Abstract

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces allows to extend the scope of the latter theory considerably. In particular, the bornological theory covers smooth convolution algebras of arbitrary locally compact groups and their duals. Another source of examples arises from deformation quantization in the sense of Rieffel. Apart from describing these examples we obtain some general results on bornological quantum groups. In particular, we construct the dual of a bornological quantum group and prove the Pontrjagin duality theorem.

Keywords

multiplier Hopf algebras, quantum groups, bornological vector spaces

Mathematical Subject Classification

Primary: 16W30, 81R50

Authors