Lie Group S-Expansions and Infinite-dimensional Lie algebras

TL;DR

This paper extends the S-expansion method of Lie algebras to act on the group manifold itself, enabling new interpretations of infinite-dimensional algebras like loop algebras and their applications in gauge theories.

Contribution

It generalizes the S-expansion method to the group level and connects it to infinite-dimensional Lie algebras such as loop algebras, with implications for Yang-Mills theories.

Findings

S-expansion applied directly to group manifolds.

Loop algebras interpreted as S-expanded Lie algebras.

Construction of Yang-Mills theory for infinite-dimensional algebra.

Abstract

The expansion method of Lie algebras by a semigroup or S-expansion is generalized to act directly on the group manifold, and not only at the level of its Lie algebra. The consistency of this generalization with the dual formulation of the S-expansion is also verified. This is used to show that the Lie algebras of smooth mappings of some manifold M onto a finite-dimensional Lie algebra, such as the so called loop algebras, can be interpreted as a particular kind of S-expanded Lie algebras. We consider as an example the construction of a Yang-Mills theory for an infinite-dimensional algebra, namely loop algebra.