伪酉矩阵与Radford代数的*-结构
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摘要
复数域上8维Radford代数是一个Hopf代数,其*-结构由一个满足■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 ■. 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.
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 ■. 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.
引文
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[2] Woronowicz S L.Twisted SU(2) group.An example of non-commutative differential calculus [J].Publ Res Inst Math Sci,1987,23:117-181.
[3] Woronowicz S L.Tannaka-Krein duality for compact matrix pseudogroups.Twisted SU(N) groups [J].Invent Math,1988,93:35-76.
[4] Kassel C.Quantum Groups [M].New York:Springer-Verlag,1995.
[5] Masuda T,Mimachi K,Nakagami Y,Noumi M,Saburi Y,Ueno K.Unitary representations of the quantum SUq(1,1):Structure of the dual space of Uq(sl(2)) [J].Lett Math Phys,1990,19:187-194.
[6] Mohammed H S E,Li T,Chen H X.Hopf *-algebra structures on H(1,q) [J].Front Math China,2015,10:1415-1432.
[7] Mohammed H S E,Chen H X.The structures of Hopf *-algebra on Radford algebras.arXiv:1903.02254v1[math.RA].
[8] Sweedler M E.Hopf Algebras [M].New York:Benjamin,1969.
[9] Montgomery S.Hopf Algebras and Their Actions on Rings [M].CBMS Regional Conf Ser Math,Amer Math Soc,Providence,RI,1993.