伪酉矩阵与Radford代数的*-结构

Pseudo-unitary Matrices and the *-structures of Radford Algebra

复数域上8维Radford代数是一个Hopf代数,其*-结构由一个满足??AA =I 的2级复数矩阵A所确定,这样的矩阵称为伪酉矩阵,而且由2个2级伪酉矩阵所确定的*-结构等价的充要条件是这2个伪酉矩阵满足一个等价关系~.研究了2级伪酉矩阵及其关于~的等价分类,证明了任一个2级伪酉矩阵关于~等价于2级单位矩阵,由此得到在*-结构等价的意义下,8维Radford代数有唯一的一个Hopf*-代数结构.

The 8-dimensional Radford algebra over the complex number field is a Hopf algebra whose-structures are determined by complex 2×2-matrices A satisfying A=I.Such matrices are called pseudo-unitary matrices.The two-structures determined by two pseudo-unitary matrices are equivalent if and only if the two pseudo-unitary matrices satisfy an equivalence relation~.In this paper,the pseudo-unitary 2×2-matrices are studied and classified with respect to the equivalence relation~.It is shown that any pseudo-unitary 2×2-matrix is equivalent to the identity matrix with respect to~.Consequently,up to the equivalence of-structures,the 8-dimensional Radford algebra has a unique Hopf-algebra structure.