Mad subalgebras and lie subalgebras of an enveloping algebra
Document Type
Article
Publication Date
1-1-2010
Abstract
Let µ(?(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra ?(1) over a base field K. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of µ(?(1)). We first prove that the set of noncentral elements of µ(?(1)) admits the Dixmier partition, µ(?(1))?K=? 5i=1 ?i, and establish characterization theorems for elements in ?i, i=1,3,4. Then we determine the elements in ?i, i=1,3, and describe the eigenvalues for the inner derivation ad Bx,x??i, i=3,4. We also derive other useful results for elements in ?i, i=2,3,4,5. As an application, we find all framed mad subalgebras of µ(?(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of µ(?(1)). We also study the realizations of the Lie algebra ?(1) in µ(?(1)) in detail. © 2010, Australian Mathematical Publishing Association Inc. All rights reserved.
Recommended Citation
Tang, Xin, "Mad subalgebras and lie subalgebras of an enveloping algebra" (2010). College of Health, Science, and Technology. 83.
https://digitalcommons.uncfsu.edu/college_health_science_technology/83