Quantum differentials on cross product Hopf algebras

Abstract

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products $A\hookrightarrow A\bowtie H\hookleftarrow H$, double cross coproducts $A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft} H\twoheadrightarrow H$, biproducts $A{\buildrel\hookrightarrow\over \twoheadleftarrow}A{\cdot\kern-.33em\triangleright\!\!\!<} B$ and bicrossproducts $A\hookrightarrow A{\blacktriangleright\!\!\triangleleft} H\twoheadrightarrow H$ on the assumption that the factors have strongly bicovariant calculi $\Omega(A),\Omega(H)$ (or a braided version $\Omega(B)$). We use super versions of each of the constructions. Moreover, the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable. In the case of the Drinfeld double $D(A,H)=A^{\rm op}\bowtie H$ (where $A$ is dually paired to $H$), we show that its canonical actions on $A,H$ are differentiable. Examples include are a canonical $\Omega(\Bbb C_q[GL_2\ltimes \Bbb C^2])$ for the quantum group of affine transformations of the quantum plane and $\Omega(\Bbb C_\lambda[{\rm Poinc_{1,1}}])$ for the bicrossproduct Poincar\'e quantum group in 2 dimensions. We also show that $\Omega(\Bbb C_q[GL_2])$ itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra.