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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Characters of Representations, Composition Series and Tensor Products

Let $(\pi, V)$ be a finite-dimensional representation of $G$. Prove the following: Suppose that $(\pi, V)$ has as a composition series $\{0\} \subset V_{1} \subset \dots \subset V_{r}=V$ with the ...
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Representing a linear operator on $V$ with an element of $V \otimes V^*$

I got interested by the first sentence of this wikipedia subsection. It claims that any linear operator $f:V\to V$ can be represented by an element of $V\otimes V^*$ in a very concrete way: the ...
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42 views

Tensor of cocomplete categories

Let $C$, $D$ and $E$ be cocomplete categories. Is there a construction $C \otimes D$ such that there is a correspondence between functors $C \otimes D \to E$ preserving colimits and functors $C ...
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53 views

Basis for Tensor products/bilinear maps

What is the delta in the image? How do I check it is a basis? Why are $i$ and $k$ the same, and $j$ and $l$? Need some clarification.
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49 views

Tensor product of Hom-module and another ring

Let $A$ be a local noetherian ring, $B$ and $C$ are finitely generated $A$-algebras and $M$ is a finitely generated $B$-module. Is the natural morphism $\mathrm{Hom}_B(M,B) \otimes_A C \to ...
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Express the operator of a tensor product by a form containing an inner product term

Assume $A,X,Y\in \mathbb{R}^{I_1\times\cdots\times I_N}$ are three arbitrary N-th order tensors. How to prove the following equation: $$\langle A,X\rangle Y=(X\otimes Y)(A),$$ where $\langle\cdot ...
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1answer
26 views

How to Prove $V\otimes sl(k)=sl(V)$?

Let V be a vector space over a field $k$. Let $sl(n)$ be the set of all matrices elements from $k$ with trace zero. Is it true that $V\otimes _k sl(n)=sl(V)= \text{set of all $n\times$ n matrices ...
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1answer
33 views

Tensor Product on Hilbert Spaces (well definedness)

Let $H_1$, $H_2$,...,$H_n$ be $n$ Hilbert Spaces. For each $\phi_i \in H_i$, Let $$ \phi_1 \otimes \phi_2 \otimes... \otimes \phi_n:= \text{Conjugate multilinear form which acts on $H_1 \times H_2.. ...
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27 views

how to prove an element is non-zero in a tensor-product

I was studying the following example from Atiyah & MacDonald's Introduction to Commutative Algebra: let $x$ be the non-zero element in $N := \mathbf{Z}/ 2\mathbf{Z}$, $M := \mathbf{Z}$, and $M' := ...
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31 views

Associativity of tensor product over various rings

From Atiyah-MacDonald: Exercise 2.15. Let $A$, $B$ be rings, let $M$ be an $A$-module, $P$ a $B$-module and $N$ an $(A,B)$-bimodule (that is, $N$ is simultaneously an $A$-module and a $B$-module ...
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tensor identity for cross product

I've read somewhere the following identity for a tensor rank 2 $ \nabla \times \nabla v =0 $ where $v$ is a vector of "j" components and $\nabla = \frac{\partial}{\partial x_i}$, such that $ \nabla ...
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$R[S]\cong \mathbb{Z}[S]\otimes_{\mathbb{Z}} R$, $R\otimes_{\mathbb{Z}} \mathbb{Z}\cong R?$

I don't know much about tensor products in general but for for some lectures I need basics about tensor products of rings, modules and abelian groups. I have the following questions: 1)Let $R$ be a ...
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1answer
62 views

different approaches of defining tensors

This Wikipedia article says that tensor can be defined as miltilinear maps or be defined using tensor products. Could anybody explain with a simple example why these two approaches give the same ...
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27 views

Universal Linearizer of Alternating Multi-$F[x]$-Linear Maps is Same as that of Multi-$F$-Linear Maps.

