Repairing the Lie operad in characterstic 2? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:44:00Z http://mathoverflow.net/feeds/question/42508 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42508/repairing-the-lie-operad-in-characterstic-2 Repairing the Lie operad in characterstic 2? Harry Gindi 2010-10-17T14:54:56Z 2010-10-26T21:01:30Z <p>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:</p> <ul> <li>$\theta+\theta\tau=0$</li> <li>$\theta(1,\theta)+\theta(1,\theta)\sigma + \theta(1,\theta)\sigma^2$.</li> </ul> <p>where $\tau$ and $\sigma$ are 2-cycles and 3-cycles respectively. </p> <p>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). </p> <p>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. </p> http://mathoverflow.net/questions/42508/repairing-the-lie-operad-in-characterstic-2/43723#43723 Answer by Benoit Fresse for Repairing the Lie operad in characterstic 2? Benoit Fresse 2010-10-26T21:01:30Z 2010-10-26T21:01:30Z <p>In fact, several monads can naturally be associated to an operad $P$ and this might be used to answer your question.</p> <p>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 $P$.</p> <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 <a href="http://math.univ-lille1.fr/~fresse/PartitionHomology.pdf" rel="nofollow">http://math.univ-lille1.fr/~fresse/PartitionHomology.pdf</a> (ref.: <a href="http://www.ams.org/mathscinet-getitem?mr=2005g:18015" rel="nofollow">http://www.ams.org/mathscinet-getitem?mr=2005g:18015</a>)</p> <p>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.</p>