Locally compact quantum groups are intensively investigated in Warsaw as
examples of noncompact quantum spaces, and studied from the point of view
of unitary representations and duality theory. Although at the moment
there is no commonly accepted concept of a locally compact quantum group,
there are promising approaches and interesting examples worked out.
One of them is due to J.Kustermans and S.Vaes who define a locally compact
quantum group taking advantage of von Neumann algebras. This approach is in
a remarkable agreement with the Hopf symmetry principle discovered by A.Connes
and H.Moscovici.
Syllabus:
topological quantum groups, subcategory of compact quantum groups,
unbounded operators affiliated to a C*-algebra as coordinate functions
on locally compact quantum groups, representation theory and duality,
double-group construction, manageable multiplicative unitary operators,
short exact sequences and bicross-products
of locally compact quantum groups, modular pairs in involution from
Haar weights, examples including quantum ``ax+b", ``az+b" and
GL(2,C) groups and C*-algebraic deformations at roots of unity.