## Advanced Studies in Pure Mathematics

### Crystal Bases and Diagram Automorphisms

#### Abstract

We prove that the action of an $\omega$-root operator on the set of all paths fixed by a diagram automorphism $\omega$ of a Kac–Moody algebra $\mathfrak{g}$ can be identified with the action of a root operator for the orbit Lie algebra $\breve{\mathfrak{g}}$. Moreover, we prove that there exists a canonical bijection between the elements of the crystal base $\mathcal{B}(\infty)$ for $\mathfrak{g}$ fixed by $\omega$ and the elements of the crystal base $\breve{\mathcal{B}}(\infty)$ for $\breve{\mathfrak{g}}$. Using this result, we give twining character formulas for the "negative part" of the quantized universal enveloping algebra $U_q(\mathfrak{g})$ and for certain modules of Demazure type.

#### Article information

Dates
First available in Project Euclid: 3 January 2019

https://projecteuclid.org/ euclid.aspm/1546542391

Digital Object Identifier
doi:10.2969/aspm/04010321

Mathematical Reviews number (MathSciNet)
MR2074598

Zentralblatt MATH identifier
1090.17013

#### Citation

Naito, Satoshi; Sagaki, Daisuke. Crystal Bases and Diagram Automorphisms. Representation Theory of Algebraic Groups and Quantum Groups, 321--341, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04010321. https://projecteuclid.org/euclid.aspm/1546542391