Lie Algebra - Category Theoretic Definition

Category Theoretic Definition

Using the language of category theory, a Lie algebra can be defined as an object A in Vec_k, 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 Vec_k, such that

where τ (a ⊗ b) := b ⊗ a and σ is the cyclic permutation braiding (id ⊗ τ_A,A) ° (τ_A,A ⊗ id). In diagrammatic form: