Let ${ \mathbf U}_q^-$ be the negative part of the quantum enveloping algebra, and $\sigma$ the algebra automorphism on ${ \mathbf U}_q^-$ induced from a diagram automorphism. Let $\underline{ \mathbf U}_q^-$ be the quantum algebra obtained from $\sigma$, and $\widetilde{ \mathbf B}$ (resp. $\widetilde{ \underline{ \mathbf B}}$) the canonical signed basis of ${ \mathbf U}_q^-$ (resp. $\underline{ \mathbf U}_q^-$). Assume that ${ \mathbf U}_q^-$ is simply-laced of finite or affine type. In our previous papers [SZ1, 2], we have proved by an elementary method, that there exists a natural bijection $\widetilde{ \mathbf B}^{ \sigma} \simeq \widetilde{ \underline{ \mathbf B}}$ in the case where $\sigma$ is admissible. In this paper, we show that such a bijection exists even if $\sigma$ is not admissible, possibly except some small rank cases.
Categories: math.QA