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# The transfer of a commutator law from a nil-ring to its adjoint group

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Published:1997-03-01
Printed: Mar 1997
• David M. Riley
• Vladimir Tasić
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## Abstract

For every field $F$ of characteristic $p\geq 0$, we construct an example of a finite dimensional nilpotent $F$-algebra $R$ whose adjoint group $A(R)$ is not centre-by-metabelian, in spite of the fact that $R$ is Lie centre-by-metabelian and satisfies the identities $x^{2p}=0$ when $p>2$ and $x^8=0$ when $p=2$. The existence of such algebras answers a question raised by A.~E.~Zalesskii, and is in contrast to positive results obtained by Krasilnikov, Sharma and Srivastava for Lie metabelian rings and by Smirnov for the class Lie centre-by-metabelian nil-algebras of exponent 4 over a field of characteristic 2 of cardinality at least 4.
 MSC Classifications: 16U60 - Units, groups of units 17B60 - Lie (super)algebras associated with other structures (associative, Jordan, etc.) [See also 16W10, 17C40, 17C50]

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