Characters of Hopf algebras
Characters of Hopf algebras
复制标题
DOI：
10.1016/0021-8693(71)90018-4
复制
发表时间：
1971-03
影响因子：
0.9
通讯作者：
R. Larson
中科院分区：
文献类型：
--
作者：
R. Larson
在有限群的特征标理论的发展过程中，证明群的正交关系的一个关键事实是cg E kG的不变性。(See例如，引理5.1的证明。[2]中的3个和等式。（31.8）in [l].）由于有限维Hopf代数[5]和某些无限维Hopf代数[7]的对偶代数包含具有类似于Cg的性质的元素，因此可以合理地期望正交关系对于此类Hopf代数的特征标成立。在本文的第二节中，我们证明了这样的正交关系。为了包括一个字符理论的无限维Hopf代数，我们必须考虑字符作为元素的Hopf代数是与余模的Hopf代数，而不是作为泛函的Hopf代数是与模块的Hopf代数。这种观点允许同时处理紧李群和完全可约仿射代数群的特征（把代表函数的代数作为Hopf代数），讨论了特征为0的代数闭域上半单李代数的特征（以U（L）”为Hopf代数;这里研究的字符与[6]的Expose 18的字符相差标量倍数），以及有限群的特征（取群代数的对偶Hopf代数作为Hopf代数）然后利用我们的结果证明了代数闭域上对合余半单Hopf代数的单余模的维数为不能被场的特性所分割。第三节证明了余半单Hopf代数的对极y是双射的，并且y ~ 2将每个单子余代数映射到自身上。第四节给出Maschke定理的一个推广：如果域的特征不整除有限维对合Hopf代数的维数，则该Hopf代数及其对偶是半单的;如果域的特征整除有限维对合Hopf代数的维数，则该对合Hopf代数是半单的。
In the development of the character theory of finite groups, one of the key facts used in proving the orthogonality relations is the invariance property of cg E kG.(See, eg, the proofs of Lemma 5.1. 3 in [2] and of Eq.(31.8) in [l].) Since finite-dimensional Hopf algebras [5] and the dual algebras of certain infinite-dimensional Hopf algebras [7] contain elements with properties analogous to those of Cg, it is reasonable to expect that an orthogonality relation will hold for characters of such Hopf algebras. In Section 2 of this paper we prove such an orthogonality relation. In order to include a character theory for infinite dimensional Hopf algebras, we must consider characters as elements of the Hopf algebra which are associated with comodules over the Hopf algebra, rather than as functionals on the Hopf algebra which are associated with modules over the Hopf algebra. This point of view allows a simultaneous treatment of the characters of compact Lie groups and completely reducible affine algebraic groups (taking as the Hopf algebra the algebra of representative functions), of the characters of semisimple Lie algebras over an algebraically closed field of characteristic 0 (taking U (L)” as the Hopf algebra; the characters studied here differ from those of Expose 18 of [6] by a scalar multiple), and of the characters of finite groups (taking as the Hopf algebra the dual Hopf algebra to the group algebra).We then use our results to prove that the dimension of a simple comodule of an involutory cosemisimple Hopf algebra over an algebraically closed field is not divisible by the characteristic of the field. In Section 3 we prove that the antipode y of a cosemisimple Hopf algebra is bijective, and that y2 maps each simple subcoalgebra onto itself. In Section 4 we give a generalization of Maschke’s Theorem: If the characteristic of the field does not divide the dimension of a finite dimensional involutory Hopf algebra, then the Hopf algebra and its dual are semisimple; if the characteristic does divide the