Weak Hopf Algebras: II. Representation Theory, Dimensions, and the Markov Trace
Weak Hopf Algebras: II. Representation Theory, Dimensions, and the Markov Trace
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DOI：
10.1006/jabr.2000.8379
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发表时间：
1999-06
影响因子：
0.9
通讯作者：
G. Bòhm;K. Szlachányi
中科院分区：
文献类型：
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作者：
G. Bòhm;K. Szlachányi
摘要 如果 A 是弱 C *-Hopf 代数，则 A 的有限维酉表示范畴是幺半群 C * 范畴，其幺半群单位是与单位 e 关联的 GNS 表示 De e。该范畴具有同构的左对偶和右对偶对象，这通常导致维度函数的概念。然而，如果 e 不是纯的，则维度函数是矩阵值，行和列由 De 中包含的不可约标记。当包含 A L ⊂ A 和 A R ⊂ A 不相连时，就会发生这种情况。尽管如此，A 上仍然存在一条迹线，它是两个夹杂物的马尔可夫迹线。我们为平凡超中心的每个 C *-WHA 推导出两个数值不变量。这些是 Haar 的共同指数 I 和 δ，分别是马尔可夫、包含 A L/R ⊂ A 或 A L/R ⊂ A 之一的条件期望。在一般情况下，I > δ。在弱 Kac 代数的特殊情况下，我们通过证明 I = δ 并且始终是整数来重现 D. Nikshych 的结果（2000 年，J. 算子理论，即将出现）。
Abstract If A is a weak C *-Hopf algebra then the category of finite-dimensional unitary representations of A is a monoidal C *-category with its monoidal unit being the GNS representation D e associated to the counit e. This category has isomorphic left dual and right dual objects, which leads, as usual, to the notion of a dimension function. However, if e is not pure the dimension function is matrix valued with rows and columns labeled by the irreducibles contained in D e . This happens precisely when the inclusions A L ⊂ A and A R ⊂ A are not connected. Still, there exists a trace on A which is the Markov trace for both inclusions. We derive two numerical invariants for each C *-WHA of trivial hypercenter. These are the common indices I and δ, of the Haar, respectively Markov, conditional expectations of either one of the inclusions A L/R ⊂ A or A L/R ⊂ A. In generic cases I > δ. In the special case of weak Kac algebras we reproduce D. Nikshych's result (2000, J. Operator Theory, to appear) by showing that I = δ and is always an integer.