Lie operad

Example 12.1.6. For every field \(K\) there is an operad enriched in \(K\)-vector spaces called the Lie operad, denoted \(\Lie \), which captures the operations present in a Lie algebra. The vector space \(\Lie (n)\) is the subspace of the free Lie algebra \(L(x_1, \dots , x_n)\) spanned by the ‘multilinear’ elements: formal sums of iterated Lie brackets that use each variable \(x_i\) exactly once. Lie algebras are algebras over \(\Lie \) in the enriched sense; forgetting the enrichment gives an underlying set-valued operad.