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Algebraic quantum groups and duality I
Let be a finite-dimensional Hopf algebra. The linear dual of
is again a finite-dimensional Hopf algebra. The duality is given by an
element , defined by where and . We use for the pairings. In the introduction of this paper, we
recall the various properties of this element as sitting in the algebra
. More generally, we can consider an algebraic quantum group
. We use the term here for a regular multiplier Hopf algebra with
integrals. For we now take the dual of . It is again an
algebraic quantum group. In this case, the duality gives rise to an element
in the multiplier algebra . Still, most of the properties of
in the finite-dimensional case are true in this more general setting. The focus
in this paper lies on various aspects of the duality between and its dual
. Among other things we include a number of formulas relating the
objects associated with an algebraic quantum group and its dual. This note is
meant to give a comprehensive, yet concise (and sometimes simpler) account of
these known results. This is part I of a series of three papers on this
subject. The case of a multiplier Hopf -algebra with positive integrals is
treated in detail in part II and part III
Analytic aspects of the shuffle product
There exist very lucid explanations of the combinatorial origins of rational
and algebraic functions, in particular with respect to regular and context free
languages. In the search to understand how to extend these natural
correspondences, we find that the shuffle product models many key aspects of
D-finite generating functions, a class which contains algebraic. We consider
several different takes on the shuffle product, shuffle closure, and shuffle
grammars, and give explicit generating function consequences. In the process,
we define a grammar class that models D-finite generating functions
Varieties of Cost Functions.
Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring
Weighted Logics for Nested Words and Algebraic Formal Power Series
Nested words, a model for recursive programs proposed by Alur and Madhusudan,
have recently gained much interest. In this paper we introduce quantitative
extensions and study nested word series which assign to nested words elements
of a semiring. We show that regular nested word series coincide with series
definable in weighted logics as introduced by Droste and Gastin. For this we
establish a connection between nested words and the free bisemigroup. Applying
our result, we obtain characterizations of algebraic formal power series in
terms of weighted logics. This generalizes results of Lautemann, Schwentick and
Therien on context-free languages
Exponential Renormalization II: Bogoliubov's R-operation and momentum subtraction schemes
This article aims at advancing the recently introduced exponential method for
renormalisation in perturbative quantum field theory. It is shown that this new
procedure provides a meaningful recursive scheme in the context of the
algebraic and group theoretical approach to renormalisation. In particular, we
describe in detail a Hopf algebraic formulation of Bogoliubov's classical
R-operation and counterterm recursion in the context of momentum subtraction
schemes. This approach allows us to propose an algebraic classification of
different subtraction schemes. Our results shed light on the peculiar algebraic
role played by the degrees of Taylor jet expansions, especially the notion of
minimal subtraction and oversubtractions.Comment: revised versio
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