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On the Representation Theory of Solvable Lie Algebras
Author(s) -
Brown Kenneth A.,
Cloux Fokko Du
Publication year - 1988
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-57.2.284
Subject(s) - mathematics , isomorphism (crystallography) , nilpotent , lie algebra , ideal (ethics) , representation theory , pure mathematics , algebraic group , corollary , combinatorics , vector space , algebraic number , algebra over a field , discrete mathematics , mathematical analysis , philosophy , chemistry , crystal structure , epistemology , crystallography
Let U be the enveloping algebra of a finite‐dimensional solvable complex Lie algebra g . We study the cliques in the primitive spectrum Prim( U ) of U , describing them in terms of the Dixmier map I : g * | A → Prim( U ), where A is the adjoint algebraic group of g . For f ε g * , we establish a canonical isomorphism between the clique of I ( f ) and the clique of the augmentation ideal of U ( g ― ( f )), where g ― ( f ) is the orthogonal in g to the tangent space at f to A . f . (When g is algebraic, g ― ( f ) is the usual stabilizer g ― ( f ) of f .) A corollary of this description is: I ( f ) is localizable if and only if g ― ( f ) is nilpotent. The proof depends in part on using the connections between links of primitive ideals and extensions of ‘standard’ irreducible ( U ‐modules—these being the modules that occur in the definition of the Dixmier map.