On the formula for the PI-exponent of Lie algebras
Abstract
We prove that one of the conditions in M.V. Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite dimensional semisimple Lie algebra acts by derivations on a finite dimensional Lie algebra over a field of characteristic , then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial -identities of a finite dimensional Lie algebra with a rational action of a connected reductive affine algebraic group by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for -module Lie algebras whose solvable radical is nilpotent, assuming only the -invariance of the radical, i.e. under weaker assumptions on the -action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for -codimensions of all finite dimensional Lie -algebras whose solvable radical is nilpotent, for an arbitrary group .
Keywords
Cite
@article{arxiv.1211.1272,
title = {On the formula for the PI-exponent of Lie algebras},
author = {Alexey Sergeevich Gordienko},
journal= {arXiv preprint arXiv:1211.1272},
year = {2014}
}
Comments
15 pages. Section 2 (def. of the free H-alg., H-ident., and H-codim.) is the same as Subs. 1.3 in arXiv:1207.1699 and Subs. 3.1 in arXiv:1210.2528. Subs. 3.1-3.2 (def. of an H-nice alg. and a formula for the Hopf PI-exp) coincide with Subs. 1.7-1.8 of arXiv:1207.1699. Lemmas 5 and 6 are adaptations of Lemmas 20 and 21 from arXiv:1207.1699 for a different case