Repairing the Lie operad in characterstic 2?

2010-10-17T14:54:56Z

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 by Benoit Fresse for Repairing the Lie operad in characterstic 2?

2010-10-26T21:01:30Z

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.

Repairing the Lie operad in characterstic 2? - MathOverflow

Repairing the Lie operad in characterstic 2?

Answer by Benoit Fresse for Repairing the Lie operad in characterstic 2?