The Cluster Structures of Simple Complex Lie Group <i>so</i> (4, ℂ) and Drinfeld Double Of <i>so</i> (4, ℂ)

For the cluster algebras of a simple complex Lie group, Gekhtman, Shapiro and Vainshtein posed a conjecture: for a Belavin-Drinfeld triple of a simple complex Lie group [Formula: see text], there is a classification of regular cluster structures on [Formula: see text] that is completely parallel to the Belavin-Drinfeld classification of solutions of the classical Yang-Baxter equation. To prove this conjecture is true for the Lie groups of Type D, in this paper, we’ll start with the Lie group [Formula: see text](4;[Formula: see text]), and construct the generalized cluster structures of the Lie group [Formula: see text](4;[Formula: see text]) and Drinfeld double of [Formula: see text](4;[Formula: see text]).