Representations of the symmetric group and varieties of linear algebras

The representation theory of the symmetric group is used to study varieties of linear algebras over a field of characteristic 0. The relatively free algebras and the lattice of subvarieties of the variety of Lie algebras $\mathfrak{AN}_2\cap\mathfrak N_2\mathfrak A$ are described. An example of an almost finitely based variety of linear algebras if constructed. A continuous set of locally finite varieties forming a chain with respect to inclusion is indicated. Information is obtained on the variety of Lie algebras (resp., associative algebras with 1) generated by the second-order matrix algebra. In particular, distributivity of the lattice of subvarieties is proved, and in the Lie case a relatively free algebra is described.
Bibliography: 16 titles.