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

Abstract

Abstract. In this paper, we give a polyhedral realization of the highest weight crystals B(λ) associated with the highest weight modules V (λ) for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of ranks 2, 3, and Monster algebras.

The trascendence of Kac-Moody algebras

“…In the last years can be cited: Jeong, Kang and other joint collaborators in 2005, 07, 09 and 14 (see [134][135][136]145], respectively), D.U. Shin in 2007 and 2008 ( [237] and [236], respectively), N.C. Leung and M. Xu [170], C. Lenart and A. Postnikov [169] in 2008 and A. Savage [224], A. Joseph and P. Lamprou [137] and S.J.…”

Crystals and Nakajima monomials for quantum generalized Kac–Moody algebras

“…Note that the connected component of K p containing · · · ⊗ b 2 (0) ⊗ b 1 (0) ⊗ r λ is isomorphic to the crystal B J (λ) [3,13]. Therefore, the connected component of…”

Nakajima monomials, Young walls and Kashiwara embedding for Uq(An(1))

“…Moreover, in [27], for a dominant integral weight λ, Nakashima gave the embedding of crystal Ψ λ ι : B(λ) → Z ∞ ⊗ R λ , and described the explicit form of Im Ψ λ ι . Recently, in [30,31], the second author extended their theory to the quantum generalized KacMoody algebras. That is, he gave the polyhedral realizations of the crystals B(∞) and B(λ) over the quantum generalized Kac-Moody algebras.…”