Overview of the Heisenberg–Weyl Algebra and Subsets of Riordan Subgroups

Silvia Goodenough (1) , Christian Lavault (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with differential operators and the associated one-parameter group. Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e. substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the `striped' Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.

Mots clés: Striped Riordan subgroups.; Operations for striped quasigroups and semigroups; Riordan arrays and groups; Combinatorial algebra; Models of creation--annihilation; Heisenberg--Weyl algebra

Domaines: Combinatoire [math.CO]; Algèbres quantiques [math.QA]; Mathématique discrète [cs.DM]; Algèbres d'opérateurs [math.OA]

Silvia Goodenough, Christian Lavault. Overview of the Heisenberg–Weyl Algebra and Subsets of Riordan Subgroups. The Electronic Journal of Combinatorics, 2015, Volume 22, 22 (4), pp.Id: 16. ⟨hal-00974929v3⟩