Bernstein algebras and Burnside-type problems

In the last decade, the connection with the Jacobian problem (after Yagzhev) attracts new interest to nilpotence-type identities in symmetric non-associative algebras. We briefly discuss this connection. Then, we will concentrated on algebraicity, nilpotence, and Engel conditions for a class of algebras that has the origin in works of Sergei Bernstein in theoretical genetics. We show that every Bernstein algebra has maximal algebraic ideal, such that the quotient is a zero-multiplication algebra. The algebraic Bernstein algebras are characterized in terms of algebraicity of multiplication operators. Moreover, the Bernstein algebra is train (that is, algebraic in the sense of for principal powers) if and only if its baric ideal is a nil-subalgebra (equivalently, Engel subalgebra). The generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras. However, both Kurosh problem and Burnside problem have negative solutions in this class. The talk is mainly based on a common paper with F. Zitan.