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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A Hilbert-Nagata theorem in
noncommutative invariant theory


Authors: Mátyás Domokos and Vesselin Drensky
Journal: Trans. Amer. Math. Soc. 350 (1998), 2797-2811
MSC (1991): Primary 16W20; Secondary 16R10, 16P40, 16W50, 13A50, 15A72
DOI: https://doi.org/10.1090/S0002-9947-98-02208-9
MathSciNet review: 1475681
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Abstract: Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.


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  • 1. G. Almkvist, W. Dicks, E. Formanek, Hilbert series of fixed free algebras and noncommutative classical invariant theory, J. Algebra 93 (1985), 189-214.MR 86k:16001
  • 2. S.A. Amitsur, The T-ideals of the free ring, J. London Math. Soc. 30 (1955), 470-475.MR 17:122c
  • 3. M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publ. Comp., Reading, Mass., 1969.MR 39:4129
  • 4. J.A. Dieudonné, J.B. Carrell, Invariant Theory, Old and New, Academic Press, New York-London, 1971.MR 43:4828
  • 5. V. Drensky, On the Hilbert series of relatively free algebras, Comm. Alg. 12 (1984), 2335-2347.MR 86f:16005
  • 6. V. Drensky, Computational techniques for PI-algebras, in ``Topics in Algebra'', Part 1 ``Rings and Representations of Algebras'', Banach Center Publ. 26, Polish Scientific Publishers, Warshaw, 1990, 17-44.MR 93e:16036
  • 7. V. Drensky, Commutative and noncommutative invariant theory, in ``Mathematics and Education in Mathematics'', Union of Bulgarian Mathematicians, Sofia, 1995, 14-50.
  • 8. E. Formanek, Noncommutative invariant theory, Contemp. Math. 43 (1985), 87-119.MR 87d:16046
  • 9. P.J. Higgins, Lie rings satisfying the Engel condition, Proc. Cambridge Phil. Soc. 50 (1954), 8-15. MR 15:596b
  • 10. G. Higman, On a conjecture of Nagata, Proc. Cambridge Phil. Soc. 52 (1956), 1-4. MR 17:453c
  • 11. V.K. Kharchenko, Noncommutative invariants of finite groups and Noetherian varieties, J. Pure Appl. Alg. 31 (1984), 83-90. MR 85j:16052
  • 12. O.G. Kharlampovich, M.V. Sapir, Algorithmic problems in varieties, Intern. J. Algebra and Computation 5 (1995), 379-602. MR 96m:20045
  • 13. A.N. Krasil'nikov, Finite basis property of some varieties of Lie algebras, (Russian), Vestnik. Moskov. Univ. Ser. I, No 2 (1982), 34-38. MR 83i:17019
  • 14. V.N. Latyshev, A generalization of Hilbert's theorem on the finiteness of bases, (Russian), Sibirsk. Mat. Zhur. 7 (1966), 1422-1424.
  • 15. I.V. L'vov, Maximality conditions in algebras with identity, (Russian), Algebra i Logika 8 (1969), 449-459. MR 43:4853
  • 16. Yu.N. Mal'tsev, On varieties of algebras, (Russian), Algebra i Logika 15 (1976), 579-584. MR 58:5457
  • 17. L.A. Vladimirova, Codimensions of T-ideals containing an identity of fourth degree, (Russian), Serdica 14 (1988), 82-94. MR 89m:16025
  • 18. N. Vonessen, Actions of linearly reductive groups on affine PI-algebras, Mem. Amer. Math. Soc. No. 414 (1989). MR 90i:16014
  • 19. E.I. Zelmanov, Engelian Lie algebras, (Russian), Sibirsk. Mat. Zhur. 29 (1988), 112-117. MR 90a:17010

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Additional Information

Mátyás Domokos
Affiliation: Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O. Box 127, H-1364, Hungary
Email: domokos@math-inst.hu

Vesselin Drensky
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
Email: drensky@math.acad.bg

DOI: https://doi.org/10.1090/S0002-9947-98-02208-9
Keywords: T-ideals, algebras with polynomial identity, noncommutative invariant theory, algebra of invariants, rational Hilbert series
Received by editor(s): June 5, 1996
Additional Notes: The first author was partially supported by Hungarian National Foundation for Scientific Research Grant no. F023436.
The second author was partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.
Article copyright: © Copyright 1998 American Mathematical Society

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