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A quasiHopf algebra freeness theorem
, 204
"... Abstract. We prove the quasiHopf algebra version of the NicholsZoeller theorem: A finitedimensional quasiHopf algebra is free over any quasiHopf subalgebra. 1. ..."
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Cited by 9 (1 self)
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Abstract. We prove the quasiHopf algebra version of the NicholsZoeller theorem: A finitedimensional quasiHopf algebra is free over any quasiHopf subalgebra. 1.
Lagrange’s theorem for Hopf monoids in species
, 2012
"... Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras, ..."
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Cited by 4 (3 self)
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Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras,
Freeness of Hopf Algebras
, 2006
"... In 1975 Kaplansky discussed “Ten conjectures on Hopf algebras ” during a lecture at the University of Chicago. The first of these conjectures concerned freeness of a Hopf algebra as a module over a subHopfalgebra. Specifically he conjectured that “a Hopf algebra is free as a module over any subHopfa ..."
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In 1975 Kaplansky discussed “Ten conjectures on Hopf algebras ” during a lecture at the University of Chicago. The first of these conjectures concerned freeness of a Hopf algebra as a module over a subHopfalgebra. Specifically he conjectured that “a Hopf algebra is free as a module over any subHopfalgebra”. Although this was quickly shown to be false in the infinite dimensional case, the finite dimensional case turned out to be true, and was proven 14 years later by Nichols and Zoeller. This result is the heart of this paper. The NicholsZoeller freeness theorem states that a finite dimensional Hopf algebra is free as a module over any subHopfalgebra. We will prove this theorem, as well as the first significant generalization of this theorem, which was proven three years later. This generalization says that if the Hopf algebra is infinite dimensional, then the Hopf algebra is still free if the subHopfalgebra is finite dimensional and semisimple. We will also look at several other significant generalizations that have since been proven. iv Acknowledgements First an foremost, I would like to thank God for the abilities he has given me. It is my hope that I will use them for the purpose he intended. I would next like to thank Dr. Davida Fischman for all of her guidance and insight on this project and throughout my studies at this university. I would also like to thank all of the professors in the department for sharing their mathematical knowledge with me over the years. Finally, I would like to thank my family for all there support over the last three years as I worked toward my goals. Without them, none of this would be possible. To Nela v
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"... These notes contain the material presented in a series of five lectures at the University of Córdoba in September 1994. The intent of this brief course was to give a quick introduction to Hopf algebras and to prove as directly as possible (to me) some recent results on finitedimensional Hopf algebra ..."
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These notes contain the material presented in a series of five lectures at the University of Córdoba in September 1994. The intent of this brief course was to give a quick introduction to Hopf algebras and to prove as directly as possible (to me) some recent results on finitedimensional Hopf algebras conjectured by Kaplansky in 1975. In particular, in the last part of the course I gave a complete proof from scratch of Zhou’s theorem (1994): Any finitedimensional Hopf algebra over the complex numbers of prime dimension p is isomorphic to the group algebra of the group of order p. I would like to thank Nicolás Andruskiewitsch and the members of the Mathematics Department of the FaMAF (University of Córdoba) for the kind invitation to visit Córdoba and for their warm hospitality. I also wish to thank the students in my class for their attention and for most enjoyable hours after the lectures. My special thanks are due to Sonia Natale who has written up the notes of a condensed course with great care and insight.
ON EPIMORPHISMS AND MONOMORPHISMS OF HOPF ALGEBRAS
, 907
"... Abstract. We notice that for a Hopf algebra H, its antipode S is both an epimorphism and a monomorphism from H to H op,cop in the category of Hopf algebras over a field. Together with the existence of Hopf algebras with noninjective or nonsurjective antipode, this proves the existence of nonsurje ..."
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Abstract. We notice that for a Hopf algebra H, its antipode S is both an epimorphism and a monomorphism from H to H op,cop in the category of Hopf algebras over a field. Together with the existence of Hopf algebras with noninjective or nonsurjective antipode, this proves the existence of nonsurjective epimorphisms and noninjective monomorphisms in the category of Hopf algebras. Using Schauenburg’s free Hopf algebra with a bijective antipode on a Hopf algebra and its “dual ” universal construction, we then show that one can even find examples of nonsurjective (noninjective) epis (resp. monos) of Hopf algebras K → H such that the antipode of H (resp. K) is bijective. Also, we emphasize some connections between the epi/surjective problem and Kaplansky’s 1’st conjecture.