扩张Hamilton李超代数的Borel子代数

Borel Subalgebras of Extension Hamiltonian Lie Superalgebras

设为特征零的代数闭域上秩为5的有限维Z-分次Hamilton单李超代数H通过添加次数导子得到的扩张李超代数.本文通过对正则元的分类,证明了关于典范环面共有160个正根系,从而得到160个Borel子代数;通过单根以及连接的定义,确定了每一个正根系的单根系,进而刻画了任意两个Borel子代数的连接关系;最后证明了共有48个Borel子代数是极大可解子代数.本文所得结果可用于进一步研究Cartan型单李超代数的结构与表示.

Let H be the extended Hamiltonian Lie superalgebra obtained by adding the degree derivation to the finite-dimensional Z-graded Hamiltonian simple Lie superalgebra of rank 5 over an algebraically closed field of characteristic zero. By means of classifying all the regular elements, we show that H has exactly 160 positive root systems with respect to the standard torus and thereby obtain 160 Borel subalgebras; by means of the definitions of simple roots and connections, we determine the simple root system for every positive root system and then describe the connection relation between any two Borel subalgebras. Finally, we prove that H has precisely 48 Borel subalgebras which are maximal solvable subalgebras. The main results may be used in the future for studying the structures and representations of simple Lie superalgebras of Cartan type.