Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
  
  
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft type type-theory universal variational-calculus

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 for any ring generated by a single element in degree satisfying the following identities:

   • .

   where and are 2-cycles and 3-cycles respectively.

   However, in characteristic , this fails to characterize Lie algebras, since it also includes the following axiom: .

   The proper axiom to include is that , 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
   • PermaLink
   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 but for any ring in which is not invertible. (If you assume that is a field, then this reduces to characteristic .) So for example, any counterexample to the claim that every model of the operad above is a Lie algebra over a field of characteristic (and what is the simplest such counterexample?) is also a counterexample to the claim that every model is a Lie algebra over .

   Also, there’s nothing wrong with the axiom 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 regardless of whether 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?