Lie algebras MOC

Direct product of Lie algebras

The direct product is the categorical product in Category of Lie algebras. Given two Lie algebras , their product consists of the direct product vector space¹ together with a bracket given by

Hence regarded as subspaces and commute and are ideals.

Internal direct product

Let be subalgebras such that and . Then is the internal direct product . Equivalently, with both ideals.

See also

• Semidirect product of Lie algebras, which generalizes the direct product

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Footnotes

1. for two (or by induction, finite) operands this is naturally isomorphic to the direct sum of vector spaces ↩