Crystal bases for quantum generalized kac¿moody algebras

• Autores: Kyeonghoon Jeong, Seok-Jin Kang, Masaki Kashiwara
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 90, Nº 2, 2005, págs. 395-438
• Idioma: inglés
• DOI: 10.1112/s0024611504015023
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• Resumen

  • 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 $U_q^-(\mathfrak{g})$.