Let A be a symmetrizable generalized Cartan matrix. Let g be the corresponding Kac-Moody algebra. Let G(R) be Tits’ Kac-Moody group functor over commutative rings R associated to g. Then G(R) is given by five axioms KMG1-KMG5 which are natural extensions of the properties of Chevalley-Demazure group schemes. We give a construction of the Tits functor for symmetrizable Kac-Moody groups G using i...

We introduce real Lösungen as an analogue of roots. For each mutation sequence arbitrary skew-symmetrizable matrix, we define a family reflections along with associated vectors which are and set curves on Riemann surface. The matrix consisting these is called L-matrix. explain how the L-matrix naturally arises in connection C-matrix. Then conjecture that depends (up to signs row vectors) only s...

We construct a family of two parameter quantum grou-\ps‎ ‎$U_{r,s}(mathfrak{g})$ associated with a generalized Kac-Moody‎ ‎algebra corresponding to symmetrizable admissible Borcherds Cartan‎ ‎matrix‎. ‎We also construct the $textbf{A}$-form $U_{textbf{A}}$ and‎ ‎the classical limit of $U_{r,s}(mathfrak{g})$‎. ‎Furthermore‎, ‎we‎ ‎display the equitable presentation for a subalgebra‎ ‎$U_{r...