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

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