Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of Orderp3

✍ Scribed by N Andruskiewitsch; H.-J Schneider

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 337 KB
• Volume: 209
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7643

✦ Synopsis

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 0

The 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.