广义相交矩阵李代数及其相关的Kac-Moody代数研究

Generalized intersection matrix algebras (GIM algebras for short) form an important class of Lie algebras which have close relations to singularity theory, representation theory and Kac-Moody algebra theory. Among them GIM algebras determined by multi-fold affinizations of Cartan matrices are extensively studied since they are related to toroidal Lie algebras. We will study a class of GIM algebras and their quantum algebras with structural matrix being a generalized intersection matrix in the sense of P. Slodowy and given by the 2-fold affinization of Cartan matrices of ADE type. Since the covering matrix is an indefinite generalized Cartan matrix such that the GIM algebra can be realized as an involutory subalgebra of the Kac-Moody algebra given by the covering matrix of the GIM via an involution, and another involution related closely to it also gives rise to an indefinite Kac-Moody algebra, the study of the GIM algebras and their quantum algebras would be an interesting expansion of the theory of Kac-Moody algebras.

广义相交矩阵李代数, 简称GIM李代数, 是李理论中的一类重要李代数, 与奇异理论、表示论和Kac-Moody代数理论有密切联系。我们主要研究一类GIM李代数及其量子代数, 其结构矩阵是P. Slodowy意义下的广义相交矩阵, 由ADE型Cartan矩阵A的二重仿射化给出。由于该类广义相交矩阵的覆盖矩阵是不定型的广义Cartan 矩阵，使得它所对应的广义相交矩阵可以实现为它的覆盖矩阵给出的Kac-Moody代数的对合子代数；而且，与该对合映射密切相关的另一个对合映射的不动点子代数也是一个不定型的Kac-Moody 代数，所以研究该类广义相交矩阵李代数及其量子代数是对不定型Kac-Moody代数的研究的重要补充。我们将研究这类GIM李代数的结构理论，包括它的根系结构，虚根的性质，Braid变换下广义相交矩阵及其覆盖矩阵的变化规律，及其对应的李代数之间的关联，并从量子群的角度进行相关研究。