The Pantheon of Derivatives - Lie Derivatives And Others (IV)

[fresh_42]

Lie Derivatives​

A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications since a tensor field includes a variety of instances, e.g. vectors, functions, or differential forms. In the case of vector fields, we additionally get a Lie algebra structure. This is, although formulated in a modern language, the actual reason why Lie algebras have been considered in the first place: as the tangent bundle of Lie groups which are themselves the invariants that appear as symmetry groups in the standard model of particle physics or more generally in the famous theorem of Emmy Noether, which is actually a theorem about invariants of differential equations (see [9],[10]). The Jacobi identity, e.g., which together with anti-commutativity defines a Lie algebra is simply a manifestation of the Leibniz rule of differentiation.

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[zwierz]

The definition of the Lie algebra $g$ of a Lie group $G$ is strange (near formula (18)). It looks like the operaton $[\cdot,\cdot]$ defined in $g$ does not have any relation to the group $G$.

Actually one of the standard definitions is as follows. Consider two vectors $a,b\in g$. And let $L_a:G\to G$ be the left shift. Then one can construct vector fields $$A(x)=(dL_x)a,\quad B(x)= (dL_x)b$$ Let $C(x)$ be the commutator of the vector fields $A(x),B(x)$. Then by definition $[a,b]=C(e)$, here $e$ is the identical element.

It would be good to note that equations (40),(41) determine $\nabla_X$ uniquely

 

[fresh_42]

That's right and I removed "on G" as the intention was to define a Lie algebra without a (unnecessary) reference to a group, which already happened prior to the formal definition.

 

[zwierz]

Formula (42) does not follow from the sentence above. The operator $\nabla_X$ has been previously defined on the vector fields. Thus the left side of this formula is indefinite. Actually this formula is a definition of the operator $\nabla_X$ on tensors of type (2,0). The fact that the manifold is Riemann as well as the dimension of the manifold have not relation to this formula. One can put it by definition for any connection on any manifold.
Accept the following axiom: $$\nabla _X\langle f,u\rangle=\langle \nabla _X f,u\rangle+\langle f, \nabla _X u\rangle,$$
for any vector fields $u,X$ and for any covector field $f$. This axiom
defines the operator $\nabla _X$ for covector fields and then formula similar to (42) extends the operator $\nabla _X$ to tensor fields of any type $(p,q)$.

ps In (42) the symbol $\times$ is tensor product I guess. $\otimes$ is commonly used

 

[fresh_42]

ps In (42) the symbol ××\times is tensor product I guess. ⊗⊗\otimes is commonly used

Nope, it's said and meant to be the cross-product.

 

[zwierz]

then what is cross product of vector fields?

upd: o, I see! ok