Lie structure of smash products
An associative algebra \(A\) is called Lie solvable (respectively Lie nilpotent) if the associated Lie algebra with the Lie bracket \([a,b]=ab-ba\) is solvable (respectively nilpotent). Let \(L\) be either a restricted Lie algebra over a field \(\mathbb{F}\) of positive characteristic, or an ordinary Lie algebra over an arbitrary field \(\mathbb{F}\), and let \(G\) be a group acting on \(L\). The authors discuss when is the smash product \(u(L)\# \mathbb{F}G\) in the first situation, or the smash product \(U(L)\# \mathbb{F}G\) in the second one, Lie solvable or Lie nilpotent. Here \(u(L)\) denotes the restricted enveloping algebra of \(L\) in the first case, and the universal enveloping algebra of \(L\) in the second case. For the Lie solvable problem, the authors work under the assumption that the characteristic of \(\mathbb{F}\) is \(\neq 2\). Using the Cartier-Kostant-Milnor-Moore Theorem, it is obtained as a consequence that a cocommutative Hopf algebra \(H\) over a field of characteristic zero is Lie solvable if and only if \(H\) is commutative.
Lie solvable (nilpotent) associative algebra
scientific article; zbMATH DE number 6709001
scientific article