Free Lie Algebra

A free Lie algebra on a set over a field is a Lie algebra with a map such that every map from to any -Lie algebra extends uniquely to a Lie algebra homomorphism from . It can be constructed from formal brackets in the generators by imposing bilinearity, alternation, and the Jacobi identity, with no additional relations. For one generator, it is one-dimensional and has zero bracket. For two generators and , the element need not vanish.

Assigning positive integer weights to the generators makes graded, with the weight of a bracket equal to the sum of its input weights. The Deligne-Drinfeld conjecture concerns freeness with one generator in each odd weight at least 3.