Kornél Szlachányi

Bialgebroid actions on depth two extensions and duality

Abstract A general notion of depth two for ring homomorphism N → M is introduced. The step two centralizers A =End N M N and B=(M⊗ N M) N in the Jones tower above N → M are shown in a natural way via H-equivalence to be dual bimodules for Morita equivalent endomorphism rings, the step one and three centralizers, R = C M ( N ) and C =End N – M ( M ⊗ N M ). We show A and B to possess dual left and right R -bialgebroid structures which generalize Lus fundamental bialgebroids over an algebra. There are actions of A and B on M and E ′= End N M with Galois properties. If M | N is depth two and Frobenius with R a separable algebra, we show that A and B are dual weak Hopf algebras fitting into a duality-for-actions tower extending previous results in this area for subfactors and Frobenius extensions.

QUANTUM CHAINS OF HOPF ALGEBRAS WITH QUANTUM DOUBLE COSYMMETRY

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

Skew-monoidal categories and bialgebroids☆

Abstract 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.

The monoidal Eilenberg–Moore construction and bialgebroids

Abstract Strong monoidal functors U : C → M with left adjoints determine, in a universal way, monoids T in the category of opmonoidal endofunctors on M . Treating such opmonoidal monads as abstract “quantum groupoids” we derive Tannaka duality between right adjoint strong monoidal functors and opmonoidal monads. Bialgebroids, i.e., Takeuchis ×R-bialgebras, appear as the special case when T has also a right adjoint. Streets 2-category of monads then leads to a natural definition of the 2-category of bialgebroids.

Hopf algebroid symmetry of abstract Frobenius extensions of depth 2

Abstract We study Frobenius 1-cells ι in an additive bicategory 𝒞 satisfying the depth 2 condition. We show that the rings of 2-cells 𝒞2 and can be equipped with dual Hopf algebroid structures. We prove also that a Hopf algebroid appears as the solution of the above abstract symmetry problem if and only if it possesses a two sided non-degenerate integral.

Weak c*-Hopf algebras: the coassociative symmetry of non-integral dimensions

By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C∗-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C∗-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplicative unitary of Baaj and Skandalis with a partial isometry. The algebraic properties, the existence of the Haar measure and representation theory are briefly discussed. An algorithm is explained how to construct examples (in particular ones with non-integral dimensions) from non-Abelian cohomology.

On the tensor product of modules over skew monoidal actegories

Abstract This paper is about skew monoidal tensored V -categories (= skew monoidal hommed V -actegories) and their categories of modules. A module over 〈 M , ⁎ , R 〉 is an algebra for the monad T = R ⁎  _ on M . We study in detail the skew monoidal structure of M T and construct a skew monoidal forgetful functor M T → M E to the category of E -objects in M where E = M ( R , R ) is the endomorphism monoid of the unit object R . Then we give conditions for the forgetful functor to be strong monoidal and for the category M T of modules to be monoidal. In formulating these conditions a notion of ‘self-cocomplete’ subcategories of presheaves appears to be useful which provides also some insight into the problem of monoidality of the skew monoidal structures found by Altenkirch, Chapman and Uustalu on functor categories [ C , M ] .