基于上三角矩阵李代数的李超代数

Lie superalgebra based on the Lie algebra of upper triangular matrices

利用李超代数理论和线性代数方法从矩阵视角研究基于二阶上三角矩阵李代数L及其自然模的李超代数G的结构。给出了L的自同构和导子的矩阵形式,并说明了李代数L上的局部自同构都是自同构,李代数L上的局部导子都是导子。利用李代数L上的结果刻画了李超代数G的自同构和超导子的矩阵形式,证明了李超代数G的局部自同构都是自同构,局部超导子都是超导子。

Using the theory of Lie superalgebra and linear algebra,the structure of Lie superalgebra G based on Lie algebra L consists of 2×2 upper triangular matrices and its natural modules are studied in matrix version.The matrix forms of automorphisms and derivations of Lie algebra L are given.It is shown that local automorphisms of L are all automorphisms,and local derivations on Lie algebras are all derivations.Using the results of Lie algebra,the matrix form of automorphisms and superderivations of Lie superalgebra G are characterized.It is proved that local automorphisms of Lie superalgebra are all automorphisms,and local superderivations are all superderivations.