scholarly journals A combinatorial Hopf algebra for the boson normal ordering problem

2018 ◽  
Vol 5 (1) ◽  
pp. 61-102 ◽  
Author(s):  
Imad Eddine Bousbaa ◽  
Ali Chouria ◽  
Jean-Gabriel Luque
Keyword(s):  
Hopf Algebra ◽  
Normal Ordering ◽  
Combinatorial Hopf Algebra

scholarly journals Lattice of combinatorial Hopf algebras: binary trees with multiplicities

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jean-Baptiste Priez
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Lattice Structure ◽  
Binary Trees ◽  
Hook Length ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Hopf Algebra ◽  
International Audience ◽  
Comme Application ◽  
Premiere Partie

International audience In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees. Dans une première partie, nous formalisons la construction d’algèbres de Hopf combinatoires à partir d’une réalisation polynomiale et de monoïdes de type monoïde plaxique. Grâce à cette construction, nous mettons à jour une structure de treillis sur ces algèbres de Hopf combinatoires. Comme application, nous construisons une nouvelle algèbre de Hopf sur des arbres binaires à multiplicités et on l’utilise pour démontrer une formule des équerressur ces arbres.

scholarly journals A polynomial realization of the Hopf algebra of uniform block permutations

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Rémi Maurice
Keyword(s):  
Hopf Algebra ◽  
Poster Session ◽  
Combinatorial Hopf Algebra ◽  
Noncommutative Polynomials ◽  
International Audience

Poster session International audience We investigate the combinatorial Hopf algebra based on uniform block permutations and we realize this algebra in terms of noncommutative polynomials in infinitely many bi-letters.

scholarly journals Integer points enumerator of hypergraphic polytopes

2021 ◽  
pp. 1-1
Author(s):  
Marko Pesovic
Keyword(s):  
Positive Integer ◽  
Hopf Algebra ◽  
Symmetric Function ◽  
Quasisymmetric Function ◽  
Quasisymmetric Functions ◽  
Simple Graphs ◽  
Integer Points ◽  
Combinatorial Hopf Algebra ◽  
Chromatic Symmetric Function

For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley chromatic symmetric function of simple graphs. We consider a certain combinatorial Hopf algebra of hypergraphs and show that universal morphism to quasisymmetric functions coincides with this enumerator function. We calculate the f-polynomial of uniform hypergraphic polytopes.

scholarly journals Hopf algebra methods in graph theory

1995 ◽  
Vol 101 (1) ◽  
pp. 77-90 ◽  
Author(s):  
William R. Schmitt
Keyword(s):  
Graph Theory ◽  
Hopf Algebra

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

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