Nil restricted Lie algebras of oscillating intermediate growth

Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth

“…In [30] Petrogradsky constructed nil restricted Lie algebras with oscillating growth over fields of positive characteristic; our constructions are not subject to this restriction. Moreover, the upper bounds in our constructions are arbitrarily rapid (subexponential), unlike the upper bounds in [30]. However, our lower boundsthough polynomial, at least in the countable case -do not get close to linear as Petrogradsky's examples.…”

“…Petrogradsky [30] constructed a far-reaching Lie-theoretic analogy of these phenomena. Namely, he constructed nil (restricted) Lie algebras over fields of positive characteristic, whose growth functions oscillate between a function very close to linear and a function very close (though not arbitrarily close) to exponential.…”

Growth of finitely generated simple Lie algebras

“…The behavior of growth functions of Lie algebras is rather different compared to growth of associative algebras, and seems, from certain points of view, even more enigmatic. For instance, they need not obey Bergman's gap theorem (which asserts that the growth of an associative algebra cannot be super-linear and subquadratic; an extreme case is demonstrated in [16], where restricted Lie algebras whose growth is oscillating between quasilinear and almost exponential are constructed); polynomial growth rate for complex (finitely) Z-graded simple Lie algebras is very rigid, in the sense that it forces one of several concrete structures (by Matiheu's classification, solving Kac's conjecture [12]); there exist examples of nil Lie algebras of polynomial growth for arbitrarily large base fields [21] (the analog for associative algebras is unknown for uncountable fields); etc.…”