On Heisenberg–Weyl algebra and subsets of Riordan subgroups

Silvia Goodenough (1) , Christian Lavault (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

(The present paper is an extended and corrected second version of [25].) 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 dif-ferential 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" Ri-ordan 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: pre-function striped matrix; Operations for striped quasigroups and semigroups; Riordan arrays and groups; Substitutions with prefunctions; Heisenberg–Weyl algebra; Combinatorial models of creation–annihilation; Combinatorial algebra; Striped Riordan subgroups

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Algèbres d'opérateurs [math.OA]; Anneaux et algèbres [math.RA]; Physique Quantique [quant-ph]

Silvia Goodenough, Christian Lavault. On Heisenberg–Weyl algebra and subsets of Riordan subgroups: On subsets of Riordan subgroups and Heisenberg--Weyl algebra. 2015. ⟨hal-00974929v2⟩