Talk:Tensor algebra

Untitled

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)

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- and Weyl 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 (= exterior)- and symmetric algebras, Clifford- and Weyl algebras, universal enveloping algebras and Schur functors. 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)

Fock space

Not even a remark on the link with Fock space? Link or should I say that it is identical? or maybe specialized to Hilbert spaces

In physics, Fock spaces describe states with undetermined number of particles, it is really surprising that the same construction appears. What is the reason?? — Preceding unsigned comment added by Noix07 (talk • contribs) 18:27, 15 February 2014 (UTC)

A Fock space is a kind of tensor algebra with extra conditions: the vector space is a Hilbert space and it is a direct sum of symmetrized or antisymmetrized tensor products of the Hilbert spaces. So they aren't identical. Would be worth a mention as an example. --Mark viking (talk) 19:16, 15 February 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Tensor algebra/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

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)

Last edited at 19:28, 28 June 2007 (UTC). Substituted at 02:38, 5 May 2016 (UTC)

Bialgebra/Hopf algebr definitions can't be right.

The sections on the bialgebras cannot possible be right, as currently written. I can see glimmers of correctness, but... anyway, I can't reconstruct the correct definitions from memory and I don't have a reference. Here's one problem, this definition:

$\Delta (v_{1}\otimes \dots \otimes v_{m}):=\sum _{i=0}^{m}(v_{1}\otimes \dots \otimes v_{i})\otimes (v_{i+1}\otimes \dots \otimes v_{m})$

what does it mean for the index to from from i=0 ? Is the definition trying to define v₀ to be 1 and ... !? Huh? Changing the index to i=1 just makes the right side equal to m times the left side, which is pointless. What was the idea here?

For m=1, I expect to get Δ : V → V ⊗ V = T²V, yeah? I don't seem how to plug in m=1 into the above, unless I use that weird v₀=1 trick which is maybe OK, but seems to fail for m=2...

Next, this definition also has problems:

$\Delta (x_{1}\otimes \dots \otimes x_{m})=\sum _{p=0}^{m}\sum _{\sigma \in \mathrm {Sh} _{p,m-p}}\left(x_{\sigma (1)}\otimes \dots \otimes x_{\sigma (p)}\right)\otimes \left(x_{\sigma (p+1)}\otimes \dots \otimes x_{\sigma (m)}\right)$

Again there is the bit with the sum p=0? Is it trying to say that $x_{\sigma (0)}=1$??? so as to recover the m=1 case as

$\Delta (x)=x\otimes 1+1\otimes x$

That almost makes sense, but the m=2 case can't be right, the dimensions are wrong.

This also doesn't make sense:

$\Delta (x_{1}\otimes \dots \otimes x_{m})=\Delta (x_{1})\Delta (x_{2})\cdots \Delta (x_{m}).$

lets examine the m=2 case:

$\Delta (x_{1}\otimes x_{2})=\Delta (x_{1})\Delta (x_{2}).$

what does the right hand even mean? Does it mean this?

$\Delta (x_{1}\otimes x_{2})=\Delta (x_{1})\otimes \Delta (x_{2})?$

that cannot possibly be correct, it completely fails to satisfy the connecting axiom for a bialgebra, you have to swap the terms around, as you multiply. This is one of the basic axioms of a bialgebra, -- the very first one on the bialgebra page, you can't just blow it off.

I imagine there's some formula that looks kind-of-like this, but this is not it.67.198.37.16 (talk) 04:20, 18 September 2016 (UTC)

I'm blanking that entire section. As far as I can tell, its non-sense. If someone cares, they can go back to this date, and see what it said. 67.198.37.16 (talk) 05:24, 18 September 2016 (UTC)