关于与ADE图的路和本质路相关的双代数
On bialgebras associated with paths and essential paths on ADE graphs
摘要：
本文定义了图$G$（树）的基本路向量空间上的分次乘法，并证明了它是结合的。在大多数有趣的应用中，这棵树是一个ADE Dynkin图。本质路的保长自同态的向量空间具有由路的长度得到的分次，并具有几个有趣的双代数结构。其中之一是A. Ocneanu，是一种特殊的量子广群（弱Hopf代数），在其他地方也有研究;它的余积给出了对偶向量空间上的过滤卷积积。另一个双代数结构是用一个分次结合积代替这个过滤卷积积得到的，它可以从前者通过在最大等级子空间上的投影得到，但有趣的是直接定义它，而不使用DTA。所得到的是一个弱双代数，而不是一个弱Hopf代数。
英文摘要：
We define a graded multiplication on the vector space of essential paths on a graph $G$ (a tree) and show that it is associative. In most interesting applications, this tree is an ADE Dynkin diagram. The vector space of length preserving endomorphisms of essential paths has a grading obtained from the length of paths and possesses several interesting bialgebra structures. One of these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct gives a filtrated convolution product on the dual vector space. Another bialgebra structure is obtained by replacing this filtered convolution product by a graded associative product.It can be obtained from the former by projection on a subspace of maximal grade, but it is interesting to define it directly, without using the DTA. What is obtained is a weak bialgebra, not a weak Hopf algebra.
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