arXiv subjects

Kornel Szlachanyi

Publications and source records attributed to Kornel Szlachanyi.

5 recordsLinked to original sources

arXiv2012

Skew-monoidal categories and bialgebroids

Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are precisely the right bialgebroids over the ring R. These skew-monoidal structures induce quotient skew-monoidal structures on the category of R-R-bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure by means of a natural distributive law. Pursuing a Theorem of Day and Street we also discuss monoidal lax comonads to describe the comodule categories of bialgebroids beyond the flat case.

math.QA↗

arXiv2004

The double algebraic view of finite quantum groupoids

This is an introduction to double algebras which is the structure modelled by the properties of the convolution product in Hopf algebras, weak Hopf algebras and in Hopf algebroids. We show that Hopf algebroids with a Frobenius integral can be characterized as the double algebras in which the two multiplications satisfy distributivity. We discuss questions of duality, antipode, Maschke theorem and examples.

math.QA↗

arXiv1998

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

We apply the theory of finite dimensional weak C^*-Hopf algebras A as developed by G. Böhm, F. Nill and K. Szlachányi to study reducible inclusion triples of von-Neumann algebras N \subset M \subset (M\cros\A). Here M is an A-module algebra, N is the fixed point algebra and \M\cros\A is the crossed product extension. ``Weak'' means that the coproduct Δon 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 E_l: M-->N, where under certain regularity conditions this correspondence is one-to-one. Associated with such left integrals we construct ``Jones projections'' e_l\in\A obeying the Jones relations as an identity in M\cros\A. Finally, we prove that N\subset M always has finite index and depth 2 and that the basic Jones construction is given by the ideal M_1:=M e_l M \subset M\cros\A, where under appropriate conditions M_1 = M\cros\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.

math.QA↗

arXiv1997

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable algebra of a generalized quantum spin chain with H-order and H^-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H=\CC G is a group algebra then A becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). To achieve this we construct localized coactions ρ: A \to (A \otimes D(H)) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetrty.

hep-th↗

arXiv1995

Quantum Chains of Hopf Algebras with Order-Disorder Fields and Quantum Double Symmetry

Given a finite dimensional C-*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H x ... and study its representations. A is the observable algebra of a generalized spin model with H-order and H^-disorder symmetries. By pointing out that A possesses a certain compressibility property we can classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). Complete, irreducible, translation covariant field algebra extensions F > A are shown to be in one-to-one correspondence with cohomology classes of 2-cocycles u in D(H) @ D(H).

hep-th↗