Classical R-Matrix and Semi-Simple Lie Algebras

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Let g be a Lie algebra. A linear operator R ε End(g) is called a r-matrix (see [5]) if the bracket given by $${\left[ {X,Y} \right]_R} = \left[ {RX,Y} \right] + \left[ {X,RY} \right]$$ (1) is also a Lie bracket on g. Such a pair (g,R) is called a double Lie algebra. Moreover, if there is a nondegenerate invariant bilinear form on g and R is skew-symmetric (g,R) becomes a Lie bialgebra ([1], [5]). It is known that (g.R) is a double Lie algebra iff the following bilinear map BR:gxg→g given by $${B_R}(X,Y) = \left[ {RX,RY} \right] - R({\left[ {X,Y} \right]_R})$$ (2) is ad-invariant, i.e., the equation $$\left[ {X,{B_R}(X,Y)} \right] + \left[ {Y,{B_R}(Z,X)} \right] + \left[ {Z,{B_R}(X,Y)} \right] = 0$$ (3) holds for all X,Y,Z ∈ g. Particularly, the equation $${B_R}(X,Y) = 0,\forall X,Y \in g$$ is the Yang-Baxter equation. The modified Yang-Baxter equation was defined in [5] as follows: $${B_R}(X,Y) = - \left[ {X,Y} \right]$$ (4) Obviously, the equation (4) means (3) hold.KeywordsToda LatticeCartan DecompositionPeriodic Toda LatticeInvolution TheoremNondegenerate Invariant Bilinear FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.