Group actions on C*-algebras, 3-cocycles and quantum field theory

We study group extensions , where acts on a C^*-algebraA. Given a twisted covariant representation ,V of the pairA, we construct 3-cocycles on with values in the centre of the group generated byV(). These 3-cocycles are obstructions to the existence of an extension of byV() which acts onA compatibly with . The main theorems of the paper introduce a subsidiary invariant which classifies actions of onV() and in terms of which a necessary and sufficient condition for the the cohomology class of the 3-cocycle to be non-trivial may be formulated. Examples are provided which show how non-trivial 3-cocycles may be realised. The framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories. We show how to interpret our results in that setting in two ways, one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups. In order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a Lie algebra version of the invariant and the 3-cocycle.     

A stereoselective 3-endo-alkyl-5,6-dehydroisoquinuclidine synthesis.

Cycloaddition of N-carbethoxy-2-alkyl-1,2-dihydropyridines $\text{4}$ with phenylvinylsulfone $\text{5}$ provides adducts $\text{6}$, which upon desulfonylation afford stereoselectively N-carbethoxy-3-$\text{endo}$- alkyl-5,6-dehydroisoquinuclidines $\text{1}$.