Abstract
Let Uq− be the negative part of the quantum enveloping algebra, and σ the algebra automorphism on Uq− induced from a diagram automorphism. Let U_q− be the quantum algebra obtained from σ, and B˜ (resp. B_˜) the canonical signed basis of Uq− (resp. U_q−). Assume that Uq− is simply-laced of finite or affine type. In our previous papers [10], [11], we have proved by an elementary method, that there exists a natural bijection B˜σ≃B_˜ in the case where σ is admissible. In this paper, we show that such a bijection exists even if σ is not admissible, possibly except some small rank cases.
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