The structure of complete left-symmetric algebras (Q1204230)

Let \(L\) be a Lie algebra over \(k\;(\text{char}(k)=0)\). A \(k\)-bilinear map \(\cdot :L\times L\to L\) is called a left symmetric structure (LSA) if the left regular representation \(\lambda: L\to\text{gl}(L)\), defined by \(\lambda(a)b=a\cdot b\) is a Lie algebra homomorphism satisfying \(\lambda(a)b-\lambda(b)a=[a,b]\). It is called complete if the right regular map \(\rho:L\to\text{gl}(L)\), defined by \(\rho(b)a=a\cdot b\) satisfies \(id_ L+\rho(b)\) bijective \(\forall b\in L\). The main result of the paper under review is that an LSA is complete if and only if \(\rho(b)\) is nilpotent for all \(b\in L\). A key role in the proof is played by the Lie algebra \(k^ n\rtimes \text{gl}(k^ n)\) and its solvable subalgebras.