Loop groups and related affine Lie algebras (Q916769)

Let \(G\) be a connected, simply connected complex simple Lie group, \({\mathfrak g}\) its Lie algebra. The Lie algebra \(\tilde{\mathfrak g}\) of algebraic loops in the Lie algebra \({\mathfrak g}\) of \(G\) has a universal central extension \(\hat{\mathfrak g}\) called an affine Lie algebra. In this paper, the Lie group \(\tilde G_k\) of \(C^k\)-loops in \(G\) and the Lie algebras \(\tilde{\mathfrak g}_k\), \(\hat{\mathfrak g}_k\) are discussed. A completed version of some facts on \(\tilde G\) and \(\tilde{\mathfrak g}\) for \(\tilde G_k\) and \(\hat{\mathfrak g}_k\) is constructed.