Repairing the Lie operad in characterstic 2?

Question

Recall: We present an operad (with $S_n$-action) in $R-Mod$ for any commutative ring $R$ not of characteristic 2 generated by a single element in degree $2$ satisfying the following identities:

• $\theta+\theta\tau=0$
• $\theta(1,\theta)+\theta(1,\theta)\sigma + \theta(1,\theta)\sigma^2$.

where $\tau$ and $\sigma$ are 2-cycles and 3-cycles respectively.

However, in characteristic $2$, this fails to characterize Lie algebras in the obvious way, since the first equation says that $\theta$ is skew-symmetric (and hence symmetric in characteristic 2).

The proper axiom to include is that $[x,x]=0$, i.e. that $[-,-]$ is alternating rather than skew-symmetric. Can we present this relation operadically? It seems like on the face of it, we can't, but I'd be happy to be surprised.

Answer 1

In fact, several monads can naturally be associated to an operad $P$ and this might be used to answer your question.

In the usual setting, one considers a generalized symmetric algebra $S(P,X) = \bigoplus_n (P(n)\otimes X^{\otimes n})_{\Sigma_n}$ where we form coinvariants under the action of the symmetric groups $\Sigma_n$. But we can also take invariants instead of coinvariants and form another functor $\Gamma(P,X) = \bigoplus_n (P(n)\otimes X^{\otimes n})^{\Sigma_n}$ associated to $P$. The image of the norm map from coinvariants to invariants still gives another functor $\Lambda(P,X)$ associated to $P$.

Under the assumption $P(0) = 0$, we have a monad structure on $\Lambda(P): X\mapsto\Lambda(P,X)$ and $\Gamma(P): X\mapsto\Gamma(P,X)$ inherited from the operadic composition structure of $P$. See (1.2.12-1.2.17) in http://math.univ-lille1.fr/~fresse/PartitionHomology.pdf (ref.: http://www.ams.org/mathscinet-getitem?mr=2005g:18015)

For the operad $P = Lie$, the algebra category associated to $\Gamma(Lie)$ can be identified with the category of $p$-restricted Lie algebras (where $p$ is the cateristic of the ground ring), while the algebra category associated to $\Lambda(Lie)$ can be identified with the category of Lie algebras equipped with an alternating Lie bracket.