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MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

  • XIN TANG (a1)
Abstract
Abstract

Let 𝒰(𝔯(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra 𝔯(1) over a base field 𝕂. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of 𝒰(𝔯(1)). We first prove that the set of noncentral elements of 𝒰(𝔯(1)) admits the Dixmier partition, 𝒰(𝔯(1))−𝕂=⋃ 5i=1Δi, and establish characterization theorems for elements in Δi, i=1,3,4. Then we determine the elements in Δi, i=1,3 , and describe the eigenvalues for the inner derivation ad Bx,x∈Δi, i=3,4 . We also derive other useful results for elements in Δi, i=2,3,4,5 . As an application, we find all framed mad subalgebras of 𝒰(𝔯(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of 𝒰(𝔯(1)) . We also study the realizations of the Lie algebra 𝔯(1) in 𝒰(𝔯(1)) in detail.

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References
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[1]Alev J. and Dumas F., ‘Invariants du corps de Weyl sous l’action de groupes finis (in French) [Invariants of the Weyl field under the action of finite groups]’, Comm. Algebra 25 (1997), 16551672.
[2]Bavula V. V., ‘Dixmier’s problem 5 for the Weyl algebra’, J. Algebra 283 (2005), 604621.
[3]Bavula V. V., ‘Dixmier’s problem 6 for somewhat commutative algebras and Dixmier’s problem 3 for the ring of differential operators on a smooth irreducible affine curve’, J. Algebra Appl. 4 (2005), 577586.
[4]Bavula V. V., ‘Dixmier’s problem 6 for the Weyl algebra (the generic type problem)’, Comm. Algebra 34 (2006), 13811406.
[5]Beĭlinson A. and Bernstein J., ‘Localisation de g-modules’, C. R. Acad. Sci. Paris 292 (1981), 1518.
[6]Berest Y. and Wilson G., ‘Mad subalgebras of rings of differential operators on curves’, Adv. Math. 212 (2007), 163190.
[7]Bernšteĭn I. N., ‘Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients’, Funct. Anal. Appl. 5 (1971), 89101.
[8]Bernšteĭn I. N., ‘The analytic continuation of generalized functions with respect to a parameter’, Funct. Anal. Appl. 6 (1972), 273285.
[9]Dixmier J., ‘Sur les algèbres de Weyl’, Bull. Soc. Math. France 96 (1968), 209242.
[10]Dixmier J., ‘Sur les algèbres de Weyl. II’, Bull. Sci. Math. (2) 94 (1970), 289301.
[11]Igusa J., ‘On Lie algebras generated by two differential operators’, Progr. Math. 14 (1981), 187195.
[12]Joseph A., ‘A characterization theorem for realizations of sl(2)’, Proc. Cambridge Philos. Soc. 75 (1974), 119131.
[13]Joseph A., ‘The Weyl algebra—semisimple and nilpotent elements’, Amer. J. Math. 97 (1975), 597615.
[14]Kashiwara M., ‘B-functions and holonomic systems. Rationality of roots of B-functions’, Invent. Math. 38 (1976), 3353.
[15]Rausch de Traubenberg M., Slupinski M. and Tanasa A., ‘Finite-dimensional Lie subalgebras of the Weyl algebra’, J. Lie Theory 16 (2006), 427454.
[16]Simoni A. and Zaccaria F., ‘On the realization of semi-simple Lie algebras with quantum canonical variables’, Nuovo Cimento A (10) 59 (1969), 280292.
[17]Smith M. K., ‘Automorphisms of enveloping algebras’, Comm. Algebra 11 (1983), 17691802.
[18]Weyl H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1950).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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