Crystal Bases for Quantum Generalized Kac-Moody Algebras

Jeong, Kyeonghoon; Kang, Seok-Jin; Kashiwara, Masaki

Crystal Bases for Quantum Generalized Kac-Moody Algebras

Authors

Jeong, Kyeonghoon

Kang, Seok-Jin

Kashiwara, Masaki

Description

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 $U_q^-(\mathfrak g)$.
60 pages, no figures