Aspects of Quantum Groups and Integrable Systems

Many q-differential operators arise in the study of q-special functions, Casimir operators, quantum groups, and representation theory. There are also natural origins via quantum integrable systems and the quantization of classical integrable systems. The latter is often expressed via a q-hierarchy picture akin to the standard Hirota–Lax–Sato formulation and this has many canonical aspects. On the other hand one can produce a great number of q-differential operators by more or less ad hoc manipulation of noncommutative differential calculi or by variations of classical Lie group methods applied to quantum groups. We examine first the hierarchy picture briefly and notice that although the standard methods generalize quite readily the resulting KP or KdV equations for example seem to have an infinite number of terms whereas many “ad hoc” derivations from differential calculi or e.g. Maurer–Cartan arguments have only a finite number of terms and there is no clear way to determine if in fact such equations have any intrinsic meaning. We show that the standard derivation of KdV via vector fields on the unit circle and the smooth dual space of Lax operators can be extended to a q-situation using a q-Virasoro algebra and we produce a corresponding qKdV equation with an infinite number of terms; this seems to be a fairly canonical derivation and we suggest that it could be equivalent to the hierarchy qKdV equation (not yet proved). This approach to KdV via the unit circle with its attendant UrKdV–mKdV equations, Schwarzian derivatives, projective geometry, etc. has another connection to quantum mechanics (QM) via the beautiful equivalence principle of Faraggi–Matone and we use this as a motivational background (cf. [1]). In this connection we remark that in fact KP for example can already be considered as a Moyal quantization of dKP (dispersionless KP) and it is not clear just what role the qKP or qKdV equations can play in quantum mechanics; however for completeness we also indicate a few approaches to Moyal type integrable equations.