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 $mathcal{G}otimesmathcal{G}$ was a $mathrm{Leibniz}$ algebra when $mathcal{G}$ was a $ mathrm{Lie}$ algebra. We also proved that $mathcal{G}otimesmathcal{G}$ 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 $mathcal{G}otimesmathcal{G}$. $mathcal{G}boxtimesmathcal{G}$ is one of the important $mathrm{Lie}$ algebra induced by $mathcal{G}otimesmathcal{G}$, and $mathcal{G}boxtimesmathcal{G}$ is isomorphic to $mathcal{G}$ when $mathcal{G}$ is a finite dimensional semi-simple $mathrm{Lie}$ algebra.}