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  • Bulletin of the Australian Mathematical Society, Volume 82, Issue 3
  • December 2010, pp. 401-423

MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

  • XIN TANG (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972710000407
  • Published online: 01 August 2010
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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[1]J. Alev and F. Dumas , ‘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.

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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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