The universal enveloping algebra is a functor from Lie algebras to unital associative algebras, and is left adjoint to the functor which sends a unital associative algebra to a Lie algebra with bracket given by the commutator. Being a left adjoint, the universal enveloping algebra construction is obviously right exact, but is it left exact? It would be nice if it was, but I have a feeling it isn't. Unfortunately I don't know enough about Lie algebras to think of a counterexample.
2026-04-07 14:45:54.1775573154
Is the universal enveloping algebra functor exact?
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No; it already fails to preserve finite products. It sends a product of Lie algebras to a tensor product of algebras.