Central elements and evaluation map for the quantum queer superalgebras

We consider the $R$-matrix presentations of the quantum queer superalgebra $U_q(q_n)$ and its affine counterpart $U_q(\widehat q_n)$. We derive crossing symmetry relations for the $R$-matrices and use them to construct central elements in both superalgebras. We also produce an epimorphism $ev:U_q(\widehat q_n)\to U_q(q_n)$ identical on the subalgebra isomorphic to $U_q(q_n)$.