Automorphic Orbits in Free Algebras of
Schreier Varieties of Algebras

Dr. Alexander A. Mikhalev

The University of Hong Kong



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.


November 24, 2000 (Friday) 


4:00 - 5:00pm


Room 517, Meng Wah Complex




All are welcome