Abstract Crystals for Quantum Generalized Kac-Moody Algebras

Abstract

Abstract. 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).

Polyhedral realization of the highest weight crystals for generalized Kac-Moody algebras

“…We recall the definition of abstract crystals introduced in [8] for quantum generalized Kac-Moody algebras. Definition 1.1 ([8]).…”

“…Note that by the definition above, B 1 ⊗ B 2 is a crystal. Moreover, the associativity law for the tensor product holds [8].…”

“…, i 2 , i 1 ) in I, such that every i ∈ I appears infinitely many times, and let Z ∞ ≥0,ι be the crystal introduced in [19]. (Indeed, the crystal Z ∞ ≥0,ι is isomorphic to the crystal B(ι), the (set-theoretical) inductive limit of [8].) By taking the composition of the above crystal embeddings repeatedly, we have Proposition 2.2 ( [8,19]).…”

“…In [9], Kang introduced the quantum generalized Kac-Moody algebras as an analogue of quantum groups, and in [7], Jeong, Kang, and Kashiwara developed the crystal basis theory for quantum generalized Kac-Moody algebras. In [8], Jeong, Kang, Kashiwara and the author introduced the notion of abstract crystals and an analogue Ψ ι of the Kashiwara embedding of crystals for quantum generalized Kac-Moody algebras. Moreover, in [19], the author gave the generalized version of polyhedral realization of the crystal B(∞).…”

“…More precisely, we introduce the crystal R λ for generalized Kac-Moody algebras and show that the connected component containing u ∞ ⊗ r λ is also isomorphic to the highest weight crystal B(λ) as in the Kac-Moody case. Moreover, on the basis of the embedding of crystals given in [8] and the crystal R λ for generalized Kac-Moody algebras, we introduce the crystal structure Z ∞ ≥0 [λ], and we give explicit images of Kashiwara embedding Ψ λ ι . As applications, we give explicit descriptions of the crystals over Kac-Moody algebras of ranks 2 and 3.…”