Tensor products allow us to build "linear" objects from "multilinear" ones. It can refer to: basic ones from linear algebra/module theory, or more sophisticated versions from differential/algebraic geometry (bundles/sheaves), functional analysis (Hilbert/Banach/locally convex spaces), or in their ...

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Non commutative tensor products / simple example?

In O'Neil's "Semi-Riemannian Geometry" it is stated that if $A$ is a $(r,0)$-tensor field and $B$ is $(0,s)$-tensor field then it is ""from the definition'' that $A \otimes B = B \otimes A$. I still ...
4
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1answer
119 views

O'Neil's problem 2.10 on Tensor Derivations. Literature on the subject.

I am having real trouble trying to understand a problem in O'Neil's "Semi-Riemannian Geometry" and I can't find much literature on the subject. I will expose the problem and I will be grateful to ...
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1answer
137 views

How can this linear map be injective?

Studying the Peter-Weyl theorem, I've come across the following linear maps: $\theta_E: E'\otimes E \rightarrow$ Hom$(E,E)$ Where $ E'\otimes E$ denotes the tensor product of a finite-dimensional ...
3
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2answers
161 views

Tensor Product of Submodules

In Keith Conrad's text on Tensor Products, he states on Page 8 of the text that for a fixed ring $R$ and $R$-modules $M$, $M'$, $N$ and $N'$, and maps $f:M\to M'$, $g:N\to N'$ that ...
2
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1answer
316 views

Universal Property of Tensor Algebra

I am trying to prove the Universal property of the Tensor Algebra $T(V)$, which states that given any unital associative algebra $\mathcal{A}$ and a linear transformation $\varphi:V\rightarrow ...
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2answers
137 views

Embedding a module into its quotient module

I've got a very basic question on tensor products. Let $R$ be a commutative integral domain, $K$ its quotient field and let $M$ be a $R$-module. Is the map $M \rightarrow K\otimes_R M$ given by ...
3
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0answers
71 views

What is $H^1([0,1]) \otimes H^1([0,1])$?

Let $H^1([0,1])$ denote the Sobolev space $H^1$ on the interval $[0,1]$. What is $H^1([0,1]) \otimes H^1([0,1])$? Here, $\otimes$ the tensor product of Hilbert spaces. In particular, how is that ...
3
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1answer
112 views

Characterization of Linear Transformations between Tensor Products

I am seeking to show that $$\mathcal{L}(V\otimes X,W\otimes Y) \cong \mathcal{L}(V,W)\otimes \mathcal{L}(X,Y)$$ where $\mathcal{L}(V,W)$ is the space of linear transformations from $V$ into $W$. To ...
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1answer
101 views

Equivalence of tensor reps & tensor products of reps

Let a finite-dimensional vector space $V$ over $\mathbb R$ or $\mathbb C$ with dual $V^*$ and a group $G$ be given. Let $\rho:G\to\mathrm{GL}(V)$ be a representation, and let $T_kV$ and $V^{\otimes ...
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1answer
51 views

Divergenceless proving

We can define a topological current, \begin{equation} J_{top}^u = \frac{1}{2v} \epsilon^{\mu \nu} \partial_\nu \phi \end{equation} How to prove it divergenceless? where the the condition is ...
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1answer
2k views

Divergence of stress tensor in momentum transfer equation

Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of $\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u}) I ...
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1answer
224 views

Understanding the right-exactness of the tensor product using *only* its universal property and the Yoneda lemma

I would like to get an intuition for why $(-)\otimes N$ is right-exact using its universal property involving bilinear maps, not by appealing to higher-level observations such as "left-adjoints ...
0
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2answers
54 views

Do $\operatorname{Hom}( - , R)$ and $ - \otimes_R B$ commute when applied to $A\cong R^d?$

Let $R$ be a commutative ring with identity. Let $A$ and $B$ are $R$-modules, and further suppose that $A$ is free with finite rank. Is it true that $$ \operatorname{Hom} (A \otimes_R B , R) \cong ...
0
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1answer
51 views

Determinant of symmetric matrix of the form $v\otimes v$

Note that for $V=\mathbf{R}^n$, $$S^2V = \{ v\otimes w \mid v, w\in V\text{ and }v\otimes w=w\otimes v \} =\{ A\in \mathrm{M}_2(\mathbf{R}) \mid A=A^T \}.$$ Clearly, $S^2V $ contains $O=\{ v\otimes v ...
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0answers
95 views

