0. Introduction 1. Affine Weyl groups (Reduction modulo W ) 2. Double Hecke algebras (Automorphisms, Demazure-Lusztig operators) 3. Macdonald polynomials (Intertwining operators) 4. Fourier transform on polynomials (Four transforms) 5. Jackson integrals (Macdonald’s η-identities) 6. Semisimple and pseudo-unitary representations (Main Theorem) 7. Spherical representations (Semisimple spherical r...

Miriam Cohen raised the question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra for...

In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras W, S, H.

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend open-closed two-dimensional topological field theories to nonorientable surfac...

A properly in nite C∗-algebra has no non-zero traces. We construct properly in nite Banach ∗-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly `close' to the class of C∗-algebras, in the sense that they can be hermitian or ∗-semisimple.