Brown, K.A. , Gilmartin, P. and Zhang, J.J. (2019) Connected (graded) Hopf algebras. Transactions of the American Mathematical Society, 372, pp. 32833317. (doi: 10.1090/tran/7686)

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Abstract
We study algebraic and homological properties of two classes of infinite dimensional Hopf algebras over an algebraically closed field k of characteristic zero. The first class consists of those Hopf kalgebras that are connected graded as algebras, and the second class are those Hopf kalgebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel'fandKirillov dimension. It is shown that if the Hopf algebra H is a connected graded Hopf algebra of finite Gel'fandKirillov dimension n, then H is a noetherian domain which is CohenMacaulay, ArtinSchelter regular and Auslander regular of global dimension n. It has S2 = IdH, and is CalabiYau. Detailed information is also provided about the Hilbert series of H. Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf kalgebras of finite Gel'fandKirillov dimension n with connected coalgebra, the underlying coalgebra is shown to be ArtinSchelter regular of global dimension n. Both these classes of Hopf algebra share many features in common with enveloping algebras of finite dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf kalgebra H of Gel'fandKirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U(g) for any Lie algebra g.
Item Type:  Articles 

Keywords:  Hopf algebra, connected coalgebra, graded algebra. 
Status:  Published 
Refereed:  Yes 
Glasgow Author(s) Enlighten ID:  Gilmartin, Paul and Brown, Professor Ken 
Authors:  Brown, K.A., Gilmartin, P., and Zhang, J.J. 
Subjects:  Q Science > QA Mathematics 
College/School:  College of Science and Engineering > School of Mathematics and Statistics > Mathematics 
Journal Name:  Transactions of the American Mathematical Society 
Publisher:  American Mathematical Society 
ISSN:  00029947 
ISSN (Online):  10886850 
Published Online:  05 November 2018 
Copyright Holders:  Copyright © 2018 American Mathematical Society 
First Published:  First published in Transactions of the American Mathematical Society 372:32833317 
Publisher Policy:  Reproduced in accordance with the publisher copyright policy 
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