Trivial Lie modules and Abelian Lie algebras #
The action of a Lie algebra
L on a module
M is trivial if
⁅x, m⁆ = 0 for all
x ∈ L and
m ∈ M. In the special case that
M = L with the adjoint action, triviality corresponds to the
concept of an Abelian Lie algebra.
In this file we define these concepts and provide some related definitions and results.
Main definitions #
lie algebra, abelian, commutative, center
A Lie (ring) module is trivial iff all brackets vanish.
The kernel of the action of a Lie algebra
L on a Lie module
M as a Lie ideal in
The largest submodule of a Lie module
M on which the Lie algebra
L acts trivially.
max_triv_submodule is functorial.
The maximal trivial submodules of Lie-equivalent Lie modules are Lie-equivalent.
A linear map between two Lie modules is a morphism of Lie modules iff the Lie algebra action on it is trivial.
The center of a Lie algebra is the set of elements that commute with everything. It can be viewed as the maximal trivial submodule of the Lie algebra as a Lie module over itself via the adjoint representation.