Normal enveloping algebras (Q442047)

Let \(\mathbb{F}\) be a field. An \(\mathbb{F}-\)algebra \(A\) with an involution \(\ast\) is called normal if \(xx^{\ast}=x^{\ast}x\) for each \(x\in A\). Let \(L\) be a Lie algebra over the field \(\mathbb{F}\). We denote the enveloping algebra of \(L\) by \(U(L)\) if \(L\) is an ordinary Lie algebra and the restricted enveloping algebra of \(L\) by \(u(L)\) if \(L\) is a restricted Lie algebra. Note that \(U(L)(\text{resp.}\, u(L))\) can be associated with the principal involution: \(x^{\ast}=-x\) for any \(x\in U(L)\) (resp. \(x\in u(L)\)). In this interesting paper, the authors completely settle the question of characterizing all normal ordinary enveloping algebras \(U(L)\) and restricted enveloping algebras \(u(L)\) with respect to their principal involutions. The main results of this paper are stated as Theorem 1.1 (for the restricted case) and Theorem 1.2 (for the ordinary case).