Ambiskew Hopf algebras

Abstract

a b s t r a c tNecessary and sufficient conditions are obtained for an ambiskew polynomial algebra A over a Hopf k-algebra R to possess the structure of a Hopf algebra extending that of R, in which the added variables X + and X − are skew primitive. The coradical filtration is calculated, many examples are described, and properties are determined.

On Category O over triangular generalized Weyl algebras

“…Both classical and quantum down-up algebras are special cases of ambiskew polynomial rings, which are the class of triangular GWAs where z 1 ∈ F × . Ambiskew polynomial rings are the focus of recent and continuing interest [9,19,24,26]. Generalized Weyl algebras can also arise from other constructions.…”

Hopf-Galois structures on ambiskew polynomial rings

“…When the element c is assumed to be central, or equivalently when ω = τ −1 (so σ = id), this question has been addressed in [4]. Our result reads as follows.…”

“…In Section 2 we recall the definition of Hopf-Galois algebras (R, µ) and their realization as Hopf-Galois objects over a Hopf algebra H(R); we discuss the concepts of group-like and skew-primitive elements in this setting. Section 3 is devoted to the answer of item (A) above: in Theorem 3.1 we find necessary and sufficient conditions so that a generalized ambiskew polynomial algebra A R extends the structure of a Hopf algebra R in a way such that the added variables are skew-primitive; we recover some of the results in [4] as a corollary. Next, in Section 4 we find necessary and sufficient conditions so that a generalized ambiskew polynomial algebra extends the structure of a Hopf-Galois algebra R as in (1.2) and we provide an answer for item (B) in Theorem 4.1.…”

“…Another interesting question is to find necessary and sufficient conditions on these data so that A preserves some extra structure on R. When R is a Hopf algebra, Brown-Macaulay [4] discuss the case in which the Hopf structure on R extends to a Hopf structure on the ambiskew polynomial Hopf algebra A R = A(R, X, Y, τ, c, ξ) via the formula…”

“…Given a Hopf-Galois algebra (R, µ) as above, there exists a canonical Hopf algebra H(R) such that R is an H(R)-Galois object [10,15]. As for the first item (A), the need of a generalization of the results in [4] becomes evident when one considers the Hopf algebra associated to the Hopf-Galois algebra A R = A(R, X, Y, τ, c, ξ) extending a Hopf-Galois algebra R: this Hopf algebra is an extension of H(R) not in terms of an ambiskew polynomial algebra but of a generalized ambiskew polynomial algebra A. See Corollary 5.5 and Subsection 6.5 for an example.…”