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A Version of the Poincaré‐Birkhoff‐Witt Theorem
Author(s) -
MakarLimanov L.
Publication year - 1994
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/26.3.273
Subject(s) - mathematics , ideal (ethics) , witt algebra , codimension , bounded function , pure mathematics , zero (linguistics) , universal enveloping algebra , field (mathematics) , algebra over a field , algebraic number , algebra representation , cellular algebra , mathematical analysis , philosophy , linguistics , epistemology
In this note I shall prove that if L is a finite‐dimensional Lie algebra over a field F of characteristic zero which is generated as an algebra by a set of elements { e 1 , e 2 ,…, e k }, then the universal enveloping algebra U ( L ) of L is linearly generated by monomials spanned by the elements { e i } of an a priori bounded width. As an application, a criterion of Kostant for a left ideal of U ( L ) to be of finite codimension is proved by purely algebraic means.