Lie operad

In mathematics, the Lie operad is an operad whose algebras are Lie algebras. The notion (at least one version) was introduced by {{harvtxt|Ginzburg|Kapranov|1994}} in their formulation of Koszul duality.

Definition à la Ginzburg–Kapranov

Fix a base field k and let $\mathcal{Lie}(x_1, \dots, x_n)$ denote the free Lie algebra over k with generators $x_1, \dots, x_n$ and $\mathcal{Lie}(n) \subset \mathcal{Lie}(x_1, \dots, x_n)$ the subspace spanned by all the bracket monomials containing each $x_i$ exactly once. The symmetric group $S_n$ acts on $\mathcal{Lie}(x_1, \dots, x_n)$ by permutations of the generators and, under that action, $\mathcal{Lie}(n)$ is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, $\mathcal{Lie} = \{ \mathcal{Lie}(n) \}$ is an operad.{{harvnb|Ginzburg|Kapranov|1994|loc=§ 1.3.9.}}

Koszul-Dual

The Koszul-dual of $\mathcal{Lie}$ is the commutative-ring operad, an operad whose algebras are the commutative rings over k.

Notes

References

{{citation

| last1 = Ginzburg | first1 = Victor

| last2 = Kapranov | first2 = Mikhail

| doi = 10.1215/S0012-7094-94-07608-4

| issue = 1

| journal = Duke Mathematical Journal

| mr = 1301191

| pages = 203–272

| title = Koszul duality for operads

| volume = 76

| year = 1994}}

External links

Todd Trimble, [http://math.ucr.edu/home/baez/trimble/trimble_lie_operad.pdf Notes on operads and the Lie operad]
https://ncatlab.org/nlab/show/Lie+operad

Category:Abstract algebra

Category:Category theory