The construction of the bialgebra here parallels the construction in the tensor algebra article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.
22.
They give rise to a " tensor " only by identifications with objects of the tensor algebra-but likewise there are identifications with concepts in the Clifford-algebra.
23.
The free algebra on a set " X " is for practical purposes the same as the tensor algebra on the vector space with " X " as basis.
24.
Conversely, symmetric tensors are defined as invariants : given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially.
25.
There is a general relativity book-that appears to be about half-finished-but it looks too complex for 14-16 year olds because it goes into tensor algebra and such.
26.
All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point "'x " '.
27.
Another example is when T is the endofunctor on the category of vector spaces which maps a vector space V to its tensor algebra T ( V ), and which maps linear maps to their tensor product.
28.
Where the sum is taken over the symmetric group of permutations on the symbols This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T ( " V " ).
29.
One can then show that the Lie bracket can be consistently lifted to the entire tensor algebra : it obeys both the product rule, and the Jacobi identity of the Poisson bracket, and thus is the Poisson bracket, when lifted.
30.
The tensor product on the tensor algebra induces a product K _ m \ times K _ n \ rightarrow K _ { m + n } making K ^ M _ * ( F ) a graded ring which is graded-commutative.
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