The braid group action on quantum queer superalgebra

Abstract

In his seminal work [22], Lusztig introduced a braid group action by automorphisms on the quantum group $U_v\left({\mathrm{sl}}_n\right)$. This result and its generalizations have since become fundamental to the study of quantum algebras. In this paper, we extend the braid group action to the queer superalgebra $U\left(q_n\right)$ and its quantization $U_v\left(q_n\right)$, which promises to be crucial for investigating the structure of these algebras. In particular, we are able to define root vectors for each, together with explicit expressions in terms of standard generators. For $U\left(q_n\right)$, we moreover obtain their super commutation formulas. As a consequence, we construct PBW-type bases for both $U\left(q_n\right)$ and $U_v\left(q_n\right)$ involving products of these root vectors, further strengthening our understanding of their structure.