Groups Defined by Extended Affine Lie Algebras with Nullity $2$

Abstract

After certain completions, we define adjoint groups of extended affine Lie algebras with nullity $2$. Then we show that such groups have Tits systems with affine Weyl groups (Part I). This idea allows us to consider linear groups over some completed quantum tori. By the same argument, we can prove that these linear groups also have Tits systems with affine Weyl groups. Using this fact we will study their universal central extensions as well as associated $K_1$-groups and $K_2$-groups (Part II). We will discuss some relationship among our groups constructed here (Part III).