Advanced Studies in Pure Mathematics

Crystal Bases and Diagram Automorphisms

Satoshi Naito and Daisuke Sagaki

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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

Source
Representation Theory of Algebraic Groups and Quantum Groups, T. Shoji, M. Kashiwara, N. Kawanaka, G. Lusztig and K. Shinoda, eds. (Tokyo: Mathematical Society of Japan, 2004), 321-341

Dates
Received: 2 March 2002
First available in Project Euclid: 3 January 2019

Permanent link to this document
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


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