Crystal Bases for Quantum Generalized Kac-Moody Algebras

Details Crystal Bases for Quantum Generalized Kac-Moody Algebras Crystal Bases for Quantum Generalized Kac-Moody Algebras

2003-05-28

arxiv.org/abs/math/0305390v1

Mathematics

Quantum Algebra

60 pages, no figures

Scientific paper

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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