A research on algebraic groups and Kac-Moody groups, and their applications

(1)引入了Uq（sl_2）的一个新公式，并给出了其表示可积的一个充分条件. (2)给出了广义仿射李代数的Kac-猜想的一个证明。(3)我们引入了与给定的一维镶嵌相关联的双代数，并利用我们的双代数给出了这种镶嵌是局部不可区分的一个特征。(4)构造了零值为2的完备扩张仿射李代数的伴随群的泛中心扩张。(5)我们证明了由一维平铺定义的群和李代数具有标准高斯分解和加性高斯分解。(6)我们证明了域上UFD代数的s1_2的许多共轭定理，特别是Cartan子代数和TDS的共轭定理，作为Jacobson-Morozov定理的类似定理。

(1) We introduced a new formulas for U q(sl_2), and gave a sufficient condition for its representation to be integrable.(2) We gave a proof of the so-called Kac-conjecture for extended affine Lie algebras.(3) We introduced certain bialgebras associated with given one-dimensional tilings, and gave a characterization for such tilings to be locally indistinguishable using our bialgebras.(4) We constructed universal central extensions of certain adjoint groups associated with completed extended affine Lie algebras with nullity 2.(5) We showed the fact that the groups and Lie algebras defined by one-dimensional tilings have standard and additive Gauss decompositions.(6) We proved many conjugacy theorems for sl_2 of UFD algebras over fields, especially for Cartan subalgebras and for TDS as some analogue of Jacobson-Morozov theorem.