Universal central extensions of Chevalley algebras over Laurent polynomial rings and GIM Lie algebras (Q1068180)

The authors give an explicit description of the universal central extension of the Chevalley algebra \(R\otimes {\mathfrak g}\), where R is the Laurent polynomial ring of n variables over a field F of characteristic zero and \({\mathfrak g}\) is a finite dimensional split Lie algebra over F. Define the n-fold extended Cartan matrix by \(A^{[n]}=(2(\alpha_ i,\alpha_ j)/(\alpha_ i,\alpha_ i))_{1\leq i,j\leq \ell +n}\) where \(\alpha_ i\) (1\(\leq i\leq \ell)\) are the simple roots of \({\mathfrak g}\) and \(\alpha_{\ell +j}=\beta\) (1\(\leq j\leq n)\), the negative highest root of \({\mathfrak g}\). Let \(L(A^{[n]})\) be the GIM Lie algebra generated by \(e_ i,f_ i,h_ i\) \((1\leq i\leq \ell +n)\) with certain defining relations. There is an extension \(\phi\) : L(A\({}^{[n]})\to R\otimes {\mathfrak g}\) defined by \(\phi (e_ i)=E_{\alpha_ i}\), \(\phi (e_{\ell +j})=X_ j\otimes E_{\beta}\), \(\phi (f_ i)=E_{-\alpha_ i}\), \(\phi (f_{\ell +j})=X_ j^{-1}\otimes E_{-\beta}\), \(\phi (h_ i)=H_{\alpha_ i}\), \(\phi (h_{\ell +j})=H_{\beta}\) (1\(\leq i\leq \ell\), \(1\leq j\leq n)\). Put \(J_ 1=Ker \phi\) and \(J=[L(A^{[n]}),J_ 1]\). Then \(L(A^{[n]})/J\) is a universal central extension of \(R\otimes {\mathfrak g}\) (Theorem 2). This result is an n-fold generalization of the case of the standard affine Lie algebra \(L(A^{[1]}).\) All theorems are stated without proofs.