On a class of representations of quantum groups

Abstract

This paper is a short account of the construction of a new class of the infinitedimensional representations of the quantum groups. The examples include finitedimensional quantum groups U q (g), Yangian Y (g) and affine quantum groups at zero level U q (ĝ) c=0 corresponding to an arbitrary finite-dimensional semisimple Lie algebra g. At the intermediate step we construct the embedding of the quantum groups into the algebra of the rational functions on the quantum multi-dimensional torus. The explicit parameterization of the quantum groups used in this paper turns out to be closely related to the parameterization of the moduli spaces of the monopoles. As a result the proposed constructions of the representations provide a quantization of the moduli spaces of the monopoles on R 3 and R 2 × S 1 .

On The Continuous Series for $${{\widehat{sl(2,{\mathbb{R}})}}}$$ s l ( 2 , R ) ^

“…It also follows that they satisfy the commutation relations (9). One can show that e ±x(u) for any u are Hermitian operators, considered on a dense set D l A .…”

“…These unitary representations, which do not have a classical limit, but behave similarly to the finite-dimensional representations of the quantum group U q (su(2)), in particular, they form a tensor category. A generalization of these representations to higher rank quantum groups has been also obtained in [8], [9], [5], [12], [13].…”

“…In the quantum group case one had to use the representations of the multiple quantum planes [9], [14], [5], [12], [13]. In our case it seems to be a similar technical problem: the role of quantum planes will be played by the loop ax + b-group.…”

Quantum groups, quantum tori, and the Grothendieck–Springer resolution

“…These embeddings, too, are constructed by explicitly verifying the relations in the Chevalley-Serre presentation of U q (g). Subsequently, analogs of such representations for any quantum affine Kac-Moody algebra U q (g) were proposed in [14].…”

“…In particular, it would be interesting to understand the significance of quantum cluster mutations for U q (g). Such an explicit coordinatization would also allow to compare our embedding to the ones of [14,13]. Finally, in a recent work [16] a regular cluster structure on the semiclassical limit of F l (U q (gl n )) was constructed, while in [5] a different set of log-canonical coordinates, which do not extend to global regular functions, was proposed.…”