On some cocycles which represent the Dixmier-Douady class in simplicial de Rham complexes

Abstract

When a Lie group $G$ has a central $U(1)$-extension, there is a cocycle in the simplicial de Rham complex ${\Omega }^3(NG)$ which represents the Dixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed a central $U(1)$-extension $\widehat{LSU(2)}\to LSU(2)$ whose Dixmier-Douady class in ${\Omega }^3(NLSU(2))$ is a kind of transgression of the second Chern class. In this paper, we consider the case of unitary group and construct a central $U(1)$-extension of $LU(2)$. After that we construct also a cocycle in a certain triple complex.

15 pages

Cited by in corpus (1)

• A central $U(1)$-extension of a double Lie groupoid