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quadratic Lie algebra
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(Definition)
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"quadratic Lie algebra" is owned by benjaminfjones.
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(view preamble)
See Also: quadratic algebra
| Also defines: |
quadratic Lie algebra |
| Keywords: |
quadratic, invariant scalar product, Lie algebra |
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Cross-references: scalar product, vector spaces, isomorphism, map, induced, isomorphic, adjoint, implies, invariant scalar product, non-degenerate, adjoint action, representation, Lie algebra
This is version 3 of quadratic Lie algebra, born on 2005-09-20, modified 2005-09-21.
Object id is 7379, canonical name is QuadraticLieAlgebra.
Accessed 1540 times total.
Classification:
| AMS MSC: | 17B01 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Identities, free Lie algebras) | | | 17B10 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Representations, algebraic theory ) |
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Pending Errata and Addenda
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![$ ( [X,Y] \mid Z) = -(Y \mid [X,Z] ) = (Y \mid [Z,X] )$](http://images.planetmath.org:8080/cache/objects/7379/l2h/img9.png "$ ( [X,Y] \mid Z) = -(Y \mid [X,Z] ) = (Y \mid [Z,X] )$"){kind=small}
![$\displaystyle \phi(ad_X(Y))(Z) = \phi([X,Y])(Z) = ( [X,Y] \mid Z ) = ( Y \mid [Z,X] ) = ad^*_X( \phi(Y)(Z) ) $](http://images.planetmath.org:8080/cache/objects/7379/l2h/img11.png "$\displaystyle \phi(ad_X(Y))(Z) = \phi([X,Y])(Z) = ( [X,Y] \mid Z ) = ( Y \mid [Z,X] ) = ad^*_X( \phi(Y)(Z) ) $"){kind=small}