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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 17th 2010
   • (edited Oct 17th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexSo we can construct an operad in $R-Mod$ for any ring $R$ 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, since it also includes the following axiom: $[x,y]=-[y,x]=[y,x]$. The proper axiom to include is that $[x,x]=0$, i.e. that $[-,-]$ is alternating rather than skew-symmetric. Can we present this operadically? It seems like on the face of it, we can't, but I'd be happy to be surprised.

   So we can construct an operad in $R-Mod$ for any ring $R$ 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, since it also includes the following axiom: $[x,y]=-[y,x]=[y,x]$.

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

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeOct 17th 2010
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   Author: TobyBartels
   Format: MarkdownItexThis isn't just a problem in characteristic $2$ but for any ring $R$ in which $2$ is not invertible. (If you assume that $R$ is a field, then this reduces to characteristic $2$.) So for example, any counterexample to the claim that every model of the operad above is a Lie algebra over a field of characteristic $2$ (and what is the simplest such counterexample?) is also a counterexample to the claim that every model is a Lie algebra over $\mathbb{Z}$. Also, there's nothing *wrong* with the axiom $[x,y]=-[y,x]=[y,x]$ in this case; it's just *insufficient*. So we can be happy with any axiom that, together with antisymmetry (and the Jacobi identity), allows one to prove $[x,x] = 0$ regardless of whether $2$ is invertible. I don't know the answer to your question >Can we present this operadically? but I also suspect that the answer is no. Does anybody know the way to go about proving that something *can't* be phrased in operadic terms?

   This isn’t just a problem in characteristic $2$ but for any ring $R$ in which $2$ is not invertible. (If you assume that $R$ is a field, then this reduces to characteristic $2$.) So for example, any counterexample to the claim that every model of the operad above is a Lie algebra over a field of characteristic $2$ (and what is the simplest such counterexample?) is also a counterexample to the claim that every model is a Lie algebra over $\mathbb{Z}$.

   Also, there’s nothing wrong with the axiom $[x,y]=-[y,x]=[y,x]$ in this case; it’s just insufficient. So we can be happy with any axiom that, together with antisymmetry (and the Jacobi identity), allows one to prove $[x,x] = 0$ regardless of whether $2$ is invertible.

   I don’t know the answer to your question

   Can we present this operadically?

   but I also suspect that the answer is no.

   Does anybody know the way to go about proving that something can’t be phrased in operadic terms?