Quotients of Lie algebras and Lie modules #
Given a Lie submodule of a Lie module, the quotient carries a natural Lie module structure. In the special case that the Lie module is the Lie algebra itself via the adjoint action, the submodule is a Lie ideal and the quotient carries a natural Lie algebra structure.
We define these quotient structures here. A notable omission at the time of writing (February 2021) is a statement and proof of the universal property of these quotients.
Main definitions #
Tags #
lie algebra, quotient
The quotient of a Lie module by a Lie submodule. It is a Lie module.
Map sending an element of M
to the corresponding element of M/N
, when N
is a
lie_submodule of the lie_module N
.
Instances For
Given a Lie module M
over a Lie algebra L
, together with a Lie submodule N ⊆ M
, there
is a natural linear map from L
to the endomorphisms of M
leaving N
invariant.
Instances For
Given a Lie module M
over a Lie algebra L
, together with a Lie submodule N ⊆ M
, there
is a natural Lie algebra morphism from L
to the linear endomorphism of the quotient M/N
.
Instances For
Given a Lie module M
over a Lie algebra L
, together with a Lie submodule N ⊆ M
, there is
a natural bracket action of L
on the quotient M/N
.
The quotient of a Lie module by a Lie submodule, is a Lie module.
LieSubmodule.Quotient.mk
as a LieModuleHom
.
Instances For
Two LieModuleHom
s from a quotient lie module are equal if their compositions with
LieSubmodule.Quotient.mk'
are equal.
See note [partially-applied ext lemmas].
The first isomorphism theorem for morphisms of Lie algebras.