Three kinds of extremal weight vectors fixed by a diagram automorphism
A Dynkin diagram automorphism \(\omega\) induces a Lie algebra automorphism as well as an automorphism of the quantized universal enveloping algebra \(U_q(\mathfrak{g})\). The authors define an action of \(\omega\) on \(\tilde{U}_q(\mathfrak{g})\), the modified quantized universal enveloping algebra, introduced by \textit{G. Lusztig} in [Proc. Natl. Acad. Sci. USA 89, 8177-8179 (1992; Zbl 0760.17011)]. They consider a crystal base \(\mathcal B (\lambda)\) of an extremal weight module \(V(\lambda)\) and the subset \(\mathcal B^0 (\lambda)\) pointwise fixed by \(\omega\). They prove a canonical injection from \(\mathcal B^0 (\lambda)\) into the crystal base of a corresponding extremal module over the quantized universal enveloping algebra of the orbit Lie algebra.
scientific article; zbMATH DE number 1995434
scientific article