Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 335

Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers

F. Nill, K. Szlach'anyi and H.-W. Wiesbrock

We apply the theory of finite dimensional weak C*-Hopf algebras A as developed by G. Böhm, F. Nill and K. Szlach'anyi to study reducible inclusion triples of von-Neumann algebras N c M c (M x A). Here M is an A-module algebra, N is the fixed point algebra and M x A is the crossed product extension. "Weak" means that the coproduct Delta on A is non-unital, requiring various modifications of the standard definitions for (co-)actions and crossed products. We show that acting with normalized positive and nondegenerate left integrals l in A gives rise to faithful conditional expectations El: M-->N, where under certain regularity conditions this correspondence is one-to-one. Associated with such left integrals we construct "Jones projections" el in A obeying the Jones relations as an identity in M x A. Finally, we prove that N c M always has finite index and depth 2 and that the basic Jones construction is given by the ideal M1:=M el M c M x A, where under appropriate conditions M1 = M x A. In a subsequent paper we will show that converseley any reducible finite index and depth-2 Jones tower of von-Neumann factors (with finite dimensional centers) arises in this way. Subject-class: Quantum Algebra; Mathematical Physics

---

Get a gzip-compressed PostScript copy of this preprint
preprint335.ps.gz (296 kB)

---