Annular representation theory for rigid $C^{*}$-tensor categories
Annular representation theory for rigid $C^{*}$-tensor categories
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刚性$C^{*}$-张量类别的环形表示理论
DOI：
10.1016/j.jfa.2015.08.017
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Corey Jones
中科院分区：
文献类型：
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作者：
S. Ghosh;Corey Jones
我们定义了刚性C-张量范畴的环代数，为平面代数的Ocneanu管代数和Jones仿射环范畴提供了一个统一的框架。研究了环代数的表示理论，证明了范畴的所有充分大（满）环代数在张紧后与可数指标集的矩阵单位代数同构，从而具有等价的表示理论.环代数允许一个类似于群的泛C-代数的泛C-代数闭包。这些代数有有趣的角代数，这些角代数由一些对象的同构类集合索引，我们称之为中心化代数。对应于同一对象的中心化代数是规范同构的融合代数的范畴，我们证明了融合代数的Popa和Vaes的容许表示正是任意（非退化）的满环代数的X-表示的限制。这使得近似和刚性属性定义的类别波帕和Vaes被解释的背景下，环形表示理论。这种观点也允许我们定义“更高的权重”的近似性质的基础上，其他中心化代数的环形代数。利用Jones和Reznikoff对环表示的分析，我们确定了δ ≥ 2的TLJ（δ）范畴的所有中心化子代数。
We define annular algebras for rigid C⁎-tensor categories, providing a unified framework for both Ocneanu's tube algebra and Jones' affine annular category of a planar algebra. We study the representation theory of annular algebras, and show that all sufficiently large (full) annular algebras for a category are isomorphic after tensoring with the algebra of matrix units with countable index set, hence have equivalent representation theories. Annular algebras admit a universal C⁎-algebra closure analogous to the universal C⁎-algebra for groups. These algebras have interesting corner algebras indexed by some set of isomorphism classes of objects, which we call centralizer algebras. The centralizer algebra corresponding to the identity object is canonically isomorphic to the fusion algebra of the category, and we show that the admissible representations of the fusion algebra of Popa and Vaes are precisely the restrictions of arbitrary (non-degenerate)⁎-representations of full annular algebras. This allows approximation and rigidity properties defined for categories by Popa and Vaes to be interpreted in the context of annular representation theory. This perspective also allows us to define “higher weight” approximation properties based on other centralizer algebras of an annular algebra. Using the analysis of annular representations due to Jones and Reznikoff, we identify all centralizer algebras for the TLJ (δ) categories for δ≥ 2.