Problem on non abelian tensor product of groups

Let $G$ be group and $x\otimes[y,z] =([x,y]\otimes z)^2$, for any $x,y,z\in G$. Then prove $$[x,y]\otimes y=1_{\otimes},$$ for any $x,y\in G$. This is a part of Corollary 4 in page 82 in the article ...
4
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2answers
457 views

Dimension of $\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Q}$ as a vector space over $\mathbb{Q}$

The following problem was subject of examination that was taken place in June. The document is here. Problem 1 states: The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector ...
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2answers
95 views

Universal property definition from Greub's Multilinear Algebra

Hi I started studying Greub's multilinear algebra book and I found something very strange when he defines the tensor product of two vector spaces: He defines: [...] Let $E$ and $F$ be vector spaces ...
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0answers
585 views

What are these physicists talking about? Dyadic green function?

I am interested in a mathematical explanation(in the sense that you say for example: is it a mapping from A to B) what a dyadic green function and the unit dyad actually is? I am reading this as ...
4
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2answers
244 views

Lie Derivative for Wedge Product of Vector Fields

I am having trouble here. The context is: Let $X$, $Y$ and $S$ be vector fields ina a manifold (we can assume it's $\mathbb{C}^2$ though I'm pretty sure this should work in any manifold), and we can ...
4
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1answer
187 views

Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?

I have been doing some review with the goal of trying to understand as much as I can via universal properties and category theory (already feeling comfortable with the mundane way of doing things). ...
1
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1answer
85 views

Facts about the tensor product

I'm proving some statements about density operators, and would like to use two things. The problem is, that I'm not entirely sure they hold. Let $V,\; W$ be finite dimensional, complex inner product ...
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2answers
456 views

Quick question: tensor product and dual of vector space

Recall that for a finite dimensional vector space $V$ we have the natural isomorphism $\phi :V^{*} \otimes V \rightarrow Hom(V,V)$ given by $\alpha \otimes v \mapsto (x \mapsto \alpha (x)v)$. Is ...
2
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2answers
178 views

$A/ I \otimes_A A/J \cong A/(I+J)$

For commutative ring with unit $A$, ideals $I, J$ it holds $$A/ I \otimes_A A/J \cong A/(I+J).$$ A proof can be found here (Problem 10.4.16) for example. However, I'd be interested in a less ...
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1answer
75 views

Second Order Tensor power [closed]

Suppose $V$ is a vector space and $T(V)$ the tensor algebra of $V$. How can I prove $T(T(V))\simeq T(V)$ ?
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58 views

Higher Tensor powers of graded vector spaces

Suppose $V$ is a $\mathbb{Z}$-graded vector space and $T(V):=\oplus_{j\in \mathbb{Z}}\oplus_{p+q=j}V_p\otimes V_q$ the graded tensor power.(As a vector space ) 1.) Is then $T(T(V))\simeq T(V)$? 2.) ...
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1answer
103 views

Show mapping involving tensor product is well defined.

Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota : N \to S \otimes_RN$ be the $R$-module homomorphism defined by $\iota(n) = 1 \otimes n$. Suppose that $L$ is any left ...
2
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1answer
129 views

Is the tensor product of 2 free Abelian groups free?

Ok, basically, I think that is true. Let's consider $A$, and $B$ are both free $\mathbb{Z}-$modules, i.e $A = \bigoplus_{i \in I} \mathbb{Z}$, and $B = \bigoplus_{j \in J} \mathbb{Z}$, so we'll have: ...
3
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1answer
201 views

“Tensor product” $\otimes$ of monoids

Referring to the top answer in this post: http://mathoverflow.net/questions/19004/is-the-category-commutative-monoids-cartesian-closed Would it be reasonable to explain what is a tensor product in ...
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1answer
183 views

Is tensor product of Sobolev spaces dense?

My question is: is $W_2^k(\mathbb{R})\otimes W_2^k(\mathbb{R})$ dense in $W_2^k(\mathbb{R}^2)$, and more generally is this true in $\mathbb{R}^d$? I found this post: Tensor products of functions ...
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0answers
138 views

Forming the tensor product of a `real' vector space with a 'complex' vector space.

