Pointed Lie Theorem And Killing Form Of YD-Lie Algebra

It is well known that the braid equation is equivalent to the quantum equation, therefore the question how to solve the Yang-Baxter equation can through the solution on braid equation. Ones have built up the theory of braided tensor categories to solve the Yang-Baxter equation.The category of Yetter-Drinfeld modules is playing the extremely vital role in the physical quantum Yang-Baxter equation solution, and already became one of hot topics in the recent years studied.Braided Lie-algebra include Lie superal-gebra , colour algebra and Lie YD-algebra. This article first introduce definitions, the symmetric bicharacter's definition and some property of the G-graded algebra , the colour superalgebra and the symmetric bicharacter. Secondly, it has drawn out the pointed YD-Lie algebra's definition, in the category of Yetter-Drinfeld modules, let Z(G) be a torsion-free abelian group, if we have a symmetric braiding c, for each G-graded algebra V over the G-graded space, a new Lie superalgebra with an operation [ ]_c satisfying an actions, then we can obtain a new Lie superalgebra, this paper to call it pointed YD-Lie algebra.In this foundation this article obtain the Lie's Theorem hold for pointed YD-Lie algebra, let Lbe a finite-dimensional solvable pointed YD-Lie subalgebra of gl{{V_g}, k), and all of homogeneous elements in [L, L]_cbe nilpotent.If Z(G) is a torsion-free abelian group, then the matrices of L relative to a suitable homogeneous basis of V are uppertriangular.And we give an example to show that the torsion-free condition is necessary. Next, we give the definition of Killing form hold for pointed YD-Lie algebra, let L be a pointed YD-Lie algebra, for every homogeneous elements x, y ∈ L, if(x, y) = tr_q(adxady).then we call (x, y) Killing form of L. As well as we introduce some property of Killing form. Finally, we use Killing form to decide if YD-Lie algebras are semisimple. Our results will lay a solid foundation on the further research of structures of YD-Lie algebra.