Dynamical quantum groups at roots of 1. (English) Zbl 1023.17007

Etingof, Pavel; Nikshych, Dmitri

Duke Math. J. 108, No. 1, 135-168 (2001).

The article generalizes the theory of dynamical quantum groups for the quantum parameter \(q\) being a root of unity. It is shown that dynamical quantum groups at a root of unity can be described by the simpler and self-dual notion of weak Hopf algebras instead of the more general Hopf algebroids.
The authors generalize Drinfeld’s theory of twists to weak Hopf algebras. They show that twisting of a quasi-triangular weak Hopf algebra again yields a quasi-triangular weak Hopf algebra.
For each dynamical twist of a Hopf algebra two weak Hopf algebras \(D\) and \(H\) are constructed for which \(D^*\) with opposite comultiplication is isomorphic to \(H\). If the Hopf algebra is quasi-triangular the canonical homomorphism \(D\to H\) arising from the quasi-triangular structure is studied. A criterion has then been found when \(D\to H\) is an isomorphism. If the Hopf algebra is the finite-dimensional version of the quantum group \(U_q(g)\) for \(g\) a finite-dimensional simply laced Lie algebra, the methods of generating dynamical twists for generic quantum parameters can be carried over to the root-of-unity case. It \(T\) is an automorphism of the Dynkin diagram of \(g\) the weak Hopf algebras \(D\) and \(H\) associated with the twist corresponding to \(T\) are isomorphic. This in particular means that \(D_T := D\) is self-dual and it implies that the categories of representations \(\text{Rep}(D_T)\) are equivalent to each other for any two such automorphisms \(T_1\) and \(T_2\).

Reviewer: Bernhard Drabant (Mühlhausen)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
16W30	Hopf algebras (associative rings and algebras) (MSC2000)
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

dynamical quantum groups; weak Hopf algebras

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