Decomposition of skew-morphisms of cyclic groups

A skew-morphism of a group ▫$H$▫ is a permutation ▫$sigma$▫ of its elements fixing the identity such that for every ▫$x, y in H$▫ there exists an integer ▫$k$▫ such that ▫$sigma (xy) = sigma (x)sigma k(y)$▫. It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups ▫$mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, varphi (n_{2})) = (varphi (n_{1}), n_{2}) = 1$▫ (▫$varphi$▫ denotes Euler\u27s function) then all skew-morphisms ▫$sigma$▫ of ▫$mathbb Z_n$▫ are obtained as ▫$sigma = sigma_1 times sigma_2$▫, where ▫$sigma_i$▫ are skew-morphisms of ▫$mathbb Z_{n_i},i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$mathbb Z_n$▫ are automorphisms of ▫$mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, varphi(n)) = 1$▫

Dirac structures and Dixmier-Douady bundles

A Dirac structure on a vector bundle V is a maximal isotropic subbundle E of the direct sum of V with its dual. We show how to associate to any Dirac structure a Dixmier-Douady bundle A, that is, a Z/2Z-graded bundle of C*-algebras with typical fiber the compact operators on a Hilbert space. The construction has good functorial properties, relative to Morita morphisms of Dixmier-Douady bundles. As applications, we show that the `spin' Dixmier-Douady bundle over a compact, connected Lie group (as constructed by Atiyah-Segal) is multiplicative, and we obtain a canonical `twisted Spin-c-structure' on spaces with group valued moment maps.Comment: 41 page