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MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

Part of: General nonassociative rings Lie algebras and Lie superalgebras

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])
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Abstract

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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

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