeros off the first row and off the main diagonal.

Products of idempotent endomorphisms of an independence algebra of infinite rank

1993 ◽  
Vol 114 (2) ◽  
pp. 303-319 ◽  
Author(s):  
John Fountain ◽  
Andrew Lewin
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Common Generalization ◽  
Finite Dimensional Vector Space ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Linear Maps

AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

scholarly journals On Schauder Bases for Spaces of Continuous Functions1)

1960 ◽  
Vol 3 (2) ◽  
pp. 173-184 ◽  
Author(s):  
H. W. Ellis ◽  
D. G. Kuehner
Keyword(s):  
Independent Set ◽  
Vector Spaces ◽  
Hamel Basis ◽  
Infinite Dimensional ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
Schauder Bases ◽  
Image Position ◽  
Normed Vector

In a finite dimensional vector space V a set xi, i = 1, 2, …, n of vectors of V is said to be a basis, base, or coordinate system for V if the vectors xi are linearly independent and if each vector in V is a linear combination of the elements x1 with real coefficients. If a topology for V is defined in terms of a norm ||.|| then {xi} is a basis for V if and only if to each x ϵ V corresponds a unique set of constants ai such thatIn infinite dimensional normed vector spaces the above concepts of basis have different generalizations. The first or algebraic definition gives a Hamel basis which is a maximal linearly independent set [l, p. 2]. We shall be interested in the other or topological definition.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

scholarly journals GROUP GRADINGS ON FINITARY SIMPLE LIE ALGEBRAS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250046 ◽  
Author(s):  
YURI BAHTURIN ◽  
MATEJ BREŠAR ◽  
MIKHAIL KOCHETOV
Keyword(s):  
Abelian Group ◽  
Lie Algebras ◽  
Vector Spaces ◽  
Closed Field ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Special Linear ◽  
Infinite Dimensional ◽  
Group Gradings ◽  
Arbitrary Abelian Group

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.

scholarly journals Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups

1978 ◽  
Vol 21 (1) ◽  
pp. 17-19
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Unitary Representations ◽  
Vector Spaces ◽  
Closed Field ◽  
Discrete Groups ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

scholarly journals Finitarily linear wreath products

2000 ◽  
Vol 43 (1) ◽  
pp. 27-41
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Finite Dimension ◽  
Wreath Products ◽  
Vector Spaces ◽  
Normal Subgroups ◽  
Infinite Dimensional ◽  
Linear Representations ◽  
Finitary Linear ◽  
Completely Reducible ◽  
Necessary And Sufficient ◽  
Standard Wreath Product

AbstractWe consider faithful finitary linear representations of (generalized) wreath products A wrΩH of groups A by H over (potentially) infinite-dimensional vector spaces, having previously considered completely reducible such representations in an earlier paper. The simpler the structure of A the more complex, it seems, these representations can become. If A has no non-trivial abelian normal subgroups, the conditions we present are both necessary and sufficient. They imply, for example, that for such an A, if there exists such a representation of the standard wreath product A wr H of infinite dimension, then there already exists one of finite dimension.

Vector Spaces

2021 ◽  
pp. 47-102
Author(s):  
Adel N. Boules
Keyword(s):  
Invariant Subspaces ◽  
Extensive Study ◽  
Vector Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Direct Sums ◽  
Infinite Dimensional ◽  
Quotient Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional

The first three sections of this chapter provide a thorough presentation of the concepts of basis and dimension. The approach is unified in the sense that it does not treat finite and infinite-dimensional spaces separately. Important concepts such as algebraic complements, quotient spaces, direct sums, projections, linear functionals, and invariant subspaces make their first debut in section 3.4. Section 3.5 is a brief summary of matrix representations and diagonalization. Then the chapter introduces normed linear spaces followed by an extensive study of inner product spaces. The presentation of inner product spaces in this section and in section 4.10 is not limited to finite-dimensional spaces but rather to the properties of inner products that do not require completeness. The chapter concludes with the finite-dimensional spectral theory.

CONSISTENCY AND ASYMPTOTIC NORMALITY OF SIEVE ML ESTIMATORS UNDER LOW-LEVEL CONDITIONS

2014 ◽  
Vol 30 (5) ◽  
pp. 1021-1076 ◽  
Author(s):  
Herman J. Bierens
Keyword(s):  
Asymptotic Normality ◽  
Density Function ◽  
Choice Model ◽  
Likelihood Estimation ◽  
Dimensional Parameter ◽  
Infinite Dimensional ◽  
Low Level ◽  
Asymptotically Normal ◽  
Finite Dimensional ◽  
Image Position

This paper considers sieve maximum likelihood estimation of seminonparametric (SNP) models with an unknown density function as non-Euclidean parameter, next to a finite-dimensional parameter vector. The density function involved is modeled via an infinite series expansion, so that the actual parameter space is infinite-dimensional. It will be shown that under low-level conditions the sieve estimators of these parameters are consistent, and the estimators of the Euclidean parameters are$\sqrt N$asymptotically normal, given a random sample of sizeN. The latter result is derived in a different way than in the sieve estimation literature. It appears that this asymptotic normality result is in essence the same as for the finite dimensional case. This approach is motivated and illustrated by an SNP discrete choice model.

scholarly journals On a theorem of Dvoretsky, Wald, and Wolfowitz concerning Liapounov Measures

1987 ◽  
Vol 29 (2) ◽  
pp. 205-220 ◽  
Author(s):  
D. A. Edwards
Keyword(s):  
Natural Number ◽  
Convex Subset ◽  
Vector Measure ◽  
Compact Convex ◽  
Partition Of Unity ◽  
Compact Convex Subset ◽  
Vector Spaces ◽  
Measurable Partition ◽  
Infinite Dimensional ◽  
Image Position

Let ω be a non-empty set, ℱ a Boolean σ-algebra of subsets of Ω, k a natural number, and let m:ℱ→ℝk be a non-atomic vector measure. Then, by the celebrated theorem of Liapounov [11], the range m[3F] = {m(A): A ε ℱ3F} of m is a compact convex subset of ℝk. This theorem has been generalized in a number of ways. For example Kingman and Robertson [8] and Knowles [9] have shown that, under appropriate conditions, results in the same spirit can be proved for measures taking their values in infinite-dimensional vector spaces. Another type of generalization was obtained by Dvoretsky, Wald and Wolfowitz [6,7]. What they do is to take m as above together with a natural number n≥ 1. They then consider the set Knof all vectorswhere (A1 A2,…, An) is an ordered ℱ-measurable partition of Ω (i.e. a partition whose terms A, all belong to ℱ). They prove in [6] that Kn is a compact convex subset of ℝnk and moreover that Kn is equal to the set of all vectors of the formwhere (ϕ1, ϕ2…, ϕn) is an ℱ-measurable partition of unity; i.e. it is an n-tuple of non-negative ϕr on Ω such thatLiapounov's theorem can be obtained as a corollary of this result by taking n= 2.

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.

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