Derivations, Gradings, Actions of Algebraic Groups, and Codimension Growth of Polynomial Identities

@article{Gordienko2012DerivationsGA,
  title={Derivations, Gradings, Actions of Algebraic Groups, and Codimension Growth of Polynomial Identities},
  author={A. S. Gordienko and M. Kochetov},
  journal={Algebras and Representation Theory},
  year={2012},
  volume={17},
  pages={539-563}
}
Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts by derivations on a finite dimensional associative or Lie algebra A over a field of characteristic 0. We prove the $\mathfrak g$-invariant analogs of Wedderburn—Mal’cev and Levi theorems, and the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of A. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we… Expand
Co-stability of Radicals and Its Applications to PI-Theory
We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either charF = 0 or charF > dimA,Expand
Asymptotics of H-identities for associative algebras with an H-invariant radical
We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed fieldExpand
Growth of Differential Identities
In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra $L$ (and its universal enveloping algebra $U(L)$)Expand
On the formula for the PI-exponent of Lie algebras
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, weExpand
Differential identities of finite dimensional algebras and polynomial growth of the codimensions
Let $A$ be a finite dimensional algebra over a field $F$ of characteristic zero. If $L$ is a Lie algebra acting on $A$ by derivations, then such an action determines an action of its universalExpand
The Grassmann Algebra and its Differential Identities
Let G be the infinite dimensional Grassmann algebra over an infinite field F of characteristic different from two. In this paper we study the differential identities of G with respect to the actionExpand
Differential identities, 2 × 2 upper triangular matrices and varieties of almost polynomial growth
Abstract We study the differential identities of the algebra U T 2 of 2 × 2 upper triangular matrices over a field of characteristic zero. We let the Lie algebra L = Der ( U T 2 ) of derivations of UExpand
Distinguishing simple algebras by means of polynomial identities
Our main goal is to extend one of classical Razmyslov’s Theorem saying that any two simple finite-dimensional $$\Omega $$Ω-algebras over an algebraically closed field, satisfying the same polynomialExpand
Equivalences of (co)module algebra structures over Hopf algebras
We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence classExpand
Graded polynomial identities as identities of universal algebras
Abstract Let A and B be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose A and B are graded by a semigroup S so that the graded identicalExpand
...
1
2
...

References

SHOWING 1-10 OF 46 REFERENCES
Amitsur’s conjecture for polynomial -identities of -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 ofExpand
Integrality of exponents of codimension growth of finite-dimensional Lie algebras
We study the asymptotic behaviour of the codimension growth sequence of a finite-dimensional Lie algebra over a field of characteristic zero. It is known that the growth of the sequence is bounded byExpand
$G$-identities on associative algebras
Let R be an algebra over a field and G a finite group of automorphisms and anti-automorphisms of R. We prove that if R satisfies an essential G-polynomial identity of degree d, then theExpand
Graded polynomial identities and codimensions: computing the exponential growth
Abstract Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied byExpand
Graded polynomial identities, group actions, and exponential growth of Lie algebras
Abstract Consider a finite dimensional Lie algebra L with an action of a finite group G over a field of characteristic 0. We prove the analog of Amitsurʼs conjecture on asymptotic behavior forExpand
Amitsur's conjecture for associative algebras with a generalized Hopf action
We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of several generalizations of polynomial identities for finite dimensional associative algebras over a field ofExpand
G-identities of non-associative algebras
The main class of algebras considered in this paper is the class of algebras of Lie type. This class includes, in particular, associative algebras, Lie algebras and superalgebras, Leibniz algebras,Expand
Group gradings on finite dimensional Lie algebras
We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if $L$ is gradeg by a non-abelian finiteExpand
Finite-dimensional non-associative algebras and codimension growth
TLDR
This paper captures the exponential rate of growth of the sequence of codimensions of the free non-associative algebra of multialternating polynomials satisfying special properties for several classes of algebras including simple alagbras with a special non-degenerate form, finite-dimensional Jordan or alternative algeBRas and many more. Expand
Multialternating graded polynomials and growth of polynomial identities
Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we constructExpand
...
1
2
3
4
5
...