Laurent Marcoux, Department of Pure Mathematics, University of Waterloo
“Paratransitive algebras of operators”
A collection A of linear operators acting on a finite-dimensional vector space V is said to be transitiveif for any vector y ∈ V and any nonzero vector x ∈ V there exists an element T ∈ A such that Tx = y. When A is itself a linear space, it is clear that this condition is equivalent to the statement that the image of any one-dimensional subspace W of V under A intersects every other one-dimensional subspace non-trivially. A classical theorem of Burnside states that when A is an algebra, this happens precisely if A = L(V), the space of all linear maps on V.
In this talk we shall discuss the following generalization of the notion of transitivity: given an n-dimensional space V as above and integers 1 ≤ k, m ≤ n, we shall say that the algebra A is (k,m)-transitive if for every k-dimensional subspace W of V, the image A(W) = {Tw : T ∈ A,w ∈ W} of W under A intersects every m-dimensional subspace of V non-trivially. An algebra is paratransitive if it is (k,m)-transitive for some k and m. We examine the structure of minimal paratransitive algebras.
This is based upon joint work with L. Livshits, G. MacDonald and H. Radjavi.