Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of Orderp3

Abstract

We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra gr A. Then gr A is a graded Hopf algebra, since the coradical A of A is a Hopf 0 subalgebra. In addition, there is a projection : gr A ª A ; let R be the algebra of 0 coinvariants of . Then, by a result of Radford and Majid, R is a braided Hopf Ž . algebra and gr A is the bosonization or biproduct of R and A : gr A , R࠻A . 0 0The principle we propose to study A is first to study R, then to transfer the information to gr A via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classifica-3 Ž . tion of pointed Hopf algebras of order p p an odd prime over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has index p or p 2 ; and an infinite family of pointed, nonisomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky. ᮊ 1998 Academic Press

Spectral sequences for the cohomology rings of a smash product

“…For example, one can untwists a twisted Calabi-Yau algebra with a smash product [12,6], or unbraid a braided Hopf algebra [3,Section 1.5]. In a more classical context, smash products have played an integral role in a classification program proposed by Andruskiewitsch and Schneider, which began at [2].…”

Lifting via cocycle deformation

“…Notice that the coradical of A is isomorphic to H by (1.3), see [AS2]. This problem is one of the steps in the Lifting Method [AS1,AS2], see also the generalization proposed in [AC]. To deal with it, we split it into two parts:…”

“…This is Problem (b) and there have been different approaches to face up to it: the Diamond Lemma [AS1,AG2,AV1]; a reduction to the first term of the coradical filtration followed by some representation theory, assuming that the Nichols algebra is quadratic [GGI]; a combination of deformation by cocycles and an examination of the PBW basis [AS3].…”

“…The background. The family of Hopf algebras u (D, λ, µ) contains the liftings of quantum linear spaces defined in [AS1]. It was shown in [Ma2] that these liftings of quantum linear spaces are cocycle deformations of their associated graded Hopf algebras.…”

A generalization of Coxeter groups, root systems, and Matsumoto’s theorem

“…The discovery of quantized Kac-Moody algebras by Drinfel'd [Dri87] and Jimbo [Jim86] opened a new major research direction with many novel features, ideas, and connections to other research fields. Motivated by these examples, Andruskiewitsch and Schneider [AS98] launched a project to classify pointed Hopf algebras with certain finiteness conditions, and performed it successfully under some hypotheses [AS05]. Their method is based on the knowledge of the finiteness property of Nichols algebras, the latter being main ingredients for the construction of quantized KacMoody algebras.…”