The Regular Representation of the twisted queer $q$-Schur Superalgebra

We study the representation theory of the quantum queer superalgebra ${U_{\lcase{v}}(\mathfrak{\lcase{q}}_{n})}$ and obtain some properties of the highest weight modules. Furthermore, based on the realization of ${U_{\lcase{v}}(\mathfrak{\lcase{q}}_{n})}$, we study the representation theory of the twisted queer $q$-Schur superalgebra ${{\widetilde{\mathcal{Q}}}_{\lcase{v}}(\lcase{n},\lcase{r})}$, and obtain the decomposition of its regular module as a direct sum of irreducible submodules, which also means ${{\widetilde{\mathcal{Q}}}_{\lcase{v}}(\lcase{n},\lcase{r})}$ is semisimple.