Lie identities for Hopf algebras

✍ Scribed by David M. Riley; Vladimir Tasić

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 470 KB
• Volume: 122
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(96)00073-4

✦ Synopsis

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 ccx1>x2> ". ~x"I~cx"+1~x,+2~ ... ~x,+,I~x"+,+Il = 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 ccxi,x 2, ... >X"1,CX,+1>X"+2> ... rX"+,ll = 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 2".