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9.0

Jun 29, 2018
06/18

by
Ken Brown; Paul Gilmartin; James J. Zhang

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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 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...

Topics: Quantum Algebra, Rings and Algebras, Mathematics

Source: http://arxiv.org/abs/1601.06687

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3.0

Jun 29, 2018
06/18

by
Paul Gilmartin

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Let $k$ be a field and let $H$ denote a pointed Hopf $k$-algebra with antipode $S$. We are interested in determining the order of $S$. Building on the work done by Taft and Wilson $[7]$, we define an invariant for $H$, denoted $m_{H}$, and prove that the value of this invariant is connected to the order of $S$. In the case where $\operatorname{char}k=0$, it is shown that if $S$ has finite order then it is either the identity or has order $2m_{H}$. If in addition $H$ is assumed to be coradically...

Topics: Rings and Algebras, Mathematics

Source: http://arxiv.org/abs/1611.03480

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Jun 27, 2018
06/18

by
Ken Brown; Paul Gilmartin

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Let H be a connected Hopf k-algebra of finite Gel'fand-Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H. They are shown to be deformations of commutative polynomial k-algebras. A number of well-known homological and other properties follow immediately from this fact. Further properties are described, examples are considered, invariants are constructed and a number of open questions...

Topics: Mathematics, Rings and Algebras, Quantum Algebra

Source: http://arxiv.org/abs/1506.02427

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Jun 30, 2018
06/18

by
Ken A. Brown; Paul Gilmartin

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This is a brief survey of some recent developments in the study of infinite dimensional Hopf algebras which are either noetherian or have finite Gelfand-Kirillov dimension. A number of open questions are listed.

Topics: Mathematics, Quantum Algebra, Rings and Algebras

Source: http://arxiv.org/abs/1405.4105