We describe the generalized Casimir operators and their actions on the positive representations Pλ of the modular double of split real quantum groups Uqq (gR). We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters λ. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of q, and we discuss explicit examples in the lower rank cases.

Let Sg be a closed surface of genus g and Mg be the moduli space of Sg endowed with the Weil–Petersson metric. In this article we investigate the Weil–Petersson curvatures of Mg for large genus g. First, we study the asymptotic behavior of the extremal Weil–Petersson holomorphic sectional curvatures at certain thick surfaces in Mg as g → ∞. Then we prove two curvature properties on the whole space Mg as g → ∞ in a probabilistic way.

This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.

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We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordstr\"om manifolds.