Explicit Realization of Hopf Cyclic Cohomology Classes of Bicrossed Product Hopf Algebras

We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\text{cH}=(\text{cU}({\text{Fg}}_1)\text{acr}\text{cR}(G_2){)}^{\text{cop}}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\text{cA}=C_c^{\text{ify}}(G_1)⋊G_2^{\delta }$. We give an explicit way to construct Hopf cyclic cohomology classes of our Hopf algebra $\text{cH}$ and then realize these classes in terms of explicit representative cocycles in the cyclic cohomology of the convolution algebra $\text{cA}$.