Categorical studies of quantum symmetries and their applications


Workshop on 2012 May 22 at the Wigner Research Centre for Physics, Budapest ,
1121 Budapest, Konkoly Thege Miklós út 29-33 (KFKI campus), building no. 3, meeting room on the 3rd floor


    speakers
  • José Gómez Torrecillas (University of Granada)
    Double crossed products of weak Hopf algebras.    abstract
      The aim of this talk is to present some results on double crossed products of weak bialgebras. By this we mean weak bialgebras which -- as algebras -- arise as a weak wreath product of two weak bialgebras, and whose coalgebra structure comes from the tensor product coalgebra. The difficulty of the problem comes from the fact that no description of weak bialgebras as algebras in some well-chosen monoidal category is known. Hence there is no evident notion of (weak) wreath product of weak bialgebras. On the other hand, many examples of double crossed product weak bialgebras (in the above sense) are known. Our strategy is to take a weak distributive law between algebras underlying weak bialgebras. Then we look for sufficient conditions under which the corresponding weak wreath product algebra becomes a weak bialgebra with respect to the tensor product coalgebra structure. The conditions we present are only sufficient for the desired weak bialgebra to exist. Although it is possible to give the sufficient and necessary conditions, they are technically involved and so do not seem to be usable in practice. These sufficient conditions, however, have a simple form and they are capable to describe the known examples (in particular the Drinfeld double of a weak bialgebra).
      This talk is based on a joint work with Gabriella Böhm.

  • Esperanza López Centella (University of Granada)
    Duality for groupoids and weak Hopf algebras.    abstract
      The category of finite groups is anti-equivalent to the category of commutative semisimple finite-dimensional Hopf algebras over an algebraically closed field of characteristic 0 (see [1, Theorem 3.4.2]). This basic result could be understood as the simplest discrete version of Tannaka's duality theorem. Weak Hopf algebras were introduced in [2-4] in the framework of the theory of quantum groupoids, so they have been treated basically as genuine noncommutative objects since their inception. In particular, the extension of the aforementioned duality for finite grupoids seems have not been explored. The aim of this talk is to study this topic, establishing an antiequivalence between the category of finite groupoids and that of commutative semisimple finite-dimensional weak Hopf algebras over an algebraically closed field of charasteristic 0. This comes as a direct consequence of our main result, which shows that there is an adjoint pair of functors between the category of groupoids with finitely many objects and the category of weak Hopf algebras.
      This talk is based on joint work with José Gómez Torrecillas.

      [1] E. Abe, Hopf Algebras, Cambridge University Press, (1980). ISBN 0 521 22240 0.
      [2] G. Böhm, K. Szlachányi, A Coassociative C ∗ -Quantum Group with Nonintegral Dimensions, Lett. Math. Phys. 35, 437 (1996).
      [3] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf Algebras I. Integral Theory and C * -Structure , J. Algebra 221 (1999), 385-438.
      [4] K. Szlachányi, Weak Hopf Algebras , Operator Algebras and Quantum Field Theory, Eds.: S. Doplicher, R. Longo, J.E. Roberts, and
           L. Zsidó, International Press (1996).

  • Bertalan Pécsi (PhD School, Eötvös University, Budapest)
    Reflections and coreflections within profunctors.     abstract
      The so called "collage" of a profunctor F: A op × B → Set is the category which contains A and B (disjointly) and heteromorphisms from objects of A to those of B, given by elements of F(a,b). If B is a reflective subcategory therein, then the profunctor corresponds to an A → B functor (up to natural isomorphism), if A is coreflective subcategory, then it corresponds to a B → A functor, if both these are satisfied, then it is an adjunction. We sketch up a bicategory based definition of double categories, then show analogous assertions for double profunctors, colax and lax functors and colax/lax adjunctions.

  • Kornél Szlachányi (Wigner RCP, Budapest)
    Skew-monoidal categories.     abstract
      Skew-monoidal categories are introduced to describe a fragment of the structure of bialgebroids which allows to treat their modules and comodules on equal footing. The closed skew-monoidal structures on the category of right R-modules happen to be precisely the right bialgebroids over R.

  • Péter Vecsernyés (Wigner RCP, Budapest)
    Phase structure of Hopf spin chains.     abstract
      Phase equivalence of representations of two sided infinitely iterated crossed product of dually related C * -Hopf algebras ··· ∝ H∝ H ∝ H∝ ··· (Hopf spin chains) is defined by equivalent extensions provided by the so-called dual net. The equivalence classes are characterized by inequivalent non-degenerate right D(H)-module von Neumann algebras with trivial D(H)-invariant subalgebra or, equivalently, by cohomology classes of intermediate Δ D(H) -cocycles.


Interested colleagues are cordially invited to join us. For admittance to the KFKI campus, participants' names need to be submitted in advance. Hence registration is kindly asked to be made before May 18 via email to the organizer Gabriella Böhm , .