Equivalences of (co)module algebra structures over Hopf algebras

Abstract

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 class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. In particular, we apply support equivalence to study the asymptotic behaviour of codimensions of H-identities of a certain class of H-module algebras. This result proves the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture which was originally concerned with ordinary polynomial identities.

The automorphisms group and the classification of gradings of finite dimensional associative algebras

“…We shall recall the construction and the main properties of the quantum symmetry semigroup of a finite dimensional algebra A. The results below in this sections are well-known or have been recently proven in much more general cases [3,4]. The following is [18, Theorem 1.1]: we shall present a short proof since we prefer to give an explicit construction of the algebra a(A), using generators and relations implemented by the structure constants of A in the spirit of […”

$V$-universal Hopf algebras (co)acting on $Ω$-algebras

“…[5].) As the authors showed in [2], this notion can be generalized further to the case of (co)module structures on algebras. Equivalence of (co)module structures can be used, for instance, in the study of polynomial H-identities.…”

“…To start with, in the 1960s, M. E. Sweedler introduced the universal measuring coalgebra which lead to the notion of the universal measuring bialgebra [11,Chapter VII]. Unlike the universal Hopf algebra/bialgebra of a specific action defined in [2], Sweedler's universal measuring bialgebra is universal among all actions, not necessarily equivalent to a given one. Moreover, Sweedler's construction leads naturally to the notion of a universal acting Hopf algebra of an algebra (see Example 4.4).…”

“…The present paper was prompted mainly by the desire to build a unified theory, which includes as particular cases universal Hopf algebras of a given (co)action from [2] as well as the Sweedler-Manin-Tambara universal (co)acting bi/Hopf algebras. This will be achieved by introducing the notion of a V -universal (co)acting bi/Hopf algebra of an algebra A over a field F for a given subalgebra V of the algebra End F pAq of linear operators on A.…”

A new invariant for finite dimensional Leibniz/Lie algebras

“…Both objects are very important: as explain in [19], the Hopf envelope of a(A, A) plays the role of a symmetry group in non-commutative geometry. For further details we refer to [1,2,3]. A more general construction, which contains all the above as special cases, was recently considered in [4] in the context of Ω-algebras.…”