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Journal of the London Mathematical Society Advance Access originally published online on August 29, 2007
Journal of the London Mathematical Society 2007 76(1):148-164; doi:10.1112/jlms/jdm036
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© 2007 London Mathematical Society

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

V. V. Bavula

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
United Kingdom

Let A be an algebra over a field K of characteristic zero and let {delta}1, ..., {delta}sisinDer K(A) be commuting locally nilpotent K-derivations such that {delta}i(xj) equals {delta}ij, the Kronecker delta, for some elements x1, ..., xsisinA. A set of generators for the algebra Formula is found explicitly and a set of defining relations for the algebra A{delta} is described. Similarly, let {sigma}1, ..., {sigma}s isin AutK(A) be commuting K-automorphisms of the algebra A is given such that the maps {sigma}i – idA are locally nilpotent and {sigma}i (xj) = xj + {delta}ij, for some elements x1, ..., xs isin A. A set of generators for the algebra A{sigma}: = {a isin A | {sigma}1(a) = ... = {sigma}s(a) = a} is found explicitly and a set of defining relations for the algebra A{sigma} is described. In general, even for a finitely generated non-commutative algebra A the algebras of invariants A{delta} and A{sigma} are not finitely generated, not (left or right) Noetherian and a minimal number of defining relations is infinite. However, for a finitely generated commutative algebra A the opposite is always true. The derivations (or automorphisms) just described appear often in many different situations (possibly) after localization of the algebra A.

v.bavula{at}sheffield.ac.uk

2000 Mathematics Subject Classification 16W22, 13N15, 14R10, 16S15, 16D30.

Received January 26, 2006; published online August 29, 2007.


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