So let's say we have two finite dimensional algebras, $L_1$ and $L_2$, spanned respectivelt by $\ t_i$ and $g_i$. If i want to prove that those algebras are isomorphic, is it sufficient to find a isomoprhism ( for example and invertible matrix) that maps the basis into each other and that leaves invariant the structure constants?
Edit: What if i find an isomorphism that does not respect the structure constants?
[Math] Isomorphism between lie algebras and their complexification
lie-algebras
Best Answer
A Lie algebra isomorphism is a bijective linear map $\phi\colon L_1\rightarrow L_2$ satisfying $$ \phi([x,y]_{L_1})=[\phi(x),\phi(y)]_{L_2} $$ for all $x,y\in L_1$. In particular, it need not preserve the structure constants. For example, the Heisenberg Lie algebra $L_1$ with basis $(x,y,z)$ and $[x,y]=z$ is isomorphic to another non-abelian nilpotent Lie algebra, with basis $(e_1,e_2,e_3)$ where the structure constants look totally different: $$ [e_1,e_2]=[e_1,e_3]=-e_1-e_2+e_3,\quad [e_2,e_3]=e_1+e_2-e_3. $$