A Littelmann path model for crystals of Generalized Kac-Moody algebras revisited

TL;DR

This paper develops a Littelmann path model for crystals of generalized Kac-Moody algebras, addressing combinatorial challenges from imaginary roots, and establishes an isomorphism and character formula linking to existing models.

Contribution

It introduces a new path model for Borcherds-Kac-Moody crystals, overcoming combinatorial issues from imaginary roots and connecting to known crystal structures.

Findings

Constructed a Littelmann path model for Borcherds-Kac-Moody crystals.

Proved an isomorphism theorem relating the new model to existing crystal models.

Derived a character formula of Borcherds-Kac-Weyl type for these crystals.

Abstract

A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an isomorphism theorem and a character formula of Borcherds-Kac-Weyl type for the crystals. In the symmetrizable case, the isomorphism theorem implies that the crystals constructed by this path model coincide with those of Jeong, Kang, Kashiwara and Shin obtained by taking the limit at q=0 in the quantized enveloping algebra.