Affine Kac-Moody symmetric spaces related with $A_{1}^{(1)}\hbox{,}\, A_{2}^{(1)}\hbox{,}\, A_{2}^{(2)}$A1(1),A2(1),A2(2)

Symmetric spaces associated with Lie algebras and Lie groups which are Riemannian manifolds have recently got a lot of attention in various branches of Physics for their role in classical/quantum integrable systems, transport phenomena, etc. Their infinite dimensional counter parts have recently been discovered which are affine Kac-Moody symmetric spaces. In this paper we have (algebraically) explicitly computed the affine Kac-Moody symmetric spaces associated with affine Kac-Moody algebras \documentclass[12pt]{minimal}\begin{document}$A_{1}^{(1)}, A_{2}^{(1)}, A_{2}^{(2)}$\end{document}A1(1),A2(1),A2(2). We hope these types of spaces will play similar roles as that of symmetric spaces in many physical systems.