Weyl groups of the extended affine root system \(A_1^{(1,1)}\) and the extended affine \({\mathfrak {sl}}(2)\) (Q1281853)

The author studies the Weyl group of the extended affine root system \(A_1^{(1,1)}\), in particular, a presentation for this group is obtained. Further, the author defines an extended affine Lie algebra, \(\widehat{gl}_n\), proves that it carries a non-degenerate, symmetric, invariant bilinear form and derives the associated extended affine Lie algebra \(\widehat{sl}_n\). In the case \(n=2\) the author studies the corresponding Weyl group and relates it to the Weyl group of \(A_1^{(1,1)}\).