Lie algebras of associative algebras #
This file defines the Lie algebra structure that arises on an associative algebra via the ring commutator.
Since the linear endomorphisms of a Lie algebra form an associative algebra, one can define the adjoint action as a morphism of Lie algebras from a Lie algebra to its linear endomorphisms. We make such a definition in this file.
Main definitions #
lie algebra, ring commutator, adjoint action
The bracket operation for rings is the ring commutator, which captures the extent to which a ring is commutative. It is identically zero exactly when the ring is commutative.
An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator.
An associative algebra gives rise to a Lie algebra by taking the bracket to be the ring commutator.
of_associative_algebra associating a Lie algebra to an associative algebra is
A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra.
A linear equivalence of two modules induces a Lie algebra equivalence of their endomorphisms.
An equivalence of associative algebras is an equivalence of associated Lie algebras.