2021/08/15 by Ma, Ying, Shoji, Toshiaki, Zhou, Zhiping
#17B37 #81R50 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2108.06673
Let \mathbf U-q be the negative part of the quantized enveloping algebra associated to a Kac-Moody algebra \mathfrak g of symmetric type, and \underline\mathbf U-q the algebra corresponding to the orbit algebra \mathfrak gσ obtained from an admissible diagram automorphism σ on \mathfrak g. Lusztig consructed the canonical basis \mathbf B of \mathbf Uq- and the canonical signed basis \underline\widetilde\mathbf B of \underline\mathbf Uq- by making use of the geometric theory of quivers. He proved that there is a natural bijection \widetilde\mathbf Bσ → \widetilde\underline\mathbf B. In this paper, assuming the existence of the canonical basis \mathbf B of \mathbf Uq-, we construct the canonical signed basis \widetilde\underline\mathbf B of \underline\mathbf Uq-, and a natural bijection \widetilde\mathbf Bσ → \widetilde\underline\mathbf B by an elementary method.