The Strongly Symmetric Elements and Solutions of Yang-baxter Equation

It is shown that all strongly symmetric elements are solutions of constant classical Yang-Baxter equation in Lie algebra, or of quantum Yang-Baxter equation in algebra. Otherwise, all solutions of constant classical Yang-Baxter equation (CYBE) in Lie algebra L with dim L ≤ 3 over field k of characteristic 2 are obtained 0 Introduction The Yang-Baxter equation first came up in a paper by Yang as factorition condition of the scattering S-matrix in the many-body problem in one dimension and in work of Baxter on exactly solvable models in statistical mechanics. It has been playing an important role in mathematics and physics ( see [1] , [9] ). Attempts to find solutions of The Yang-Baxter equation in a systematic way have led to the theory of quantum groups. The Yang-Baxter equation is of many forms. The classical Yang-Baxter equation and quantum Yang-Baxter equation are two kinds of them. In many applications one need to know the solutions of the two equations. The author in [10] systematically studied the solutions of low dimensional Lie algebras, found strongly symmetric elements and shew that all of them are solutions of CYBE. Naturally, we would like to ask the following questions: (1) Is every strongly element in L ⊗ L a solution of CYBE for any dimensional Lie algebra L? (2) Do the conclusions in (BI2494R) hold for field of characteristic 2 ? ∗This work is supported by National Science Foundation