By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between our skein algebra and the one defined by Muller, and use it to show that the quantum trace map can be extended to the Muller skein algebra.

中文翻译：

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绞线代数的三角分解

通过引入 Kauffman 支架绞线代数的更精细版本，我们展示了如何将表面的 Kauffman 支架绞线代数分解为与表面理想三角剖分中的三角形相对应的基本块。理想三角形的新绞线代数有一个简单的表示。这为 Bonahon 和 Wong 的量子轨迹图的存在提供了一个简单的证明。我们还解释了我们的绞线代数与穆勒定义的代数之间的关系，并用它来表明量子迹图可以扩展到穆勒绞线代数。