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In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra.

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rdfs:label	Graded Lie algebra (en) Algebra di Lie graduata (it)
rdfs:comment	In matematica, un'algebra di Lie si dice graduata quando è dotata di una gradazione compatibile con le parentesi di Lie. In altre parole, essa è un'algebra di Lie che è un'algebra graduata non-associativa nel quadro dell'operazione di commutazione. Questo concetto viene esteso nella superalgebra di Lie graduata, in cui si richiede che le parentesi di Lie non siano necessariamente anticommutative. (it) In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra. (en)
dcterms:subject	Lie algebras
Wikipage page ID	5475402 (xsd:integer)
Wikipage revision ID	1074866903 (xsd:integer)
Link from a Wikipage to another Wikipage	Module (mathematics) Trace-free Bilinear form Deformation theory Homomorphism Vector space Superalgebra Lie derivative Lie superalgebra Lie algebras Commutative monoid Connection (mathematics) Mathematics Matrix (mathematics) Graded (mathematics) Graded algebra Braided monoidal category Monoid Murray Gerstenhaber Coproduct Anticommutative Lie algebra Lower central series Frölicher–Nijenhuis bracket Derivation (abstract algebra) Parabolic Lie algebra Additive group Field (mathematics) Differential graded Lie algebra Direct sum Graded vector space Universal enveloping algebra Adjoint representation of a Lie algebra Characteristic (algebra) Differential geometry Donald C. Spencer Bulletin of the AMS Free Lie algebra Free group Integer Kunihiko Kodaira Cartan decomposition Category (mathematics) Semiring Nijenhuis–Richardson bracket Nonassociative algebra Semisimple Lie algebra Supersymmetry Lie algebra-valued form Lie bracket Identity map Commutative algebra (structure) Homogeneous element
sameAs	Graded Lie algebra Graded Lie algebra Graded Lie algebra Graded Lie algebra Graded Lie algebra
dbp:wikiPageUsesTemplate	dbt:Cite_journal dbt:Main
has abstract	In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra. A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. These arise in the study of derivations on graded algebras, in the deformation theory of Murray Gerstenhaber, Kunihiko Kodaira, and Donald C. Spencer, and in the theory of Lie derivatives. A supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super -gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supersymmetric) setting, and then tensorizes to obtain the supersymmetric analog. Still greater generalizations are possible to Lie algebras over a class of braided monoidal categories equipped with a coproduct and some notion of a gradation compatible with the braiding in the category. For hints in this direction, see Lie superalgebra#Category-theoretic definition. (en) In matematica, un'algebra di Lie si dice graduata quando è dotata di una gradazione compatibile con le parentesi di Lie. In altre parole, essa è un'algebra di Lie che è un'algebra graduata non-associativa nel quadro dell'operazione di commutazione. Questo concetto viene esteso nella superalgebra di Lie graduata, in cui si richiede che le parentesi di Lie non siano necessariamente anticommutative. (it)
gold:hypernym	Algebra
prov:wasDerivedFrom	wikipedia-en:Graded_Lie_algebra?oldid=1074866903&ns=0
page length (characters) of wiki page	9570 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Graded_Lie_algebra
is rdfs:seeAlso of	Graded ring
is Link from a Wikipage to another Wikipage of	Monster Lie algebra Representation theory of the Lorentz group Lie superalgebra Graded (mathematics) Lie algebra Frölicher–Nijenhuis bracket Hall word Graded ring Kac–Moody algebra List of Lie groups topics Hermitian symmetric space Free Lie algebra Brunnian link Nijenhuis–Richardson bracket

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