Summary
Let G be a group and ≤ : G → GL(V) a representation of G in a vector space V of dimension n over a commutative field k of characteristic zero. The group ≤(G) acts by automorphisms on the algebra of regular functions k[V], and this action can be canonically extended to theWeyl algebra A n (k) of differential operators over k[V] and then to the skewfield of fractions D n(k) of A n(k). The problem studied in this paper is to determine sufficient conditions for the subfield of invariants of D n(k) under this action to be isomorphic to a Weyl skewfield D m(K) for some integer 0 ≤ m ≤ n and some purely transcendental extension K of k. We obtain such an isomorphism in two cases: (1) when ≤ splits into a sum of representations of dimension one, (2) when ≤ is of dimension two. We give some applications of these general results to the actions of tori on Weyl algebras and to differential operators over Kleinian surfaces.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliographie
J. Alev and F. Dumas, Invariants du corps de Weyl sous l’action de groupes finis, Commun. Algebra 25 (1997), 1655–1672.
—, Sur les invariants des algèbres de Weyl et de leurs corps de fractions, Lectures Notes Pure and Applied Math. 197 (1998), 1–10.
J. Alev and Th. Lambre, Comparaison de l’homologie de Hochschild et de l’homologie de Poisson pour une déformation des surfaces de Klein, in Algebra and Operator Theory (Tashkent, 1997), Kluwer Acad. Publ., Dordrecht, 1998, 25–38.
L. Chiang, H. Chu and M. Kang, Generation of invariants, J. Algebra 221 (1999), 232–241.
I. V. Dolgachev, Rationality of fields of invariants, Proc. Symposia Pure Math. 46 (1987), 3–16.
K. R. Goodearl and R. B. Warfield, An Introduction to non commutative Noetherian Rings, Cambridge University Press, London, 1985.
I.M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Inst. Hautes Etudes Sci. Publ. Math. 31 (1966), 509–523.
A. Joseph, A generalization of the Gelfand-Kirillov conjecture, Amer. J. Math. 99 (1977), 1151–1165.
—, Coxeter structure and finite group action, in Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), 185–219, Sémin. Congr. 2, Soc. Math. France, Paris, 1997.
P. I. Katsylo, Rationality of fields of invariants of reducible representations of the group SL2, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 39 (1984), 77–79.
M. Kervaire et T. Vust, Fractions rationnelles invariantes par un groupe fini, in Algebraische Transformationsgruppen und Invariantentheorie, D.M.V. Sem. 13 Birkhäuser, Basel, (1989), 157–179.
G.R. Krause and T.H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension. Revised version. Graduate Studies in Mathematics, 22 American Mathematical Society, 2000.
Th. Levasseur, Anneaux d’opérateurs différentiels, in Séminaire d’Algèbre P. Dubreil — M.-P. Malliavin (1980), Lecture Notes in Math. 867, Springer, Berlin-New York, 1981, 157–173.
T. Maeda, On the invariant field of binary octavics, Hiroshima Math. J. 20 (1990), 619–632.
J. C. McConnell and J. C. Robson, Non commutative Noetherian Rings, Wiley, Chichester, 1987.
T. Miyata, Invariants of certain groups I, Nagoya Math. J. 41 (1971), 68–73.
S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings, Lecture Notes in Math. 818, Springer-Verlag, Berlin, 1980.
I. M. Musson, Actions of tori on Weyl algebras, Commun. Algebra 16 (1988), 139–148.
D. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), 71–84.
—, Groups acting on fields: Noether’s problem, in Group Actions on Rings (Brunswick, Maine, 1984), Contemp. Math., 43, Amer. Math. Soc., Providence, RI, 1985, 267–277.
T. A. Springer, Invariant Theory, Lectures Notes in Maths 585, Springer-Verlag, Berlin, 1977.
E. B. Vinberg, Rationality of the field of invariants of a triangular group, Vestnik Mosk. Univ. Mat. 37 (1982), 23–24.
J. Zhang, On lower transcendence degree, Adv. Math. 139 (1998), 157–193.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dédié à Anthony Joseph, à l’occasion de son soixantième anniversaire
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Alev, J., Dumas, F. (2006). Opérateurs différentiels invariants et problème de Noether. In: Bernstein, J., Hinich, V., Melnikov, A. (eds) Studies in Lie Theory. Progress in Mathematics, vol 243. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4478-4_3
Download citation
DOI: https://doi.org/10.1007/0-8176-4478-4_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4342-3
Online ISBN: 978-0-8176-4478-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)