Let $G$ be a connected simply-connected complex simple algebraic group with a fixed maximal torus $H$ , a pair of Borel subgroups $B_{\pm}$ such that $B_{+}\cap B_{-}=H$ and a Weyl group $W=\mathrm{Norm}_{G}\left(H\right)/H$ and the maximal unipotent subgroups $N_{\pm}\subset B_{\pm}$. For a Weyl group element $w\in W$, we consider the unipotent cell $N_{-}\cap B_{+}\dot{w}B_{+}$, where $\dot{w}$ is a lift of $w$ in $\mathrm{Norm}_{G}\left(H\right)$. Berenstein, Fomin and Zelevinsky introduced certain automorphism on the unipotent cell, called twist automorphism, for solving “the factorization problems” which describes the inverse of “toric chart” of the associated Schubert varieties.
The quantum unipotent cell is a quantum analogue of the coordinate ring of $N_{-}\cap B_{+}\dot{w}B_{+}$ which was introduced by De Concini and Procesi and they proved an isomorphism between it and a quantum analogue of the coordinate ring of $N_{-}\left(w\right)\cap\dot{w}G_{0}$, where $N_{-}\left(w\right)=N_{-}\cap\dot{w}N_{+}\dot{w}^{-1}$ and $\dot{w}G_{0}$ is the “Gauss” cell associated with $w$.
In this talk, we construct a quantum analogue of the twist automorphism, called a quantum twist automorphism, as a composite of the De Concini-Procesi isomorphism and the twist isomorphism between $N_{-}\cap B_{+}\dot{w}B_{+}$ and $N_{-}\left(w\right)\cap\dot{w}G_{0}$ which is defined by Gaussian decomposition and study its basic properties. In fact, we proved that the the quantum twist automorphism preserves the dual canonical basis of the quantum unipotent cell. This is a joint work with Hironori Oya.