Abstract Crystals for Quantum Generalized Kac-moody Algebras

In this paper, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals B(∞) and B(λ). Introduction The purpose of this paper is to develop the theory of abstract crystals for quantum generalized Kac-Moody algebras. In [6], the third author introduced the crystal basis theory for quantum groups associated with symmetrizable Kac-Moody algebras. (In [10], Lusztig constructed canonical bases for quantum groups of ADE type.) It has become one of the most central themes in combinatorial representation theory, for it provides us with a very powerful combinatorial tool to investigate the structure of integrable modules over quantum groups and Kac-Moody algebras. The generalized Kac-Moody algebras were introduced by Borcherds in his study of Monstrous Moonshine [1]. The Monster Lie algebra, an example of generalized Kac-Moody algebras, played a crucial role in his proof of the Moonshine conjecture [2]. In [5], the second author constructed the quantum generalized Kac-Moody algebra Uq(g) as a deformation of the universal enveloping algebra of a generalized Kac-Moody algebra g. He also showed that, for a generic q, the Verma modules and the unitarizable highest weight modules ∗ This research was supported in part by KOSEF Grant # R01-2003-000-10012-0 and KRF Grant # 2005-070-C00004. ⋄ This research was supported by KOSEF Grant # R01-2003-000-10012-0. AMS Classification 2000: 17B37, 81R50. 1 2 K. JEONG, S.-J. KANG, M. KASHIWARA, D.-U. SHIN over g can be deformed to those over Uq(g) in such a way that the dimensions of weight spaces are invariant under the deformation. In [4], the first three authors developed the crystal basis theory for quantum generalized Kac-Moody algebras. More precisely, they defined the notion of crystal bases for Uq(g)-modules in the category Oint (see § 1), proved standard properties of crystal bases including the tensor product rule, and showed that there exists a crystal basis (and a global basis) of the negative part U q (g) of a quantum generalized Kac-Moody algebra and one of the irreducible Uq(g)module V (λ) with a dominant integral weight λ as its highest weight. In this paper, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and investigate their fundamental properties. We then prove the crystal embedding theorem, which yields a procedure to determine the structure of the crystal B(∞) in terms of elementary crystals. Finally, as an application of the crystal embedding theorem, we provide a characterization of the crystals B(∞) and B(λ). We also include an explicit description of the crystals B(∞) and B(λ) for quantum generalized Kac-Moody algebras of rank 2 and for the quantum Monster algebra. 1. Generalized Kac-Moody algebras Let I be a finite or countably infinite index set. A real matrix A = (aij)i,j∈I is called a Borcherds-Cartan matrix if it satisfies the following conditions: (i) aii = 2 or aii ≤ 0 for all i ∈ I, (ii) aij ≤ 0 if i 6= j, (iii) aij ∈ Z if aii = 2, (iv) aij = 0 if and only if aji = 0. In this paper, we assume that A is even and integral ; i.e., aii ∈ 2Z≤1 for all i ∈ I and aij ∈ Z for all i, j ∈ I. Furthermore, we also assume that A is symmetrizable; i.e., there exists a diagonal matrix D = diag(si ∈ Z>0; i ∈ I) such that DA is symmetric. We say that an index i ∈ I is real if aii = 2 and imaginary if aii ≤ 0. We denote by I = {i ∈ I ; aii = 2 } and I im = {i ∈ I ; aii ≤ 0 } the set of real indices and the set of imaginary indices, respectively. A Borcherds-Cartan datum (A,P,Π,Π) consists of ABSTRACT CRYSTALS FOR GENERALIZED KAC-MOODY ALGEBRAS 3CRYSTALS FOR GENERALIZED KAC-MOODY ALGEBRAS 3 (i) a Borcherds-Cartan matrix A = (aij)i,j∈I , (ii) a free abelian group P , the weight lattice, (iii) Π = {αi ∈ P ; i ∈ I }, the set of simple roots, (iv) Π = {hi ; i ∈ I } ⊂ P ∨ := Hom(P,Z), the set of simple coroots, satisfying the properties: (a) 〈hi, αj〉 = aij for all i, j ∈ I, (b) for any i ∈ I, there exists Λi ∈ P such that 〈hj ,Λi〉 = δij for all j ∈ I, (c) Π is linearly independent. We denote by P = {λ ∈ P ; λ(hi) ≥ 0 for all i ∈ I } the set of dominant integral weights. We also use the notation Q = ⊕