Current algebras on S^3 of complex Lie algebras

TL;DR

This paper extends the concept of current algebras to the three sphere using quaternion-valued cocycles, introducing new structures and analyzing their root space decompositions in the context of complex Lie algebras.

Contribution

It introduces a quaternion-valued 2-cocycle for current algebras on S^3 and explores their central extensions and root space decompositions.

Findings

Defined a quaternion-valued 2-cocycle on the current algebra.

Constructed the second central extension including a derivation.

Analyzed the root space decomposition of the extended algebra.

Abstract

This is a full revised version of the previous same titled article. The 2-cocycle of the central extension of the current algebra on the three sphere is taken the place of a new quarternion valued one, that is defined by the boundary Dirac operator. And we introduced the symmetric invariant bilinear form associated to the centrally extended current algebra. We shall extend the affine Kac-Moody algebra on the loop to a Lie algebra of smooth mappings of the three sphere into a complex simple Lie algebra. Let L be the C-algebra generated by the Laurent polynomial type harmonic spinors over the complex plane deleted the origin. Here "harmonic" means the zero-mode of the Dirac operator. For a simple complex Lie algebra g, the g-current algebra is defined as the real Lie algebra Lg that is generated by the tensor product of L and g. The real part K of L is a commutative real subalgebra. For the Cartan subalgebra h of g, the tensor product of K and h gives a Cartan subalgebra Kh of Lg. We have the weight space decomposition of Lg with respect to the Cartan subalgebra Kh. We introduce a quarternion valued 2-cocycle on the g-current algebra Lg. Then we have the central extension associated to the 2-cocycle. Adjoining a derivation coming from the radial vector field on the three sphere we obtain the second central extension. We shall investigate the root space decomposition of the centrally extended current algebra.