Editing Lie algebra (section)

==Definition using category-theoretic notation==
The definition of a Lie algebra can be reformulated more abstractly in the language of [[category theory]]. Namely, one can define a Lie algebra in terms of linear maps—that is, [[morphism]]s in the [[category of vector spaces]]—without considering individual elements. (In this section, the field over which the algebra is defined is assumed to be of characteristic different from 2.)

For the category-theoretic definition of Lie algebras, two [[tensor product#Tensor powers and braiding|braiding isomorphisms]] are needed. If {{mvar|A}} is a vector space, the ''interchange isomorphism'' <math>\tau: A\otimes A \to A\otimes A</math> is defined by
:<math>\tau(x\otimes y)= y\otimes x.</math>
The ''cyclic-permutation braiding'' <math>\sigma:A\otimes A\otimes A \to A\otimes A\otimes A </math> is defined as
:<math>\sigma=(\mathrm{id}\otimes \tau)\circ(\tau\otimes \mathrm{id}),</math>
where <math>\mathrm{id}</math> is the identity morphism. Equivalently, <math>\sigma</math> is defined by
:<math>\sigma(x\otimes y\otimes z)= y\otimes z\otimes x.</math>

With this notation, a Lie algebra can be defined as an object <math>A</math> in the category of vector spaces together with a morphism
:<math>[\cdot,\cdot]\colon A\otimes A\rightarrow A</math>
that satisfies the two morphism equalities
:<math>[\cdot,\cdot]\circ(\mathrm{id}+\tau)=0,</math>
and
:<math>[\cdot,\cdot]\circ ([\cdot,\cdot]\otimes \mathrm{id}) \circ (\mathrm{id}+\sigma+\sigma^2)=0.</math>