Tensor products of Lie modules #
Tensor products of Lie modules carry natural Lie module structures.
lie module, tensor product, universal property
It is useful to define the bracket via this auxiliary function so that we have a type-theoretic
expression of the fact that
L acts by linear endomorphisms. It simplifies the proofs in
The tensor product of two Lie modules is a Lie ring module.
The tensor product of two Lie modules is a Lie module.
The universal property for tensor product of modules of a Lie algebra: the
tensor-hom adjunction is equivariant with respect to the
A weaker form of the universal property for tensor product of modules of a Lie algebra.
Note that maps
f of type
M →ₗ⁅R,L⁆ N →ₗ[R] P are exactly those
R-bilinear maps satisfying
⁅x, f m n⁆ = f ⁅x, m⁆ n + f m ⁅x, n⁆ for all
x, m, n (see e.g,
A pair of Lie module morphisms
f : M → P and
g : N → Q, induce a Lie module morphism:
M ⊗ N → P ⊗ Q.
Given Lie submodules
M' ⊆ M and
N' ⊆ N, this is the natural map:
M' ⊗ N' → M ⊗ N.
The action of the Lie algebra on one of its modules, regarded as a morphism of Lie modules.
A useful alternative characterisation of Lie ideal operations on Lie submodules.
Given a Lie ideal
I ⊆ L and a Lie submodule
N ⊆ M, by tensoring the inclusion maps and then
applying the action of
M, we obtain morphism of Lie modules
f : I ⊗ N → L ⊗ M → M.
This lemma states that
⁅I, N⁆ = range f.