标题： 秩2的Racah代数：性质、对称性和表示 显示英文标题

标题： The Racah Algebra of Rank 2: Properties, Symmetries and Representation

摘要： 本文的目标有三个。 首先，我们提供了一个新的“通用”定义，用于秩2的Racah代数，作为秩1 Racah代数的扩展，其中生成元由子集索引，任何三个不相交的索引集合定义一个与秩1情况同构的子代数。 通过这个定义，我们探讨了代数的一些性质，包括验证这些自然假设与其他文献中的定义关系等价。 其次，我们研究了秩2 Racah代数生成元的对称性。 这些对称性使我们能够部分抽象出生成元的选择，并以不同的格式写出关系和性质。 最后，我们提供了Racah代数的一个新表示。 这种新的表示只需要一个生成元是对角的，并且基于秩1 Racah代数的分裂基表示的展开。 显示英文摘要

摘要： The goals of this paper are threefold. First, we provide a new ''universal'' definition for the Racah algebra of rank 2 as an extension of the rank-1 Racah algebra where the generators are indexed by subsets and any three disjoint indexing sets define a subalgebra isomorphic to the rank-1 case. With this definition, we explore some of the properties of the algebra including verifying that these natural assumptions are equivalent to other defining relations in the literature. Second, we look at the symmetries of the generators of the rank-2 Racah algebra. Those symmetries allows us to partially make abstraction of the choice of the generators and write relations and properties in a different format. Last, we provide a novel representation of the Racah algebra. This new representation requires only one generator to be diagonal and is based on an expansion of the split basis representation from the rank-1 Racah algebra.