Jones and Penneys showed that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of its principal graph, via a Markov towers of algebras approach. We relate several equivalent perspectives on the notion of module over a subfactor planar algebra, and show that a Markov tower is equivalent to a module over the Temperley-Lieb-Jones planar algebra. As a corollary, we obtain a classification of semisimple pivotal ${\mathrm{C}}^{⁎}$ modules over Temperley-Lieb-Jones in terms of pointed graphs with a Frobenius-Perron vertex weighting. We then generalize the Markov towers of algebras approach to show that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of the fusion graph of any of its cyclic modules.

Jones和Penneys表明，通过马尔可夫塔代数方法，有限深度子因子平面代数嵌入其主图的二部图平面代数中。我们关联了关于子因子平面代数上的模块概念的几个等效观点，并证明了马尔可夫塔等同于Temperley-Lieb-Jones平面代数上的模块。作为推论，我们获得了半简单枢轴的分类${\mathrm{C}}^{⁎}$Temperley-Lieb-Jones上的模块具有Frobenius-Perron顶点权重的有向图。然后，我们对代数的马尔可夫塔进行泛化，以证明有限深度的子因子平面代数嵌入其任何循环模块的融合图的二分图平面代数中。