Algebra seminar talk by Jonas Deré (KU Leuven KULAK)
On Wednesday the 26th of April 2023, Jonas Deré (KU Leuven KULAK) digitally payed a visit to the Charlotte Centre for Algebra and talked about his work on Rational forms in Lie algebras associated to graphs in the joint research seminar of Lincoln University and Lund University.
Title: Rational forms in Lie algebras associated to graphs
Abstract: By the classic work of Mal’cev, studying finitely generated torsion-free nilpotent groups up to commensurability is equivalent to studying finite dimensional rational nilpotent Lie algebras. Given a field and a graph, one can construct a 2-step nilpotent Lie algebra over the field, where the Lie bracket between vertices is fully described in terms of the edges. This interesting class of Lie algebras intermediates between abelian and free 2-step nilpotent Lie algebras. For example, both the Heisenberg algebra and the direct sum of two Heisenberg algebras lies in this class. It is well-known that the direct sum of two real Heisenberg Lie algebras contains many different rational forms, i.e. rational subalgebras such that every basis over Q forms a basis for the real Lie algebra as well. In this talk, we present a general method to describe all the rational forms in these Lie algebras associated to graphs (and thus also of all the corresponding torsion-free nilpotent groups) by using the so-called coherent components of the graph. This is joint work with Thomas Witdouck.
Charlotte Scott Centre for Algebra 
Reblogged this on Maths & Physics News.
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