# Differential and pre-differential of Jacobi identity

Let M be a manifold.

To what extent all Lie algebra structures with tensorial property on $$\chi^{\infty}(M)$$ are studied?

That is a Lie algebra structure for which $$[X,fY]=f[X,Y]$$.

(For every parallelizable $$n$$-manifold $$M$$ and every Lie algebra structure on $$\mathbb{R}^{n}$$, one can introduce a Lie algebra structure on $$\chi^{\infty}(M)$$ with this tensorial property. So what is an example of non parallelizable case?)

Any such structure on a manifold is a vector valued 2-form, that is an element of $$\Omega_{1,2}(M)$$. Such vector valued 2-forms are called Jacobi 2-form. The Jacoby identity on $$\Omega_{1,2}(M)$$ is denoted by $$I_{2}$$

What identities $$I_{1}$$ and $$I_{3}$$ can be defined on $$\Omega_{1,1}(M)$$ and $$\Omega_{1,3}(M)$$ such that they are invariant under the differentiation? That is: The differential of every $$I_{2}$$- form, satisfies $$I_{3}$$ or the differential of every $$I_{1}$$-form satisfy $$I_{2}$$(The Jacoby I dentity). Can one extend these (possible) identities for arbitrary $$k$$ on $$\Omega_{1,k}$$?(Then study the subcomplex and its cohomology of vector valued differential forms which satisfy $$I_{k}$$, for all $$k$$? May be Such cohomology can be named "Jacobi cohomology")

For compact manifolds we may pose some other questions as follows:

1. Are there some obstructions for existence of a Jacobi 2-form whose fibre wise structure is a simple Lie algebra?

2)Is there an example of a Jacobi 2-form such that we obtain different(perhaps infinit number of) Lie algebra structures on the tangent spaces?

• The condition on "tensorial Lie algebra structures" means that the structure comes from point-wise operations, i.e. your bundle is a locally trivial bundle of Lie algebras. I am not aware of a natural example of this situation, though there are cases in which the cotangent bundle $T^*M$ naturally is a bundle of Lie algebras. There is no natural notion of "differentiation" on vector valued forms unless you choose an additional structure there, e.g. a linear connection on $TM$. The main natural operation there is the Froelicher-Nijenhuis bracket. Jun 24, 2015 at 12:49
• @AndreasCap Thank you very much for your very interesting comment. As I wrote in the question, such tensorial Lie algebra struct exists, when M is parallelizable. As you said we need a connection. For example put $M=\mathbb{R}^{3}$ then the vector valued two form $P(x,y,z)(dy\wedge dz, dz\wedge dx, dx\wedge dy)$ satisfies the jacoby identity.It is the natural Lie algebra of $\mathbb{R}^{3}$. The differential of this two form(with flat connection) is the standard volum form muliplied with a gradient vector field. Jun 24, 2015 at 21:26
• As you said a manifold which fibres of the tangent bundle are equiped with a given(fixed)Lie algebra structure possess 2-forms which satisfies the jacoby identity. but it would be possible that a 2-form satisfies Jacoby identity but the structure of Lie algebras is not fixed, when the base point varies. I think that for vector bundle which fibres are matrix ALGEBRA, there is an invariant which is introduced by Douady. But I do not know whether there is a Lie algebra analogy? Any way, in my question, I do not fix a given lie algebra structure. I just consider 2-forms with Jacoby identity. Jun 24, 2015 at 21:37
• Finally, according to your comment, we need a connection. So we can associate to a Riemannian manifold(with LC connection), such type of cohomology(provided we can extent the Jacoby identity to higher order forms. Does this idea leads one to triviality? Thanks again for your comment. Jun 24, 2015 at 21:40
• @AndreasCap In line of this post one can ask the following particular question: Is there a vector valued 3-form $(\alpha, \beta, \gamma)$ on $\mathbb{R}^{3}$ which is not the differential of a jacobi 2-form? Jun 25, 2015 at 20:45