Talk:Tensor algebra

From Wikipedia, the free encyclopedia

Mathematics Portal This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating:	Stub Class	Mid Priority	Field: Algebra
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Close to start but needs references, links, an explanation of the use of this fundamental construction in mathematics in general, and a more unifying approach throughout. Arcfrk 19:28, 28 June 2007 (UTC)

I removed the following statement:

To have the complete algebra of tensors, contravariant as well as covariant, one should take T(W) where W is the direct sum of V and its dual space - this will consist of all tensors T_I^J with upper indices J and lower indices I, in classical notation.

Classically, a tensor over V is an element of V⊗...⊗V⊗V^*⊗...⊗V^*; with the construction given above, one would also get mixed terms such as V⊗V^*⊗V. To get the true algebra of all classical tensors, one would have to impose relations so that the elements of V commute with those of V^*. 21:13, 5 Sep 2004 (UTC)

[edit] Ambiguous sentence about Quotients

This sentence may produce a misunderstanding:

Because of the generality of the tensor algebra, many other algebras of interest are constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain quotients of T(V). Examples of this are the exterior algebra, the symmetric algebra, other Schur functors, Clifford algebras and universal enveloping algebras.

From an historical standpoint, as far as I know, tensor algebra was defined by Ricci about 10 years after Clifford developed his algebra. In turn, 30 years before that, Grassmann had published his Extension Theory, which is the basis of both the modern exterior algebra (aka Grassmann algebra) and Clifford algebra. See http://modelingnts.la.asu.edu/html/evolution.html. The above inserted quoted text is ambiguous because the reader may interpret the words "are constructed by starting with" as related with the history of mathematics. Paolo.dL 16:13, 8 June 2007 (UTC)

Point taken, and I also feel that perhaps "are constructed" is a bit limiting (there are other constructions out there). How about "can be constructed"? Also, I will link quotients to make it clear that we mean quotient algebras rather than some other sort of quotient. Silly rabbit 16:21, 8 June 2007 (UTC)

Thank you, Silly Rabbit. My suggestion (changes in bold):

Because of the generality of the tensor algebra, many other algebras of interest can be constructed by ... Examples of this are the Grassmann algebra, the symmetric algebra, other Schur functors, Clifford algebras and universal enveloping algebras. However, historically tensor algebra was developed after Clifford algebra, which in turn was based on Grassmann's Extension theory.

The last sentence may need refinements. I am not an expert in this field, therefore I won't edit and I will leave the final decision to others. Paolo.dL 16:32, 8 June 2007 (UTC)