Growth in varieties of Lie algebras.

*(English. Russian original)*Zbl 0718.17004
Russ. Math. Surv. 45, No. 6, 27-52 (1990); translation from Usp. Mat. Nauk 45, No. 6(276), 25-45 (1990).

Let F(X) be the free Lie algebra freely generated by \(X=\{x_ 1,x_ 2,...\}\) over a field of characteristic 0 and let \(P_ n\) be the set of all multilinear in \(x_ 1,...,x_ n\) elements from F(X). For every variety \({\mathfrak V}\) of Lie algebras let V be the ideal of all polynomial identities for \({\mathfrak V}\). By definition the growth in \({\mathfrak V}\) is the growth of the codimension sequence \(c_ n(V)=\dim P_ n/(P_ n\cap V)\), \(n=1,2,..\).. The main purpose of the paper under review is to survey some recent results on the growth of varieties of Lie algebras. Some new results are obtained as well.

In Section 1 the author deals with the background from the representation theory of the symmetric group and its applications to algebras with polynomial identities. Sections 2 and 3 are devoted to varieties of polynomial and exponential growth and Section 4 gives examples of varieties of superexponential growth. Finally, in Sections 5 and 6 the author investigates varieties of solvable Lie algebras of exponential growth such that all the proper subvarieties are of polynomial growth as well as the finite basis property of such varieties.

In Section 1 the author deals with the background from the representation theory of the symmetric group and its applications to algebras with polynomial identities. Sections 2 and 3 are devoted to varieties of polynomial and exponential growth and Section 4 gives examples of varieties of superexponential growth. Finally, in Sections 5 and 6 the author investigates varieties of solvable Lie algebras of exponential growth such that all the proper subvarieties are of polynomial growth as well as the finite basis property of such varieties.

Reviewer: V.Drensky (Sofia)

##### MSC:

17B01 | Identities, free Lie (super)algebras |

17B30 | Solvable, nilpotent (super)algebras |

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |