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.
We can regard a module over an associative ring
A as a Lie ring module over
A with Lie
bracket equal to its ring commutator.
Note that this cannot be a global instance because it would create a diamond when
M = A,
specifically we can build two mathematically-different
has_bracket A As:
@ring.has_bracket A _which says
⁅a, b⁆ = a * b - b * a
(@lie_ring_module.of_associative_module A _ A _ _).to_has_bracketwhich says
⁅a, b⁆ = a • b(and thus
⁅a, b⁆ = a * b)
An associative algebra gives rise to a Lie algebra by taking the bracket to be the ring commutator.
A representation of an associative algebra
A is also a representation of
A, regarded as a
Lie algebra via the ring commutator.
See the comment at
lie_ring_module.of_associative_module for why the possibility
M = A means
this cannot be a global instance.
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.