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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Connected (graded) Hopf algebras


Authors: K. A. Brown, P. Gilmartin and J. J. Zhang
Journal: Trans. Amer. Math. Soc. 372 (2019), 3283-3317
MSC (2010): Primary 16T05, 16W50; Secondary 17B37, 20G42
DOI: https://doi.org/10.1090/tran/7686
Published electronically: November 5, 2018
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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 0. The first class consists of those Hopf $ k$-algebras that are connected graded as algebras, and the second class are those Hopf $ k$-algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel'fand-Kirillov dimension. It is shown that if the Hopf algebra $ H$ is a connected graded Hopf algebra of finite Gel'fand-Kirillov dimension $ n$, then $ H$ is a noetherian domain which is Cohen-Macaulay, Artin-Schelter regular, and Auslander regular of global dimension $ n$. It has $ S^2 = \mathrm {Id}_H$, and it is Calabi-Yau. 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 $ k$-algebras of finite Gel'fand-Kirillov dimension $ n$ with connected coalgebra, the underlying coalgebra is shown to be Artin-Schelter regular of global dimension $ n$. Both these classes of Hopf algebras 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 $ k$-algebra $ H$ of Gel'fand-Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to $ U(\mathfrak{g})$ for any Lie algebra $ \mathfrak{g}$.


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Additional Information

K. A. Brown
Affiliation: School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, Scotland
Email: ken.brown@glasgow.ac.uk

P. Gilmartin
Affiliation: School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, Scotland
Email: p.gilmartin.1@research.gla.ac.uk

J. J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: zhang@math.washington.edu

DOI: https://doi.org/10.1090/tran/7686
Received by editor(s): June 25, 2017
Received by editor(s) in revised form: July 31, 2018, and August 4, 2018
Published electronically: November 5, 2018
Article copyright: © Copyright 2018 American Mathematical Society