Seminar
Automorphic Orbits in Free
Algebras of
Schreier Varieties of Algebras
Dr. Alexander A. Mikhalev
The University of Hong Kong
Abstract
A variety of algebras over a
field is a class of algebras closed under taking subalgebras,
homomorphic images and direct products. Birkhoff's theorem says that for any variety of algebras
there exists a set of identities such that the variety consists of all algebras
satisfying these identities. A variety of algebras is said to be Schreier if any subalgebra of a
free algebra of this variety is free in the same variety of algebras.
The main types of Schreier varieties of algebras are the variety of all
algebras; the variety of Lie algebras; varieties of Lie superalgebras;
the variety of Lie p-algebras (over a field of positive characteristic);
varieties of Lie p-superalgebras; varieties of
all commutative and all anticommutative algebras. We
consider combinatorial properties of automorphic
orbits of elements in free algebras of these varieties of algebras.
A system of elements of a
free algebra F is primitive if it is a subset of some set of free
generators of F. The rank of a system of elements of a free algebra F
with a free generating set X is the minimal number of generators from X
on which automorphic images of these elements can
depend. We expose matrix criteria for a system of elements to be primitive (to
have given rank, respectively). Using these results we obtain a series of
algorithms for symbolic computation in automorphic
orbits. We show also some recent results about generalized primitive elements
of free algebras.