A Comment on the calculation of the Lie algebra cocycle in the book Loop Groups

TL;DR

This paper critiques the derivation of the Lie algebra cocycle in 'Loop Groups', highlighting ambiguities in the trace operation and emphasizing the need for proper definition to ensure correct evaluation.

Contribution

It clarifies the ambiguities in the trace operation within the cocycle expression, proposing a more precise approach for its evaluation.

Findings

Identifies ambiguities in the existing cocycle expression

Highlights the importance of proper trace definition

Suggests corrections for accurate computation

Abstract

An expression for the Lie algebra cocycle corresponding to the central extension of the group GLres(H) is derived in the book Loop Groups [1]. It will be shown in this paper that some of the terms in this expression are ambiguous and in order to be evaluated correctly the trace operation must be properly defined.