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Crystal bases for quantum generalized kac–moody algebras

  • Kyeonghoon Jeong (a1), Seok-Jin Kang (a1) and Masaki Kashiwara (a2)

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 $U_q^-(\mathfrak{g})$.

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Crystal bases for quantum generalized kac–moody algebras

  • Kyeonghoon Jeong (a1), Seok-Jin Kang (a1) and Masaki Kashiwara (a2)

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