Free Lie algebra
In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity. == Definition == The definition of the free Lie algebra generated by a set X is as follows: Let X be a set and i : X → L {\displaystyle i\colon X\to L} a morphism of sets (function) from X into a Lie algebra L. The Lie algebra L is called free on X if i {\displaystyle i} is the universal morphism; that is, if for any Lie algebra A with a morphism of sets f : X → A {\displaystyle f\colon X\to A} , there is a unique Lie algebra morphism g : L → A {\displaystyle g\colon L\to A} such that f = g ∘ i {\displaystyle f=g\circ i} .