Memorial University of Newfoundland

Atlantic Association for Research in the Mathematical Sciences

Atlantic Algebra Centre

Mini-course

Finite Algebras

February 11 - 17, 2026

 Abstract: The mini-course will give an introduction to the theory of finite nonassociative algebras, with the stress on polynomial identities, gradings and generic properties. Because of time constraints, very few results will be proven. The prerequisite is an advanced undergraduate algebra and linear algebra course. A detailed course description follows. 

Lecture I "Basic concepts of structure and identities of finite algebras": Types of finite algebras, including nilpotent algebras, solvable algebras, simple algebras. Varieties: locally finite, locally nilpotent, finitely generated, minimal. Critical algebras and minimal representation of an algebra in a finitely generated variety.

Lecture II "Numerical characteristic of finitely generated varieties": Free spectrum of a variety. Computation of the number of nilpotent, solvable, etc. algebras of given dimension. Exponent of a finitely generated variety, its computation for finite simple algebras.

Lecture III "Finite basis problem”: A survey of positive and negative resluts on the finite basis problem for different varieties. A study of generic properties of finite algebras. If time permits, some related results on the gradings of finite nonassociative algebras.