Let $V$ be a an $n$-dimensional vector space over a field $F$. Let $M=F[x]\otimes_F V$. We can consider $M$ as an $F[x]$-module by extending scalars using the inclusion $F\to F[x]$. Fact 1. There is ...
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30 views

Dual of a module and base change [closed]

Let $\phi:A \to B$ be a ring homomorphism and $M$ an $A$-module (not necessarily free). Is it true that $$\mathrm{Hom}_A(M,A) \otimes_A B \cong \mathrm{Hom}_B(M \otimes_A B,B)?$$ If necessary assume ...
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34 views

reduced density matrix for the given composite system

Given the composite system of two qubits $$ |\psi^{AB}\rangle=\frac{1}{\sqrt{2}}(|0^{A}\rangle \otimes|0^{B}\rangle+|1^{A}\rangle\otimes|1^{B}\rangle) $$ with the density matrix of the composite ...
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32 views

$V(1)$ generates the tensor category of representations of $sl_2(\mathbb{C})$ - what exactly does this mean?

Does anyone know what it means for a $V(1)$ to generate the tensor category of (finite dimensional) representations of $sl_2(\mathbb{C})$. Here $V(1)$ is the 2 dimensional irreducible Lie algebra ...
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1answer
22 views

Elements of tensor product

Let $R, S$ be rings such that $R$ is a subring of $S$ and $1_R = 1_S$. Let $N$ be a left $R$-module. Let the free $\mathbb{Z}$-module on $S \times N$ be $F_\mathbb{Z}(S \times N)$. Let $H$ be the ...
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What are the non-vanishing directions of $A_{a(b} B_{cd)} - B_{a(b} A_{cd)}$? ($A$ and $B$ symmetric tensors)

Question Consider two symmetric tensors $A_{ab}=A_{ba},\, B_{ab}=B_{ba}$ which are generally of full rank (non-degenerate, all eigenvalues non-zero but of general sign) and their tensor product under ...
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23 views

Tensor product with irreducible representation has no $G$-invariant submodules

Let $\rho: G \to GL(V)$ be a finite dimensional irreducible representation of a group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and let $R$ be a commutative ring with ...
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34 views

What kind of operation is available here?

What kind of operation $\clubsuit$ is available here? $$ \underbrace{\left(v^{\mathrm T}A_{n\times n\times n}\right)}_{\in\mathbb R^{n\times n}}\underbrace{\left(v^{\mathrm T}B_{n\times n\times ...
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Books to learn tensor product on hilbert spaces

I have just started to work on Quantum Computing. I have began to read a paper which deals with tensor product on hilbert spaces. I have a had a course in functional analysis. So I don't have an ...
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80 views

If $M \otimes N \cong M' \otimes N$, is it true that $M \cong M'$? [duplicate]

I tried using the universal property of tensor products to show that there are mutually inverse maps from $M \times N$ to $M' \times N$, and use this to show that $M \cong M'$, but I didn't get far. I ...
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3answers
50 views

Showing that a ring homomorphism induces a correspondence that maps free modules to free modules

Let $f: \Lambda \rightarrow \Lambda'$ be a ring homomorphism. This induces a mapping $M \mapsto \Lambda' \otimes_{\Lambda} M$, where $M$ is a $\Lambda$-module and $\Lambda' \otimes_{\Lambda} M$ is a ...
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30 views

Cotangent bundle tensor product tangent bundle

What is the meaning of Cotangent bundle tensor product tangent bundle: $T^*M\otimes TM$? what will an element of this space be?
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47 views

Understanding of an example of “extending scalars”

The following is an example in the Abstract Algebra by Dummit and Foote: I don't understand in this example why $\iota$ is an isomorphism. By Theorem 8, I can get $$ id_N=\Phi\circ\iota $$ which ...
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1answer
85 views

$A\otimes_{\mathbb C}B$ is finitely generated as a $\mathbb C$-algebra. Does this imply that $A$ and $B$ are finitely generated?

