Composition series of affine manifolds and n-gerbes

Affine su(2) fusion rules from gerbe 2-isomorphisms

We give a geometric description of the fusion rules of the affine Lie algebra su(2)_k at a positive integer level k in terms of the k-th power of the basic gerbe over the Lie group SU(2). The gerbe can be trivialised over conjugacy classes corresponding to dominant weights of su(2)_k via a 1-isomorphism. The fusion-rule coefficients are related to the existence of a 2-isomorphism between pullbacks of these 1-isomorphisms to a submanifold of SU(2) x SU(2) determined by the corresponding three conjugacy classes. This construction is motivated by its application in the description of junctions of…

On the Calabi composition of multiple affine hyperspheres

In this paper, we explicitly construct the Calabi composition of multiple affine hyperspheres possibly including some points viewing as 0-dimensional hypersheres. Then we compute all the basic affine invariants of the composed affine hyperspheres, proving that the composed affine hypersphere is symmetric one if and only if each of its composing factors of positive dimension is symmetric.

A new characterization of Calabi composition of hyperbolic affine hyperspheres

In this paper, we mainly prove a theorem with a corollary establishing two characterizations of the Calabi composition of hyperbolic hyperspheres, where the second characterization (i.e., the corollary) has been given via a dual correspondence theorem earlier but now we would like to use a very direct method. Note that Z.J. Hu, H.Z. Li and L. Vrancken also gave a characterization of the $2$-factor Calabi composition in a different manner.

The diffential geometry of composition sequences of differentiable manifolds

Let F_0=B,...,F_n be a sequence of differentiable manifolds, G_i a Lie subgroup of diffeomorphisms of F_i, and H_i a subgroup of G_i central in G_i. We suppose also given a locally trivial bundle p_{K_i} over F_{i-1} which typical fiber is K_i the quotient of G_i by H_i. The aim of this paper is to study the differential geometry of the following problem: classify sequences M_n\to...M_1, where each map from M_i to M_{i-1} is a locally trivial fibration which typical fiber is F_i and which transition functions image are elements of G_i. We associate to this problem a tower of gerbes and define …

Regular Poisson manifolds of compact types (PMCT 2)

This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine orbifolds. We prove that the cohomology class of the leafwise symplectic form varies linearly and that there is a distinguished polynomial function describing the leafwise sympletic volume. The leaf space of a PMCT carries a natural Duistermaat-Heckman measure and a Weyl type integration formula holds. We introduce the notion of a symplectic gerbe, and we sh…

双连接中点矩阵均值及其Gauss复合

Dual-connection midpoint matrix means and their Gauss composition

本文研究黎曼中点何时为Gauss复合均值，给出判据并推广Nakamura迭代至一族矩阵迭代，其Gauss复合仍为几何矩阵均值。