Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^3

Details Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^3 Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^3

1998-03-13

arxiv.org/abs/math/9803058v1

Mathematics

Quantum Algebra

AmsTeX, 28 pages. To be published in J. of Algebra

Scientific paper

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 grad(A). Then grad(A) is a graded Hopf algebra, since the coradical A_0 of A is a Hopf subalgebra. In addition, there is a projection \pi: grad(A) \to A_0; let R be the algebra of coinvariants of \pi. Then, by a result of Radford and Majid, R is a braided Hopf algebra and grad(A) is the bosonization (or biproduct) of R and A_0: grad(A) is isomorphic to (R # A_0). The principle we propose to study A is first to study R, then to transfer the information to grad(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 classification of pointed Hopf algebras of order p^3 (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, non-isomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky.

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