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Extensions of current groups on \(S^3\) and the adjoint representations. (English) Zbl 1298.81107

Summary: Let \(\Omega^3(\mathrm{SU}(n))\) be the Lie group of based mappings from \(S^3\) to \(\mathrm{SU}(n)\). We construct a Lie group extension of \(\Omega^3(\mathrm{SU}(n))\) for \(n\geq 3\) by the abelian group \(\exp 2\pi i {\mathcal A}_3^{\ast}\), where \(\mathcal A_3^{\ast}\) is the affine dual of the space of \(\mathrm{SU}(n)\)-connections on \(S^3\). J. Mickelsson [Commun. Math. Phys. 110, 173–183 (1987; Zbl 0625.58043)] constructed a similar Lie group extension. In this article we give several improvement of his results, especially we give a precise description of the extension of those components that are not the identity component. We also correct several argument about the extension of \(\Omega^3(\mathrm{SU}(2))\) which seems not to be exact in Mickelsson’s work, though his observation about the fact that the extension of \(\Omega^3(\mathrm{SU}(2))\) reduces to the extension by \(Z_2\) is correct. Then we shall investigate the adjoint representation of the Lie group extension of \(\Omega^3(\mathrm{SU}(n))\) for \(n\geq 3\).

MSC:

81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
22E67 Loop groups and related constructions, group-theoretic treatment
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
81T13 Yang-Mills and other gauge theories in quantum field theory
22E70 Applications of Lie groups to the sciences; explicit representations

Citations:

Zbl 0625.58043
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References:

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