Tensor product of algebras

Tensor product of a Lie algebra and a commutative algebra

Let be a Lie algebra and be a commutative algebra, both over . Then their tensor product algebra is also a Lie algebra. #m/thm/lie

Proof

That the product on is alternating follows immediately. For the Jacobi identity note

as required.

Note that if is unital, then we have the injective Lie algebra homomorphism

Functoriality

This construction forms a functor .

#state/tidy | #lang/en | #SemBr