Coherence Isomorphisms for a Hopf Category

  • Volodymyr Lyubashenko
Part of the NATO Science Series book series (NAII, volume 22)


Crane and Frenkel proposed a notion of a Hopf category in [1]. It was motivated by Lusztig’s approach to quantum groups—his theory of canonical bases. In particular, Lusztig obtains braided deformations \( {U_{q}}{\mathfrak{n}_{ + }} \) of universal enveloping algebras \( U{\mathfrak{n}_{ + }} \) for some nilpotent Lie algebras \( {\mathfrak{n}_{ + }} \) together with canonical bases of these braided Hopf algebras [2, 3, 4]. The elements of the canonical basis are identified with certain objects of equivariant derived categories, contained in semisimple abelian subcategories of semisimple complexes. Conjectural properties of these categories were collected into a system of axioms of a Hopf category, equipped with functors of multiplication and comultiplication, isomorphisms of associativity, coassociativity and coherence which satisfy four equations [1]. Crane and Frenkel gave an example of a Hopf category resembling the semisimple category encountered in Lusztig’s theory corresponding to one-dimensional Lie algebra \( {\mathfrak{n}_{ + }} \) —nilpotent subalgebra of \( \mathfrak{s}\mathfrak{l}(2) \). The mathematical framework and some further examples of Hopf categories were provided by Neuchl [5].


Hopf Algebra Quantum Group Distinguished Triangle Perverse Sheave Semisimple Complex 
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Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Volodymyr Lyubashenko
    • 1
  1. 1.Institute of MathematicsKyivUkraine

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