Integral calculus on \(E_{q}\)(2) (Q543932)

This paper first recalls that, given a differential graded algebra \(\Omega( A) =\bigoplus_{k=0}^{\infty}\Omega^{k}( A) \) over a \(\mathbb{K}\)-algebra \(A\), a hom-connection over \(A\) with respect to \(\Omega(A)\) is a \(\mathbb{K}\)-linear map \(\nabla_{0} :\mathcal{J}^{1}(A) \to A\) satisfying \(\nabla_{0}(\phi a) =\nabla_{0}( \phi) a+\phi(da)\) for all \(a\in A\) and \(\phi\in\mathcal{J}^{1}(A)\), where \(\mathcal{J}^{k}( A) :=\hom_{A}(\Omega^{k}(A),A) \) is an \(A\)-bimodule with \((a\phi b) (\omega) :=a\phi( b\omega) \) for \(a,b\in A\), \(\phi\in\mathcal{J}^{k}(A)\) and \(\omega\in\Omega^{k}( A) \), and the differential operator \(d:\Omega^{k}( A) \to\Omega^{k+1}( A) \) of \(\Omega( A) \) satisfies \(d\circ d=0\). Such a hom-connection \(\nabla_{0}\) can be naturally extended to \(\nabla_{k}:\mathcal{J}^{k+1}( A) \to \mathcal{J}^{k}( A) \) by \(\nabla_{k}( \phi) ( \omega) :=\nabla_{0}( \phi\omega) +(-1)^{k+1}\phi(d\omega)\), and, when \(\nabla\) is flat in the sense that \(\nabla_{0}\circ\nabla_{1}=0\), this yields a chain complex \(( \mathcal{J}( A) ,\nabla) \equiv(\bigoplus _{k=0}^{\infty}\mathcal{J}^{k}( A) ,\nabla) \), called the complex of integral forms. The author then presents complexes of integral forms over the algebra of the quantum Euclidean group \(E_{q}( 2) \) and over the algebra of its homogeneous space -- the quantum plane \(\mathbb{C}_{q}\), and, in the spirit of Poincaré duality, establishes the isomorphism between the complex of integral forms \(( \mathcal{J}( A) ,\nabla) \) and the corresponding de Rham complex \(( \Omega( A) ,d) \) for these two examples.