ABSTRACT. We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Saçlioglu, we present the 238 hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.
We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of [Formula: see text] and the [Formula: see text] super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.
Suppose V G is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V . We show that if all irreducible V G -modules contained in V live in some braided tensor category of V G -modules, then they generate a tensor subcategory equivalent to the category Rep G of finitedimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V . Additionally, we show that if the fusion rules for the irreducible V G -modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of V G -modules. These results do not require rigidity on any tensor category of V G -modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when V G is C2-cofinite but not necessarily rational. When V G is both C2-cofinite and rational and V is a vertex operator algebra, we use the equivalence between Rep G and the corresponding subcategory of V G -modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to Rep SU (2), up to modification by an abelian 3-cocycle.
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