Generalized Derivations of Lie Algebras
By G. Leger, Eugene M. Luks. 2000
195 citations recorded by the source corpus.
Abstract Suppose L is a finite-dimensional Lie algebra with multiplication μ: L ∧ L → L . Let Δ( L ) denote the set of triples ( f , f ′, f ″), with f , f ′, f ″ ∈ Hom( L , L ), such that μ ∘ ( f ∧ I L + I L ∧ f ′) = f ″ ∘ μ. We consider the Lie algebra GenDer( L ) = { f ∈ Hom( L , L )|∃ f ′, f ″: ( f , f ′, f ″) ∈ Δ( L )}. Well-researched subalgebras of GenDer( L ) include the derivation algebra, Der( L ) = { f ∈ Hom( L , L )|( f , f , f ) ∈ Δ( L )}, and the centroid, C( L ) = { f ∈ Hom( L , L )|( f , 0, f ) ∈ δ( L )}. We now study the subalgebra QDer( L ) = { f ∈ Hom( L , L )|∃ f ′: ( f , f , f ′) ∈ Δ( L )}, and the subspace QC( L ) = { f ∈ Hom( L , L )|( f , − f , 0) ∈ Δ( L )}. In characteristic ≠ 2, GenDer( L ) = QDer( L ) + QC( L ) and we are concerned with the inclusions Der( L ) ⊆ QDer( L ) and C( L ) ⊆ QC( L ) ∩ QDer( L ). If Z( L ) = 0 then C( L ) = QC( L ) ∩ QDer( L ) and, under reasonable conditions on Lie algebras with toral Cartan subalgebras, we show QDer( L ) = Der( L ) + C( L ); if L is a parabolic subalgebra of a simple Lie algebra of rank > 1 in characteristic 0, then we even have GenDer( L ) = ad( L ) + ( I L ). In general QC( L ) is not closed under composition or Lie bracket; however, if Z( L ) = 0 then QC( L ) is a commutative, associative algebra, and we describe conditions that force QC( L ) = C( L ) or, equivalently, GenDer( L ) = QDer( L ). We show that, in characteristic 0, GenDer( L ) preserves the radical of L , thus generalizing the classical result for Der( L ). We also discuss some applications of the main results to the study of functions f ∈ Hom( L , L ) such that f ∘ μ or μ ∘ ( f ∧ I L ) defines a Lie multiplication.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/8aaef0e27ba7864fefce97b3f8fe1a6080f342b1
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Citations in the corpus
- On δ-derivations of Lie algebras and superalgebras (2010, methodology): “A more general concept ‐ the so-called quasiderivations ‐ was studied by Leger and Luks in [ LL ].”
- Biderivations and Commuting Linear Maps on Lie Algebras (2018, unknown): “So far, with a partial exception [11] (where, however, only finite dimensional Lie algebras were treated), their description was known only for some special examples of Lie algebras.”
- Biderivations of the Parabolic Subalgebras of Simple Lie Algebras (2011, background): “13 of [7] and the main result of [8] saying that every derivation of is inner.”
- Generalized Derivations of Lie Superalgebras (2010, background): “The purpose of this article is to generalize some beautiful results in Leger and Luks [13] to the generalized derivation superalgebras of a Lie superalgebra.”
- Generalized Derivations of Lie Color Algebras (2013, background): “The most important and systematic research on the generalized derivation algebras of a Lie algebra and their Lie subalgebras was due to Leger and Luks [17].”