Path model for representations of generalized Kac–Moody algebras

In [JL], , Joseph and Lamprou (2009) generalized Littelmannʼs path model for Kac–Moody algebras to the case of generalized Kac–Moody algebras. We show that Joseph–Lamprouʼs path model can be embedded into Littelmannʼs path model for a certain Kac–Moody algebra constructed from the Borcherds–Cartan datum of a given generalized Kac–Moody algebra; note that this is not an embedding of crystals. Using this embedding, we give a new proof of the isomorphism theorem for path crystals, obtained in Joseph and Lamprou (2009) [JL, §7.4]. Moreover, for Joseph–Lamprouʼs path crystals, we give a decomposition rule for tensor product and a branching rule for restriction to Levi subalgebras. Also, we obtain a characterization of standard paths in terms of a certain monoid which can be thought of as a generalization of a Weyl group.