MULTIPARAMETER QUANTUM GROUPS, DEFORMATIONS AND SPECIALISATIONS

Speaker(s)

FABIO GAVARINI

Affiliation

Università di Roma Tor Vergata, Italy

Language of the talk

English

Date

Jan. 21, 2026, 5:15 p.m.

Information about the event

ZOOM

Title in Polish

MULTIPARAMETER QUANTUM GROUPS, DEFORMATIONS AND SPECIALISATIONS

Seminar

North Atlantic Noncommutative Geometry Seminar

---

We introduce the notion of a formal multiparameter quantum universal enveloping algebra (FoMpQUEA) as a straightforward generalization of Drinfeld’s quantum group Uℏ(g), where the underlying Lie algebra g is any symmetrizable Kac-Moody algebra. Then we show that the class of all FoMpQUEA's is closed under deformations by (“toral”) twists and deformations by (“toral”) 2-cocycles, if we suitably adapt the latter notion. As a consequence, we find that most (if not all) “multiparameter formal QUEA's” considered so far in the literature actually fall within this class. In particular, we prove that any FoMpQUEA is isomorphic to a suitable deformation, by a twist or by a 2-cocycle, of Drinfeld’s standard uniparameter QUEA. Independently, we also introduce multiparameter Lie bialgebras (MpLbA’s) and consider their deformations by twists and by 2-cocycles. The connection with FoMpQUEA's is that the semiclassical limit of every FoMpQUEA is a suitable MpLbA and, conversely, every MpLbA can be quantized to a suitable FoMpQUEA. In fact, the processes of the “specialization” of a FoMpQUEA to a MpLbA and the processes of the “deformation" (by a toral twist or toral 2-cocycle) of a MpLbA commute with each other. To end with, I will explain how all this extends to the setup where g is a simple Lie superalgebra of basic type. In addition, for type A, I will also present an alternative dual approach, which yields a suitable, FRT-like, multiparametric quantisation of the algebra of functions on the formal supergroup of type A. Based on joint work with Gastón Andrés García and Margherita Paolini.

---