Dynamic of Abelian Subgroups of GL(n, $$\mathbb{C}$$ ): A Structure Theorem

Abstract

In this paper, we characterize the dynamic of every Abelian subgroup G of GL(n, K), K = R or C. We show that there exists a G-invariant, dense open set U in K n saturated by minimal orbits with K n − U a union of at most n G-invariant vector subspaces of K n of dimension n − 1 or n − 2 over K. As a consequence, G has height at most n and in particular it admits a minimal set in K n − {0}. (2000). 37C85. Mathematics Subject Classification

Linear semigroups with coarsely dense orbits

“…We determine the structure of the open orbits of S. The following theorem will be applied for the group G * = S of theorem 3.1. Parts of these results are in [1]. Exactly the same arguments as above show that Gv = V \ H for every v / ∈ H. In particular H is the only maximal G-invariant (real or complex) vector subspace of V .…”

“…We determine the structure of the open orbits of S. The following theorem will be applied for the group G * = S of theorem 3.1. Parts of these results are in [1].…”

“…(3) Finally, if f (S) is dense in R for every non-zero continuous homomorphism f : G → R then S is a subgroup of G by (1). Also, the factor Z l in Pontryagin's structure theorem G/K ∼ = R m ⊕ Z l does not occur, so G/K ∼ = R m , and S · K = G by corollary 2.2.…”

“…This easily implies the same criterion for subsemigroups of compactly generated abelian locally compact topological groups, by Pontryagin's structure theorem, corollary 2.4. It has as corollaries a criterion for density, corollaries 2.2 and 2.5(3), and for cocompactness, corollary 2.5 (1). These statements may be regarded as duality statements for subsemigroups of such abelian groups.…”

“…The group G * contains an open subset of G, by lemma 3.7. Thus, being a group, G * is open in G and hence a closed subgroup of G. This implies(1). The group G is an abelian real linear algebraic group so it has a finite number of connected components.…”