Quantum queer superalgebra and its integral form

Abstract

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