李代数的张量积所确定的Leibniz代数

讨论了李代数(g)以及由这个李代数诱导的Leibniz代数(g)(×)(g)的一些性质,主要从不变双线性型和导子看这两个代数之间的差异,证明了在特定条件下两者的不变双线性型维数是一致的.为进一步确定李代数(g)和(g)(×)(g)的差异,讨论了由(g)(×)(g)诱导的一类重要的李代数(g)(×)(g)；最后证明了,如...

Leibniz algebras defined by tensor product of Lie algebras

By the definition of $\mathrm{Leibniz}$ algebra, we showed that \ $\mathcal{G}\otimes\mathcal{G}$\ was a $\mathrm{Leibniz}$\ algebra when \ $\mathcal{G}$\ was a $ \mathrm{Lie}$ algebra. We also proved that $\mathcal{G}\otimes\mathcal{G}$\ and $\mathcal{G}$\ have the same dimension of invariant symmetric bilinear forms in a special case, and the dimension of the derivation algebra of\ $\mathcal{G}$\ is always less than that of $\mathcal{G}\otimes\mathcal{G}$. $\mathcal{G}\boxtimes\mathcal{G}$\ is one of the important \ $\mathrm{Lie}$\ algebra induced by $\mathcal{G}\otimes\mathcal{G}$, and $\mathcal{G}\boxtimes\mathcal{G}$\ is isomorphic to $\mathcal{G}$\ when $\mathcal{G}$\ is a finite dimensional semi-simple\ $\mathrm{Lie}$\ algebra.}