I have a question that I am hoping someone could clarify for me. Context: Consider the algebra $A = (B,\circ)$, given by: \begin{align} B = \{ \begin{pmatrix} a & f\\ \overline{f} & ...
4
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5answers
107 views

Direction of map between tensor products

Suppose $A$ and $B$ are commutative rings, $A\to B$ is a ring map, and $M, N$ are $B$-modules. Is there a map $M\otimes_A N \to M\otimes_B N$, or in the other direction? This should be very ...
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1answer
84 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
5
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1answer
958 views

Proving that Tensor Product is Associative

I want to show that $X\otimes(Y\otimes Z)$ is isomorphic to $(X\otimes Y)\otimes Z$. Intuitively I think I should just choose bases $\{e_{i}\}_{i\in I}, \{f_{j}\}_{j\in J}$, and $\{g_{k}\}_{k\in K}$ ...
0
votes
1answer
428 views

Double dot product of two tensors [duplicate]

I have a problem that makes me very confused... I have two tensors that must be multiply. A is second order tensor and B is fourth order tensor. I know when multiplying two tensor with double dot ...
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1answer
57 views

What does it mean to have an $L$-basis of $L\otimes_K V$?

Exercise: I have got a vector space $V$ over $K$, and $L$ a field extension of $K$. The task is to show that if $(v_1, ..., v_n)$ is a basis of $V$, then $(1\otimes_K v_1, ... ,1\otimes_K v_n)$ is an ...
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1answer
1k views

Tensor double dot product

I have two tensors that i must calculate double dot product. matrix A is rank 2 and matrix B is rank 4. I want to multiply them with Matlab and I know in Matlab it becomes: A : B = trace (A*B) but it ...
0
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1answer
64 views

Suficient condition for tensor product of vector spaces..

Can anyone help me showing the following: Let $E$, $F$, $G$ and $H$ vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. If for every $\psi:E\times F\rightarrow H$ bilinear there is an ...
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3answers
280 views

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces? I am having a very hard time at digesting tensor products ...
2
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1answer
82 views

Show that $(B(E\times E), \otimes)$ is a tensor product of $E^*\times E^*$

Show that $(B(E\times E), \otimes)$ is a tensor product of $E^*\times E^*$ , where $E$ is a real finite dimensional vector space , $E^*$ is its dual space, $B(E\times E)$ is the space of all bilinears ...
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57 views

Tensor field notation

Let $L$ be a finite dimensional vector space. A tensor of type $(p,q)$ on $L$ is an element of the tensor product $L^{\otimes q}\otimes (L^{*})^{\otimes p}$. How to interpret the following formulation ...
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For covariant tensors, why is it $\bigwedge^k(V)$, not $\bigwedge^k(V^*)$?

In learning the very basics of differential geometry, I have seen the exterior product defined a couple of ways: First, I have seen it as the image of the covariant tensors (which I believe are ...
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1answer
250 views

Infinite tensor product definition

My question is short and popular: how to define an infinite tensor product of modules over a ring? So, there is an infinite set $I$ and $A$-modules $M_i$. I should understand what $\otimes_{i\in I} ...
3
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1answer
158 views

Alternating tensors vs $p$-vectors

Is there a reason to differentiate between alternating tensors and $p$-vectors? More precisely, is the exterior algebra always isomorphic to the subalgebra of alternating tensors? Thanks.
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Tensor Product is associative, distributive, not commutative.

Tensor Product is associative, distributive, not commutative. Here is my attempt to show tensor product is associative, is it legit? If $T$ is a $p$-tensor and $S$ a $q$ tensor, then $T \otimes ...
3
votes
1answer
103 views

Atiyah-Macdonald, Proposition 2.12, uniqueness of the tensor product.

The following is a result from Atiyah-Macdonald, defining and showing existence and uniqueness of tensor product of modules over a commutative ring. Proposition 2.12. Let $M, N$ be $A$-modules. ...
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1answer
104 views

The definition of addition on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. Construct the tensor product of $H_1$ and ...
2
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1answer
187 views

Multiplying vectors (answered own question)

I recently realised that asking a question and answering our own question is allowed here, so here is a question I've seen commonly on many sites: "How does one multiply two vectors?" This is very ...
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1answer
36 views

Show the $\mathbb{C} S_3$-module of dimension 2 has $S(V \otimes V)$ is not irreducible

Consider the $\mathbb{C} S_3$-module of dimension 2, call it $V$. I want to concretely show that $S(V \otimes V)$ is not irreducible. I found a representation for $S_3$ over $\mathbb{C}$ of degree ...
2
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1answer
112 views

Isomorphism between $\mathbb{C} X \otimes \mathbb{C} X$ and $\mathbb{C} (X \times X)$

Let $G$ be a finite group and $X$ a finite set. Denote by $\mathbb{C}X$ the vector space of functions from $X$ to $\mathbb{C}$. In a book I found the following statement: $\varphi : \mathbb{C} X ...
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3answers
442 views

My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?

Ok, I'm reading some thesis of some former students, and come up with this proof, but it doesn't really look good to me. So I guess it should be wrong somewhere. So, here it goes: Let $R$ be a ...