LIE IDENTITIES FOR HOPF-ALGEBRAS

Let R denote either a group algebra over a field of characteristic p > 3 or the restricted enveloping algebra of a restricted Lie algebra ov er 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 (2),...,x(n)],[x(n+1),x(n+2),...,x(n+m)],x(n+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 sho wn to be Lie nilpotent if they satisfy an identity of the form [[x(1), x(2),...,x(n)],[x(n+1),x(n+2),...,x(n+m)]] = 0. It was previously know n that Lie centre-by-metabelian group algebras are commutative provide d p > 3, and that a Lie soluble group algebra of derived length n is c ommutative if its characteristic exceeds 2(n). (C) 1997 Elsevier Scien ce B.V.