The locally compact (l.c.) quantum group is a relatively new C*-algebraic formalism for quantum groups, generalizing the Kac algebra, compact quantum group and Hopf algebra approaches. Earlier attempts of a unifying definition of quantum groups using e.g. multiplicative unitaries have had some success, but have also ran into several technical problems.
One of the main features distinguishing it from other approaches is the axiomatic existence of an invariant weight, giving a noncommutative analogue of the Haar measure.
The category of l.c. quantum groups allow for a dual construction, generalizing the Pontryagin duality of abelian groups.
The theory has an equivalent formulation in terms of von Neumann algebras.
Famous quotes containing the words locally, compact, quantum and/or group:
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)
“Take pains ... to write a neat round, plain hand, and you will find it a great convenience through life to write a small and compact hand as well as a fair and legible one.”
—Thomas Jefferson (1743–1826)
“The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“The boys think they can all be athletes, and the girls think they can all be singers. That’s the way to fame and success. ...as a group blacks must give up their illusions.”
—Kristin Hunter (b. 1931)