Differential Calculus on h-Deformed Spaces

Basile Herlemont (1, 2) , Oleg Ogievetsky (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E2 Géométrie, Physique et Symétries

We construct the rings of generalized differential operators on the h-deformed vector space of gl-type. In contrast to the q-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators Diffh,σ(n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings Diffh,σ(n).

Mots clés: differential operators; Yang–Baxter equation; reduction algebras; universal enveloping algebra; representation theory; Poincaré–Birkhoff–Witt property; rings of fractions

Domaines: Anneaux et algèbres [math.RA]; Physique mathématique [math-ph]; Algèbres quantiques [math.QA]; Théorie des représentations [math.RT]

Basile Herlemont, Oleg Ogievetsky. Differential Calculus on h-Deformed Spaces. Symmetry, Integrability and Geometry : Methods and Applications, 2017, Special Issue on Recent Advances in Quantum Integrable Systems, 2017 (13), pp.082. ⟨10.3842/SIGMA.2017.082⟩. ⟨hal-01696326⟩