Lie identities for Hopf algebras Original Research Article

Let R denote either a group algebra over a field of characteristic p> 3 or the restricted enveloping algebra of a restricted Lie algebra over a field of characteristic p> 2. Viewing R as a Lie Algebra in the natural way, our main result states that R satisfies a law of the form image[[x1, x2, …, xn], [xn + 1, xn + 2, …, xn + m], xn + m + 1] = 0 if and only if R is Lie nilpotent. It is deduced that R is commutative provided p> 2 max m, n. Group algebras over fields of characteristic p = 3 are shown to be Lie nilpotent if they satisfy an identity of the form image[[x1,x2,…,xn], [xn + 1, xn + 2, …, xn + m]] = 0. It was previously known that Lie centre-by-metabelian group algebras are commutative provided p> 3, and that a Lie soluble group algebra of derived length n is commutative if its characteristic exceeds 2n.