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When a Lie group $G$ has a central $U(1)$-extension, there is a cocycle inthe simplicial de Rham complex $\Omega^3(NG)$ which represents theDixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed acentral $U(1)$-extension $\widehat{LSU(2)} \rightarrow LSU(2)$ whoseDixmier-Douady class in $\Omega^3(NLSU(2))$ is a kind of transgression of thesecond 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 acertain triple complex.

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