Quantum queer superalgebra and its integral form

Abstract

In this paper, we introduce quantum root vectors for the quantum queer superalgebra ${\boldsymbol U}_{\!{v}}({\mathfrak q_n})$ via a braid-group action, compute their complete commutation relations, and construct a PBW-type basis for the Lusztig integral form ${{\boldsymbol U}_{v,\mathcal{ Z}}}$. This yields an explicit presentation of ${{\boldsymbol U}_{v,\mathcal{ Z}}}$ and provides a way to understand the structure of the quantum queer superalgebra at roots of unity.