Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-18T14:59:51.573Z Has data issue: false hasContentIssue false
  • English
  • Français

MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

Part of: Lie algebras and Lie superalgebras General nonassociative rings

Published online by Cambridge University Press:  16 August 2010

XIN TANG
XIN TANG*
Affiliation:
Department of Mathematics & Computer Science, Fayetteville State University, Fayetteville, NC 28301, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Dixmier partitionmad subalgebrasLie subalgebras

MSC classification

Secondary: 17A36: Automorphisms, derivations, other operators 17B35: Universal enveloping (super)algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 401 - 423
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[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), 1655–1672.CrossRefGoogle Scholar
[2]Bavula, V. V., ‘Dixmier’s problem 5 for the Weyl algebra’, J. Algebra 283 (2005), 604–621.CrossRefGoogle Scholar
[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), 577–586.Google Scholar
[4]Bavula, V. V., ‘Dixmier’s problem 6 for the Weyl algebra (the generic type problem)’, Comm. Algebra 34 (2006), 1381–1406.Google Scholar
[5]Beĭlinson, A. and Bernstein, J., ‘Localisation de g-modules’, C. R. Acad. Sci. Paris 292 (1981), 15–18.Google Scholar
[6]Berest, Y. and Wilson, G., ‘Mad subalgebras of rings of differential operators on curves’, Adv. Math. 212 (2007), 163–190.Google Scholar
[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), 89–101.CrossRefGoogle Scholar
[8]Bernšteĭn, I. N., ‘The analytic continuation of generalized functions with respect to a parameter’, Funct. Anal. Appl. 6 (1972), 273–285.Google Scholar
[9]Dixmier, J., ‘Sur les algèbres de Weyl’, Bull. Soc. Math. France 96 (1968), 209–242.Google Scholar
[10]Dixmier, J., ‘Sur les algèbres de Weyl. II’, Bull. Sci. Math. (2) 94 (1970), 289–301.Google Scholar
[11]Igusa, J., ‘On Lie algebras generated by two differential operators’, Progr. Math. 14 (1981), 187–195.Google Scholar
[12]Joseph, A., ‘A characterization theorem for realizations of sl(2)’, Proc. Cambridge Philos. Soc. 75 (1974), 119–131.CrossRefGoogle Scholar
[13]Joseph, A., ‘The Weyl algebra—semisimple and nilpotent elements’, Amer. J. Math. 97 (1975), 597–615.Google Scholar
[14]Kashiwara, M., ‘B-functions and holonomic systems. Rationality of roots of B-functions’, Invent. Math. 38 (1976), 33–53.CrossRefGoogle Scholar
[15]Rausch de Traubenberg, M., Slupinski, M. and Tanasa, A., ‘Finite-dimensional Lie subalgebras of the Weyl algebra’, J. Lie Theory 16 (2006), 427–454.Google Scholar
[16]Simoni, A. and Zaccaria, F., ‘On the realization of semi-simple Lie algebras with quantum canonical variables’, Nuovo Cimento A (10) 59 (1969), 280–292.Google Scholar
[17]Smith, M. K., ‘Automorphisms of enveloping algebras’, Comm. Algebra 11 (1983), 1769–1802.Google Scholar
[18]Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1950).Google Scholar