Searcharxiv⌕ Search

arXiv subjects

David E. Radford

Publications and source records attributed to David E. Radford.

14 recordsLinked to original sources
arXiv2021

Factorials and powers, a minimality result

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. We show that the smallest such $n$ is one of a pair of possibilities, or is one possibility, which we show how to calculate. There are three interesting numerical sequences which play a central role in our arguments. This paper is based on the improvement on Sterling's approximation of factorials due to Robbins \cite{Robbins} and results of \cite{Radford}

math.NT↗
arXiv2021

Factorials and powers, a minimality result, revisited

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. There are several numerical sequences associated with the study of the smallest such integer which are studied in \cite{RadFact} and \cite{RadGamma}. Here we continue the examination of one of them.

math.NT↗
arXiv2015

On Automorphisms of Biproducts

We study certain subgroups of the full group of Hopf algebra automorphisms of a biproduct. In the process interesting subgroups of certain permutation groups come into play.

math.RA↗
arXiv2014

Biproducts and Kashina's examples

We revisit a class of examples described in the original paper on biproducts, expand the class, and provide a detailed analysis of the coalgebra and algebra structures of many of these examples. Connections with the semisimple Hopf algebras of dimension a power of two determined by Kashina are examined. The finite-dimensional non-trivial semisimple cosemisimple Hopf algebras we construct are shown to be lower cosolvable. Some of these have one proper normal Hopf subalgebra and are not lower solvable.

math.RA↗
arXiv2010

Oriented Quantum Algebras and Coalgebras, Invariants of Oriented 1-1 Tangles, Knots and Links

In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between oriented quantum coalgebras and quantum coalgebras. We show that there are regular isotopy invariants of oriented 1-1 tangles and of oriented knots and links associated to oriented and twist oriented quantum coalgebras respectively. There are many parallels between the theory of oriented quantum coalgebras and the theory of quantum coalgebras

math.QA↗
arXiv2006

Biproducts and Two-Cocycle Twists of Hopf Algebras

Let $H$ be a Hopf algebra with bijective antipode over a field $k$ and suppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in the Yetter--Drinfel'd category ${}_H^H{\mathcal YD}$. We describe the bialgebras $(R{#}H)^{op}$ and $(R{#}H)^o$ explicitly as bi-products $R^{\UOP}{#}H^{op}$ and $R^{\UO}{#}H^o$ respectively where $R^{\UOP}$ is a bialgebra in ${}^{H^{op}}_{H^{op}}{\mathcal YD}$ and $R^{\UO}$ is a bialgebra in ${}^{H^o}_{H^o}{\mathcal YD}$. We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products.

math.QA↗
arXiv2006

On the irreducible representations of a class of pointed Hopf algebras

We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work of N. Andruskiewitsch and the second author. Special cases are the multiparameter deformations of the enveloping algebras of semisimple Lie algebras where the deforming parameters are not roots of unity and some of their finite-dimensional versions in the root of unity case.

math.QA↗
arXiv2006

Representations Parameterized by a Pair of Characters

Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$ on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy $(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a pair of characters $ρ\in \Alg(U, k)$ and $χ\in \Alg(A, k)$ we define a left $H$-module $L(ρ, χ)$. Under reasonable hypotheses the correspondence $(ρ, χ) \mapsto L(ρ, χ)$ determines a bijection between character pairs and the isomorphism classes of objects in a certain category ${}_H\underline{\mathcal M}$ of left $H$-modules. In many cases the finite-dimensional objects of ${}_H\underline{\mathcal M}$ are the finite-dimensional irreducible left $H$-modules. In math.QA/0603269 we apply the results of this paper and show that the finite-dimensional irreducible representations of a wide class of pointed Hopf algebras are parameterized by pairs of characters.

math.QA↗
arXiv2001

Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links

This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is formulated in terms of the biquandle, a generalization of the quandle of David Joyce (which, in turn, is a generalization of the fundamental group of the knot or link). We then introduce the concept of a bi-oriented quantum algebra which provides an algebraic context for the associated quantum invariant.

math.GT↗
arXiv2000

Oriented Quantum Algebras and Invariants of Knots and Links

In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a more algebraic perspective, and develop a more detailed theory for them and their associated invariants.

math.GT↗
arXiv1999

On Two Proofs for the Existence and Uniqueness of Integrals for Finite-Dimensional Hopf Algebras

This paper has two purposes. The first is to explicate the diagrammatic approach to Hopf algebras due to Kuperberg, and to examine his proof of the existence and uniqueness of integrals in both the diagrammatic and purely algebraic contexts. The second purpose of the paper is to show that the theory of integrals for a finite dimensional Hopf algebra A can be deduced from ideas concerning the trace function on End(A). Connections between Kuperberg's work and the trace function should be of interest to those who study three manifold invariants in relation to Hopf algebras.

math.QA↗
arXiv1994

Invariants of 3-Manifolds Derived From Finite Dimensional Hopf Algebras

This paper studies invariants of 3-manifolds derived from certain fin ite dimensional Hopf algebras. The invariants are based on right integrals for these algebras. It is shown that the resulting class of invariants is distinct from the class of Witten-Reshetikhin-Turaev invariants.

hep-th↗