Let G be a connected reductive algebraic group over a field of positive characteristic p and denote by \(\mathcal T\) the category of tilting modules for G. The higher Jones algebras are the endomorphism algebras of objects in the fusion quotient category of \(\mathcal T\). We determine the simple modules and their dimensions for these semisimple algebras as well as their quantized analogues. This provides a general approach for determining various classes of simple modules for many well-studied algebras such as group algebras for symmetric groups, Brauer algebras, Temperley–Lieb algebras, Hecke algebras and BMW-algebras. We treat each of these cases in some detail and give several examples.

中文翻译：

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高等琼斯代数及其简单模块

令G为正特性p的场上的连通还原代数群，并用\（\ mathcal T \）表示G的倾斜模块的类别。较高的Jones代数是\（\ mathcal T \）的融合商类别中的对象的内同构代数。我们为这些半简单代数及其量化的类似物确定简单的模块及其维数。这为确定许多经过研究的代数（例如，对称群的群代数，Brauer代数，Temperley–Lieb代数，Hecke代数和B M W）的简单模块的各种类别提供了一种通用方法。-代数。我们对每种情况进行了详细介绍，并给出了一些示例。