Bulletin of the Australian Mathematical Society

Research Article

MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

XIN TANGa1

a1 Department of Mathematics & Computer Science, Fayetteville State University, Fayetteville, NC 28301, USA (email: xtang@uncfsu.edu)

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.

(Received March 18 2009)

2000 Mathematics subject classification

  • primary 17A36;
  • 17B35; secondary 17B60

Keywords and phrases

  • Dixmier partition;
  • mad subalgebras;
  • Lie subalgebras

Footnotes

This research is supported in part by Fayetteville State University Faculty Development Research Grant.