2,225 research outputs found
Commutative combinatorial Hopf algebras
We propose several constructions of commutative or cocommutative Hopf
algebras based on various combinatorial structures, and investigate the
relations between them. A commutative Hopf algebra of permutations is obtained
by a general construction based on graphs, and its non-commutative dual is
realized in three different ways, in particular as the Grossman-Larson algebra
of heap ordered trees.
Extensions to endofunctions, parking functions, set compositions, set
partitions, planar binary trees and rooted forests are discussed. Finally, we
introduce one-parameter families interpolating between different structures
constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245
Combinatorial algebra for second-quantized Quantum Theory
We describe an algebra G of diagrams that faithfully gives a diagrammatic representation of the structures of both the Heisenberg–Weyl algebra H – the associative algebra of the creation and annihilation operators of quantum mechanics – and U(LH), the enveloping algebra of the Heisenberg Lie algebra LH. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(LH). While both H and U(LH) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation–annihilation system, of which it provides a concrete model
Hopf Algebras of Heap Ordered Trees and Permutations
It is known that there is a Hopf algebra structure on the vector space with
basis all heap-ordered trees. We give a new bialgebra structure on the space
with basis all permutations and show that there is a direct bialgebra
isomorphism between the Hopf algebra of heap-ordered trees and the bialgebra of
permutations.Comment: 10 pages LaTeX, minor revisio
Semisimple Hopf algebras of dimension 60
We determine the isomorphism classes of semisimple Hopf algebras of dimension
60 which are simple as Hopf algebras.Comment: amslatex, 23 page
Recursive relations in the core Hopf algebra
We study co-ideals in the core Hopf algebra underlying a quantum field
theory.Comment: 11 page
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