Category Theoretic Definition
Using the language of category theory, a Lie algebra can be defined as an object A in Veck, the category of vector spaces over a field k of characteristic not 2, together with a morphism : A ⊗ A → A, where ⊗ refers to the monoidal product of Veck, such that
where τ (a ⊗ b) := b ⊗ a and σ is the cyclic permutation braiding (id ⊗ τA,A) ° (τA,A ⊗ id). In diagrammatic form:
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