Self-similarity for Lie algebras
Laurent Bartholdi (Goettingen)
I will give the general definition of a self-similar Lie algebras: an
algebra that contains "infinitesimal", tightly bound copies of itself,
and show that important examples of Lie algebras fall into that class,
in particular constructions by Petrogradsky, Shestakov and Zelmanov.
In analogy with Burnside's problem, one is interested to understand
infinite-dimensional algebraic algebra (every element generates a
finite-dimensional subalgebra), and in particular nil algebras (every
element raised to some power becomes 0). I will show that important
such examples of algebra arise within self-similar Lie algebras, and
will give sufficient conditions for a self-similar Lie algebra to be
nil.
I will also relate these self-similar Lie algebras to self-similar
groups such as examples by Grigorchuk and Gupta-Sidki, and will also
show their connection to Caranti's classification of algebras of
maximal class.