Hodge Star as Braided Fourier Transform

TL;DR

This paper introduces a canonical Hodge star operator on braided Hopf algebras using a super-braided Fourier transform, revealing new algebraic relations and applications in quantum calculus.

Contribution

It presents a novel, canonical approach to defining the Hodge star operator on braided Hopf algebras, differing from previous methods and applicable to quantum groups and calculi.

Findings

Hodge star operator obeys a q-Hecke relation in middle degree on $k_q[SL_2]$

Order 3 in middle degree on $k[S_3]$ with 3D calculus

Provides a Hodge map on quantum plane calculi

Abstract

We study super-braided Hopf algebras $\Lambda$ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules $\Lambda^1$ over a Hopf algebra $A$ which are quotients of the augmentation ideal $A^+$ under right multiplication and the adjoint coaction. Here super-bosonisation $\Omega=A\ltimes\Lambda$ provides a bicovariant differential graded algebra on $A$. We introduce $\Lambda_{max}$ providing the maximal prolongation, while the canonical braided-exterior algebra $\Lambda_{min}=B_-(\Lambda^1)$ provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator $\sharp$ by super-braided Fourier transform on $B_-(\Lambda^1)$ 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 $k[S_3]$ with its 3D calculus and obeying the $q$-Hecke relation $\sharp^2=1+(q-q^{-1})\sharp$ in middle degree on $k_q[SL_2]$ with its 4D calculus. Our work also provided a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras $A$ whereby any subcoalgebra $L\subseteq A$ defines a sub braided-Lie algebra and $\Lambda^1\subseteq L^*$ provides the required data $A^+\to \Lambda^1$.