Abstract
We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ1 over a Hopf algebra A which are quotients of the augmentation ideal A+ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation provides a bicovariant differential graded algebra on A. We introduce Λmax providing the maximal prolongation, while the canonical braided-exterior algebra Λmin = B−(Λ1) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ♯ by super-braided Fourier transform on B−(Λ1) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S3] with its 3D calculus and obeying the q-Hecke relation ♯2 = 1 + (q − q−1)♯ in middle degree on kq[SL2] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra mathcal {L}subseteq A defines a sub-braided Lie algebra and {Lambda }^{1}subseteq mathcal {L}^{*} provides the required data A+ → Λ1.
Highlights
Differential exterior algebras on quantum groups were extensively studied in the 1990s since [39] and have a critical role as examples of noncommutative geometry more generally
One problem which has remained open since that era is the general construction of a Hodge star operator in noncommutative geometry, even in the quantum group case
Eq 2.10 for all i and proves the lemma. This fleshes out the braided-Hopf algebra interpretation of the Woronowicz exterior algebra on a Hopf algebra [39] using [35] for the direct treatment of d
Summary
Differential exterior algebras on quantum groups were extensively studied in the 1990s since [39] and have a critical role as examples of noncommutative geometry more generally. The totally antisymmetric symbol defined by 12···n = 1 takes care of the reordering inside the integral This construction extends to the whole manifold and in the presence of a metric gives the classical Hodge operator : m → n−m aside from a reversal of the order of products in the result, which amounts to a sign that depends on m (and is due to our conventions on the duality pairing). The super-braided Hopf algebra interpretation of the Woronowicz construction of bicovariant differential exterior algebras was in [28, 29] among other works In this context the universal property of B− corresponds in some sense to the minimal relations needed to ensure Poincareduality, a remark that will be reflected in our approach to the Hodge star. We will make extensive use of the theory of braided-Hopf algebras [21] including the diagrammatic notation in [19, 22]
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