9 research outputs found
Amitsur's conjecture for polynomial H-identities of H-module Lie algebras
Consider a finite dimensional H-module Lie algebra L over a field of
characteristic 0 where H is a Hopf algebra. We prove the analog of Amitsur's
conjecture on asymptotic behavior for codimensions of polynomial H-identities
of L under some assumptions on H. In particular, the conjecture holds when H is
finite dimensional semisimple. As a consequence, we obtain the analog of
Amitsur's conjecture for graded codimensions of any finite dimensional Lie
algebra graded by an arbitrary group and for G-codimensions of any finite
dimensional Lie algebra with a rational action of a reductive affine algebraic
group G by automorphisms and anti-automorphisms.Comment: 38 pages; this article is a generalization of M.V. Zaicev's paper
(Izv. Math, 2002) and the author's arXiv:1112.6245; the outline of the proof
of the main result is the same as in those articles, however we have to deal
with new phenomena that appear in H-module algebras; the introductory part,
where we give a survey of what has already been done, overlaps with
arXiv:1203.538
Semigroup graded algebras and codimension growth of graded polynomial identities
We show that if is any of four semigroups of two elements that are not
groups, there exists a finite dimensional associative -graded algebra over a
field of characteristic such that the codimensions of its graded polynomial
identities have a non-integer exponent of growth. In particular, we provide an
example of a finite dimensional graded-simple semigroup graded algebra over an
algebraically closed field of characteristic with a non-integer graded
PI-exponent, which is strictly less than the dimension of the algebra. However,
if is a left or right zero band and the -graded algebra is unital, or
is a cancellative semigroup, then the -graded algebra satisfies the
graded analog of Amitsur's conjecture, i.e. there exists an integer graded
PI-exponent. Moreover, in the first case it turns out that the ordinary and the
graded PI-exponents coincide. In addition, we consider related problems on the
structure of semigroup graded algebras.Comment: 21 pages; Minor misprints are correcte