Let k be a field, and H a Hopf algebra with bijective antipode. If H is commutative, noetherian, semisimple and cosemisimple, then the category HYD H of Yetter-Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that H is commutative.

Introduction Lecture 1. Algebraic properties of correlators in 2D topological field theory. Moduli of a 2D TFT and WDVV equations of associativity. Lecture 2. Equations of associativity and Frobenius manifolds. Deformed flat connection and its monodromy at the origin. Lecture 3. Semisimplicity and canonical coordinates. Lecture 4. Classification of semisimple Frobenius manifolds. Lecture 5. Mon...

For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus ≤ 1 Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology.

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...

In this paper, we describe the maximal bounded Z-filtrations of Artinian semisimple rings. These turn out to be the filtrations associated to finite Z-gradings. We also consider simple Artinian rings with involution, in characteristic 6= 2, and we determine those bounded Z-filtrations that are maximal subject to being stable under the action of the involution. Finally, we briefly discuss the an...

where g±d 6= 0. The positive integer d is called the depth of this Z-grading, and of the nilpotent element e. This notion was previously studied e.g. in [P1]. An element of g of the form e+ F , where F is a non-zero element of g−d, is called a cyclic element, associated with e. In [K1] Kostant proved that any cyclic element, associated with a principal (= regular) nilpotent element e, is regula...