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1. Algebraic and topological structures on rational tangles

Author

Seyed M.H. Mansourbeigi, Vida Milani, Hossein Finizadeh, and Poli Paper

Subjects
tangle convergent, Coalgebra, Structure (category theory), Bi-pseudo-module, 010103 numerical & computational mathematics, Interval (mathematics), lcsh:Analysis, Group Hopf algebra, Continued fraction, Topology, 01 natural sciences, Tangle, Interval coalgebra, Incidence algebra, Mathematics::Quantum Algebra, Physical Sciences and Mathematics, Order (group theory), 0101 mathematics, Algebraic number, Mathematics, group Hopf algebra, Computer Sciences, lcsh:Mathematics, lcsh:QA299.6-433, Tangle convergen, bi-pseudo-module, lcsh:QA1-939, Mathematics::Geometric Topology, continued fraction, 010101 applied mathematics, Algebra, incidence algebra, pseudo-tensor product, Pseudo-tensor product, Pseudo-module, pseudo-module, Geometry and Topology, tangle, Locally finite partial order, locally finite partial order, interval coalgebra
Abstract

In this paper we present the construction of a group Hopf algebra on the class of rational tangles. A locally finite partial order on this class is introduced and a topology is generated. An interval coalgebra structure associated with the locally finite partial order is specified. Irrational and real tangles are introduced and their relation with rational tangles are studied. The existence of the maximal real tangle is described in detail.

Published
2017

2. From left modules to algebras over an operad: application to combinatorial Hopf algebras

Author

Muriel Livernet, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and this paper was written during the stay of the author at Institut Mittag-Leffler (djursholm, sweden).

Subjects
Pure mathematics, twisted bialgebra, 18D50 (Primary), 16W30, 16A06 (Secondary), Context (language use), 0102 computer and information sciences, Quasitriangular Hopf algebra, 01 natural sciences, Mathematics::Algebraic Topology, Mathematics::Category Theory, Mathematics::Quantum Algebra, FOS: Mathematics, S-module, 0101 mathematics, Mathematics, operad, Algebra and Number Theory, Functor, Quantum group, Applied Mathematics, 010102 general mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics::Rings and Algebras, Tensor algebra, combinatorial Hopf algebra, Mathematics - Rings and Algebras, Hopf algebra, MSC2000: Primary: 18D50, Secondary: 16W30, 16A06, Rings and Algebras (math.RA), 010201 computation theory & mathematics, free associative algebra, Geometry and Topology, Forgetful functor, Analysis, Vector space
Abstract

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from S-modules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for these Hopf algebras. Let O denote the forgetful functor from S-modules to graded vector spaces. Left modules over an operad P are treated as P-algebras in the category of S-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad P: the functor O sends P-algebras to P-algebras. If P is a Hopf operad then O sends Hopf P-algebras to Hopf P-algebras. If the operad P is regular one gets two different structures of Hopf P-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal P-bialgebra and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as Hopf algebras on the faces of the permutohedra and associahedra., Section 4.3 removed. To appear in Annales Math\'ematiques Blaise Pascal

Published
2006