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Crystal bases for quantum generalized kac–moody algebras

Published online by Cambridge University Press:  25 February 2005

Kyeonghoon Jeong
Affiliation:
Department of Mathematics, NS30, Seoul National University, Seoul 151-747, South Korea. E-mail: [email protected], [email protected]
Seok-Jin Kang
Affiliation:
Department of Mathematics, NS30, Seoul National University, Seoul 151-747, South Korea. E-mail: [email protected], [email protected]
Masaki Kashiwara
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan. E-mail: [email protected]
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Abstract

In this paper, we develop the crystal basis theory for quantum generalized Kac–Moody algebras. For a quantum generalized Kac–Moody algebra $U_q(\mathfrak{g})$, we first introduce the category $\mathcal{O}_{int}$ of $U_q(\mathfrak{g})$-modules and prove its semisimplicity. Next, we define the notion of crystal bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$. We then prove the tensor product rule and the existence theorem for crystal bases. Finally, we construct the global bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$.

Type
Research Article
Copyright
2005 London Mathematical Society

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