Addition theorems and related geometric problems of group representation theory

Ekaterina Shulman

Banach Center Publications (2013)

  • Volume: 99, Issue: 1, page 155-172
  • ISSN: 0137-6934

Abstract

top
The Levi-Civita functional equation f ( g h ) = k = 1 n u k ( g ) v k ( h ) (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation g T g have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for which x, O(x) is contained in the sum of a fixed finite-dimensional subspace and a finite-dimensional invariant subspace? 3) what can be said about a pair L, M of finite-dimensional subspaces if T g L M 0 for all g ∈ G? 4) which finite-dimensional subspaces L have the property that for each g ∈ G there is 0 ≠ x ∈ L with T g x = x ? The problem 1) arises in the study of the Hyers-Ulam stability of the L-C equation. It leads to the theory of covariant widths - the analogues of Kolmogorov widths which measure the distances from a given set to n-dimensional invariant subspaces. The problem 2) is related to multivariable extensions of the L-C equation; the study of this problem is based on the theory of subadditive set-valued functions which was developed specially for this aim. To problems 3) and 4) one comes via the study of the equations i = 1 m a i ( g ) b i ( h g ) = j = 1 n u j ( g ) v j ( h ) . We will finish by the consideration of “fractionally-linear version” of the L-C equation which is very important for the theory of integrable dynamical systems.

How to cite

top

Ekaterina Shulman. "Addition theorems and related geometric problems of group representation theory." Banach Center Publications 99.1 (2013): 155-172. <http://eudml.org/doc/286113>.

@article{EkaterinaShulman2013,
abstract = {The Levi-Civita functional equation $f(gh) = ∑_\{k=1\}^\{n\} u_\{k\}(g)v_\{k\}(h)$ (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation $g ↦ T_\{g\}$ have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for which x, O(x) is contained in the sum of a fixed finite-dimensional subspace and a finite-dimensional invariant subspace? 3) what can be said about a pair L, M of finite-dimensional subspaces if $T_\{g\}L ∩ M ≠ \{0\}$ for all g ∈ G? 4) which finite-dimensional subspaces L have the property that for each g ∈ G there is 0 ≠ x ∈ L with $T_\{g\}x = x$? The problem 1) arises in the study of the Hyers-Ulam stability of the L-C equation. It leads to the theory of covariant widths - the analogues of Kolmogorov widths which measure the distances from a given set to n-dimensional invariant subspaces. The problem 2) is related to multivariable extensions of the L-C equation; the study of this problem is based on the theory of subadditive set-valued functions which was developed specially for this aim. To problems 3) and 4) one comes via the study of the equations $∑_\{i=1\}^\{m\} a_\{i\}(g)b_\{i\}(hg) = ∑_\{j=1\}^\{n\} u_\{j\}(g)v_\{j\}(h)$. We will finish by the consideration of “fractionally-linear version” of the L-C equation which is very important for the theory of integrable dynamical systems.},
author = {Ekaterina Shulman},
journal = {Banach Center Publications},
keywords = {functional equations on semigroups; addition theorems; representations of topological semigroups; stability in the Hyers-Ulam sense; subadditive set-valued functions on groups; elliptic functions},
language = {eng},
number = {1},
pages = {155-172},
title = {Addition theorems and related geometric problems of group representation theory},
url = {http://eudml.org/doc/286113},
volume = {99},
year = {2013},
}

TY - JOUR
AU - Ekaterina Shulman
TI - Addition theorems and related geometric problems of group representation theory
JO - Banach Center Publications
PY - 2013
VL - 99
IS - 1
SP - 155
EP - 172
AB - The Levi-Civita functional equation $f(gh) = ∑_{k=1}^{n} u_{k}(g)v_{k}(h)$ (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation $g ↦ T_{g}$ have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for which x, O(x) is contained in the sum of a fixed finite-dimensional subspace and a finite-dimensional invariant subspace? 3) what can be said about a pair L, M of finite-dimensional subspaces if $T_{g}L ∩ M ≠ {0}$ for all g ∈ G? 4) which finite-dimensional subspaces L have the property that for each g ∈ G there is 0 ≠ x ∈ L with $T_{g}x = x$? The problem 1) arises in the study of the Hyers-Ulam stability of the L-C equation. It leads to the theory of covariant widths - the analogues of Kolmogorov widths which measure the distances from a given set to n-dimensional invariant subspaces. The problem 2) is related to multivariable extensions of the L-C equation; the study of this problem is based on the theory of subadditive set-valued functions which was developed specially for this aim. To problems 3) and 4) one comes via the study of the equations $∑_{i=1}^{m} a_{i}(g)b_{i}(hg) = ∑_{j=1}^{n} u_{j}(g)v_{j}(h)$. We will finish by the consideration of “fractionally-linear version” of the L-C equation which is very important for the theory of integrable dynamical systems.
LA - eng
KW - functional equations on semigroups; addition theorems; representations of topological semigroups; stability in the Hyers-Ulam sense; subadditive set-valued functions on groups; elliptic functions
UR - http://eudml.org/doc/286113
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.