Janez Bernik

ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

A set

On semitransitive Lie algebras of minimal dimension

\mathcal{S}

On certain graded representations of filiform Lie algebras

of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists

Common fixed points and common eigenvectors for sets of matrices

A\in \mathcal{S}

On groups and semigroups of matrices with spectra in a finitely generated field

such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

Reducibility and triangularizability of semitransitive spaces of operators

Abstract Let be an -dimensional vector space over . Some structural results on Lie subalgebras of acting semitransitively and of minimal possible dimension are obtained.

On Semitransitive Collections of Operators

Abstract Let be a connected complex linear algebraic group of the same dimension as V such that the poset of the Zariski closures of the orbits for its action coincides with a full flag of subspaces of V. Using the classification of graded filiform Lie algebras, we determine the isomorphism types of the unipotent radical U of G in case G is not nilpotent and U is of maximal class. In particular, if , there are, up to isomorphism, only two such unipotent groups.

Spectral conditions and band reducibility of operators

The following questions are studied: Under what conditions does the existence of a (nonzero) fixed point for every member of a semigroup of matrices imply a common fixed point for the entire semigroup? What is the smallest number k such that the existence of a common fixed point for every k members of a semigroup implies the same for the semigroup? If every member has a fixed space of dimension at least k: What is the best that can be said about the common fixed space? We also consider analogs of these questions with general eigenspaces replacing fixed spaces.

Reduction theorems for groups of matrices

Let K be an algebraically closed field of characteristic zero and F<K a subfield which is finitely generated over the prime field. Assume is a semigroup of invertible matrices such that the spectra of all the elements of are contained in F. Then the group , generated by , contains a solvable normal subgroup of finite index. As a consequence, it follows that an irreducible semigroup such that the spectra of all the elements of are contained in F is conjugate to a subsemigroup of Mn (F).Let K be an algebraically closed field of characteristic zero and F<K a subfield which is finitely generated over the prime field. Assume is a semigroup of invertible matrices such that the spectra of all the elements of are contained in F. Then the group , generated by , contains a solvable normal subgroup of finite index. As a consequence, it follows that an irreducible semigroup such that the spectra of all the elements of are contained in F is conjugate to a subsemigroup of Mn (F).