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2013 | 99 | 1 | 155-172
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Addition theorems and related geometric problems of group representation theory

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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.
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  • Department of Mathematics, Vologda State Pedagogical University, Vologda 160035, Russia
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