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Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p3 / Nicolás Andruskiewitsch, Hans-Jürgen Schneider.

Por: Colaborador(es): Idioma: Inglés Series Trabajos de Matemática. Serie A / Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física ; no. 48Detalles de publicación: Córdoba, Argentina : Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 1997.Descripción: 28 p. ; 30 cmTema(s): Resumen: 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 A0 of A is a Hopf subalgebra. In addition, there is a projection π: gr A → A0; let R be the algebra of coinvariants of π. Then, by a result of Radford and Majid, R is a braided Hopf algebra and gr A is the bosonization (or biproduct) of Rand A0: gr A similar, equals R#A0. 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 classification of pointed Hopf algebras of order p3(pan 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 p2; 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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Series FaMAF Series FaMAF FaMAF Sección Series Famaf Trabajo Matemática Serie A CAJA 4 - F0136 1 Disponible F0136
Series FaMAF Series FaMAF FaMAF Sección Series Famaf Trabajo Matemática Serie A CAJA 4 - F0137 2 Disponible Versión corregida en Journal of Algebra vol. 209, no.2, p. 658-691(1998). F0137
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Versión corregida en Journal of Algebra vol. 209, no.2, p. 658-691(1998).

Bibliografía : p. 28.

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 A0 of A is a Hopf subalgebra. In addition, there is a projection π: gr A → A0; let R be the algebra of coinvariants of π. Then, by a result of Radford and Majid, R is a braided Hopf algebra and gr A is the bosonization (or biproduct) of Rand A0: gr A similar, equals R#A0. 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 classification of pointed Hopf algebras of order p3(pan 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 p2; 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.

Texto en inglés.


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