Hodge Star as Braided Fourier Transform

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 Open image in new windowA. We introduce Λ _{m a x} providing the maximal prolongation, while the canonical braided-exterior algebra Λ _min = 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 kS ₃] with its 3D calculus and obeying the q-Hecke relation ?² = 1 + (q ? q ^?1)? in middle degree on k _q S L ₂] 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.