Let $U_q(q_n)$ be the quantum enveloping superalgebra of the queer Lie superalgebra $q_n$. We will talk about the Howe duality between $U_{q^{-1}}(q_m)$ and $U_q(q_n)$. Namely, there is a non-commutative analogy of the symmetric superalgebra $S(C^{mn|mn})$ that is a $U_{q^{-1}}(q_m)\otimes U_q(q_n)$-supermodule admitting a multiplicity free decomposition. Such a $U_{q^{-1}}(q_m)\otimes U_q(q_n)$-supermodule was created with the use of the quantum coordinate superalgebra of type $Q$. We will also show that the $(U_{q^{-1}}(q_m), U_q(q_n))$-Howe duality implies Sergeev-Olshanski duality. This talk is based on a joint-work with Yongjie Wang.

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