On the extensions of infinite‐dimensional representations of Lie semigroups

Adolf R Mirotin

doi:10.1155/s0161171202011559

Abstract

The necessary and sufficient conditions have been obtained for extendability of a Banach representation of a generating Lie semigroup S to a local representation of the Lie group G generated by S when the tangent wedge of S is a Lie semialgebra. The most convenient conditions we obtain correspond to the case of unitary representations. In this case, we give a criterion of global extendability if G is exponential and solvable.

Highlights

If S is a subsemigroup of a topological group G with interior points and G is a left quotient group for S, it is easy to prove that every representation of S by invertible operators on a Banach space ᐂ may be extended, in a unique manner, to the representation of G on ᐂ
It is easy to prove that every finite-dimensional representation of a generating Lie semigroup S can be extended to the local representation of the Lie group G generated by S, if G is connected and the tangent wedge of S is a Lie semialgebra [14]
In this paper, using the infinitesimal method, we study the problem of extendability of a Banach representations π of a generating Lie semigroup S to the connected Lie group G generated by S when the tangent wedge of S is a Lie semialgebra

Summary

Introduction

If S is a subsemigroup of a topological group G with interior points and G is a left quotient group for S, it is easy to prove that every representation of S by invertible operators on a Banach space ᐂ may be extended, in a unique manner, to the representation of G on ᐂ (see Proposition 6.1 below). Let S be a quasi-invariant generating Lie semigroup in a connected Lie group G and let π be a representation of S on a Banach space ᐂ such that the operator π (s0) is invertible for some s0 ∈ int S.

Results

Conclusion