复正交李超代数osp(1,4)的拟型心与型心

Quasi-Centroids and Centroids of the Complex Orthogonal Lie Superalgebra osp(1,4)

摘要: 李超代数是一种Z₂-李代数，它由物理学家在研究粒子超对称性的统计规律时提出，具有深刻的数学物理意义和内涵。目前，李超代数的相关理论在物理学量子场论、核物理以及超引力等领域中应用广泛。在本文中，首先，根据复正交李超代数的定义得到了osp(1,4)的一组标准基。其次，利用待定系数法确定型心和拟型心的理论表示形式。再次，分别探讨了osp(1,4)在奇变换、偶变换情形下型心和拟型心的具体矩阵表达式。最终得到其在奇变换下的拟型心在标准基上的矩阵为0_14×14，型心在标准基上的矩阵为0_14×14；其在偶变换下的拟型心在标准基上的矩阵为λI_14×14，型心在标准基上的矩阵为λI_14×14。

Abstract: The Lie superalgebra is a kind of Z₂-graded Lie algebra, which was proposed by physicists in the study of the statistical laws of particle supersymmetry and has deep mathematical physical mean-ing and connotation. Recently, the theory related to Lie superalgebra is widely used in the fields of quantum field theory, nuclear physics, and supergravity in physics. In this paper, firstly, a set of standard bases of osp(1,4) is obtained according to the definition of the complex orthogonal Lie superalgebra. Secondly, the theoretical representations of centroid and quasi-centroid are deter-mined using the method of coefficients to be determined. Thirdly, the specific matrix expressions for the centroid and quasi-centroid of osp(1,4) in the odd transformed and even transformed cases are explored separately. The final matrix of its quasi-centroid on the standard basis under odd transformation is obtained as 0_14×14. The matrix of the centroid on the standard basis is 0_14×14. The matrix of the quasi-centroid on the standard basis under the even transformation is λI_14×14. The matrix of the centroid on the standard basis is λI_14×14.