The Lie derivative with respect to a vector field is an "'R "'- derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold.
32.
We then have a natural transformations corresponding to the embedding of V into its tensor algebra, and a natural transformation corresponding to the map from T ( T ( V ) ) to T ( V ) obtained by simply expanding all tensor products.
33.
These two are sufficient to extend the notion of the Lie bracket to the entire tensor algebra, by appealing to a lemma : since the tensor algebra is a free algebra, any homomorphism on its generating set can be extended to the entire algebra.
34.
These two are sufficient to extend the notion of the Lie bracket to the entire tensor algebra, by appealing to a lemma : since the tensor algebra is a free algebra, any homomorphism on its generating set can be extended to the entire algebra.
35.
Here, the tensor product symbol ?" is used to indicate the tensor product used to define a coalgebra; it must not be confused with the tensor product ?", which is used to define the bilinear multiplication operator of the tensor algebra.
36.
There are other general examples, as well : it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article.
37.
Then contraction operates on the full ( mixed ) tensor algebra of " M " in exactly the same way as it does in the case of vector spaces over a field . ( The key fact is that the natural pairing is still perfect in this case .)
38.
It is being used to distinguish it from the " internal " tensor product \ otimes, which is already " taken " and being used to denote multiplication in the tensor algebra ( see the section " Multiplication ", below, for further clarification on this issue ).
39.
One can, in fact, define the tensor algebra " T " ( " V " ) as the unique algebra satisfying this property ( specifically, it is unique up to a unique isomorphism ), but one must still prove that an object satisfying this property exists.
40.
Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i . e . by constructing certain quotient algebras of " T " ( " V " ).
