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On the eigenvector algebra of the product of elements with commutator one in the first Weyl algebra

Published online by Cambridge University Press:  13 July 2011

V. V. BAVULA
V. V. BAVULA*
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH. e-mail: [email protected]
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Abstract

Let A1 = K〈X, Y|[Y, X]=1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(YX) of A1 is ℤ. Let A1 → A1, X ↦ x, Y ↦ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of the Weyl algebra A1 is ℤ and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 245 - 262
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Amitsur, S. A.Commutative linear differential operators. Pacific J. Math. 8 (1958), 1–10.CrossRefGoogle Scholar
[2]Bavula, V. V.Finite-dimensionality of Extn and Torn of simple modules over a class of algebras. Funct. Anal. Appl. 25 (1991), no. 3, 229–230.CrossRefGoogle Scholar
[3]Bavula, V. V.Generalized Weyl algebras and their representations. (Russian) Algebra i Analiz 4 (1992), no. 1, 75–97; translation in St. Petersburg Math. J. 4 (1993), no. 1, 71–92.Google Scholar
[4]Bavula, V. V.Dixmier's Problem 5 for the Weyl algebra, J. Algebra 283 (2005), no. 2, 604–621.CrossRefGoogle Scholar
[5]Bavula, V. V.The inversion formula for automorphisms of the Weyl algebras and polynomial algebras, J. Pure Appl. Algebra 210 (2007), 147–159. (Arxiv:math.RA/0512215).CrossRefGoogle Scholar
[6]Bavula, V. V. The Jacobian Conjecture2n implies the Dixmier Problemn. ArXiv:math.RA/0512250.Google Scholar
[7]Belov–Kanel, A. and Kontsevich, M.The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. Mosc. Math. J. 7 (2007), no. 2, 209–218 (arXiv:math. RA/0512171).CrossRefGoogle Scholar
[8]Dixmier, J.Sur les algèbres de Weyl. Bull. Soc. Math. France 96 (1968), 209–242.CrossRefGoogle Scholar
[9]Guccione, J. A., Guccione, J. J. and Valqui, C. On the centralizers in the Weyl algebra. Arxiv:0912.5202.Google Scholar
[10]Joseph, A.The Weyl algebra–-semisimple and nilpotent elements. Amer. J. Math. 97 (1975), no. 3, 597–615.CrossRefGoogle Scholar
[11]Tsuchimoto, Y.Endomorphisms of Weyl algebra and p-curvatures. Osaka J. Math. 42 (2005), no. 2, 435–452.Google Scholar