A Borel-Tits theory was developped for almost split forms of symmetrizable Kac-Moody Lie algebras [J. of Algebra 171, 43-96 (1995)]. In this paper, we look to almost split real forms for symmetrizable hyperbolic KacMoody Lie algebras and we establish a complete list of these forms, in terms of their Satake-Tits index, for the strictly hyperbolic ones and for those which are obtained as (hyperbo...

Kac and Kazhdan conjectured a character formula of a simple module with a generic highest weight at the critical level. We prove this formula for the affine Lie superalgebras with symmetrizable Cartan matrices.

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate meth...

Let → Γ be a valued quiver. Let C be the symmetrizable generalized Cartan matrix associated to → Γ. We show that the whole quantum group associated to C can be realized from the category of the representations of the product valued quiver → Γ. This method can be used to realize the whole generalized Kac-Moody Lie algebra associated to C, as discussed in [LL].

We consider the class of viscous-dispersive and higherorder conservation laws. We generalize the work of Kawashima & Shizuta, and others, by extending to higher-order the notions of symmetrizability, strict dissipativity, and genuine coupling. We prove, for symmetrizable systems, that strict dissipativity is equivalent to both (i) genuine coupling and (ii) the existence of a skewsymmetric compe...