arXiv subjects

Maria V. Milentyeva

Publications and source records attributed to Maria V. Milentyeva.

4 recordsLinked to original sources

arXiv2014

Large and symmetric: The Khukhro--Makarenko theorem on laws --- without laws

We prove a generalisation of the Khukhro--Makarenko theorem on large characteristic subgroups with laws. This general fact implies new results on groups, algebras, and even graphs and other structures. Concerning groups, we obtain, e.g., a fact in a sense dual to the Khukhro--Makarenko theorem. A graph-theoretic corollary is an analogue of this theorem in which planarity plays the role of a multilinear identity. We answer also a question of Makarenko and Shumyatsky.

math.GR↗

arXiv2014

On the dimensions of commutative subalgebras and subgroups of nilpotent algebras and Lie groups of class 2

We obtain the functions that bound the dimensions of finite dimensional nilpotent associative or Lie algebras of class 2 over an algebraically closed field in terms of the dimensions of their commutative subalgebras. As a result, we also compute a similar function for complex nilpotent Lie groups of class 2.

math.RA↗

arXiv2014

On the dimensions of commutative subalgebras and subgroups

We consider the functions that bound the dimensions of finite-dimensional associative or Lie algebras in terms of the dimensions of their commutative subalgebras. It is proved that these functions have quadratic growth. As a result, we also get similar estimates for the dimension of a Lie group with bounded dimensions of its abelian Lie subgroups.

math.RA↗

arXiv2014

Generic properties of 2-step nilpotent Lie algebras and torsion-free groups

To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained in a proper Zariski-closed subset. We compute the maximal dimension of an abelian subalgebra of a generic Lie algebra and give a sufficient condition for a generic Lie algebra to admit no surjective homomorphism onto a non-abelian Lie algebra of a given dimension. Also we consider analogous questions for finitely generated torsion free nilpotent groups of class 2.

math.RA↗