Consider $A$ and $B$ two $\mathbb C$-algebras such that $A\otimes_{\mathbb C}B$ is finitely generated as a $\mathbb C$-algebra. Does this imply that $A$ and $B$ are finitely generated? I know that ...
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Checking if an element of tensor product is zero

I am trying to understand tensor product. Let $R$ be a ring and $R_0$ its subring. Then $R$ is a right $R_0$-module. Let $M$ be a left $R_0$-module. Is the element $1\otimes m$ of $R\otimes_{R_0} ...
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45 views

a property of the tensor product of modules

The following theorem is from the Abstract Algebra by Dummit and Foote (in the section 10.4 tensor products of modules): Would anybody illustrate how Theorem 8 is used to get $$ \textrm{ker ...
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1answer
30 views

Multiplication of tensor and vector

How to do the multiplication of the multidimensional array $A_{n\times n\times n}$ and the vector $v_n$ (indices denote dimensions)? Can you kindly give suggestions or references? Thanks in advance.
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1answer
31 views

Functor right adjoint to $.\otimes_BA$

Given a ring morphism from $B$ to $A$, we can regard an $A$-module $M$ as a $B$-module. Then how can I prove the functor $._B:\operatorname{Mod}_A \to \operatorname{Mod}_B$ is right adjoint to ...
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Does a basis for $V \otimes_{\mathbb{F}} W$ always consist of pure tensors?

Given a field $\mathbb{F}$ and two $F$-vector spaces $V$ and $W$, it's true that if $\{v_i\}$ and $\{w_j\}$ are bases for $V$ and $W$, respectively, then the set $\{v_i \otimes w_j\}$ is a basis for ...
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2answers
26 views

Notation for equal via isomorphism

I am working with tensor products and there are a lot of identities, etc. that are true after appropriate identifications (between tensor products) are made. For example, if $V$ is a vector space, ...
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58 views

Show that $a \otimes 1$ will vanish in $A\otimes_{\Bbb{Z}}\Bbb{Q}$ only when $a=0$.

Let $A$ be a $\mathbb{Z}$-module. I want to see that if $a \otimes 1=0$ in $A\otimes_{\Bbb{Z}}\Bbb{Q}$, then $a=0$. I'll assume $A$ is torsion free. Can I use that there exists an injective ...
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about the the tensor product

Because I'm a beginner in those stuffs I find some difficulties in the following (Using the universal property of the tensor product) prove that $\mathbb {Q}\otimes \mathbb{Q} \cong \mathbb{Q}$ ...
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1answer
35 views

What is considered to be the natural (injective) homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$?

Let $R$ be a ring and $I,J,L \unlhd R$ such that $J \subseteq I$. What is considered to be the natural homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$ ? Remark: It must be ...
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1answer
60 views

Induction functor from $\mathbb{K}$-mod to ($\mathbb{K}\times\mathbb{K}$)-mod

Let $\mathbb{K}$ a field. Given a $\mathbb{K}$-mod (a vector space) there is an induction functor on $\mathbb{K} \times \mathbb{K}$-mod that is, as usual, $- \otimes_\mathbb{K} (\mathbb{K} \times ...
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1answer
78 views

When considering vector spaces $A$ and $B$, what is the difference between $A \times B,$ $A \otimes B$ and $A \wedge B?$

I have looked at this resource http://hitoshi.berkeley.edu/221a/tensorproduct.pdf to instinctively differentiate between the tensor product and the direct sum of two vector spaces. I am currently ...
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19 views

Product of a matrix and a tensor

I need to know how to compute the following product: $M(x)\frac{\partial M(x)}{\partial x}M(x)$ $\quad$ where $x \in R^{n}$. Assuming the dimensions of the matrices are compatible,how do we take ...
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27 views

tensor product of a number field with $ \mathbb R $

Let $K$ be a number field, i.e. a finite extension of the field of rational numbers $ \mathbb Q .$ And consider the tensor product: $$ K \otimes_{\mathbb Q } \mathbb R $$ I have the questions: ...
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1answer
38 views

Confusion regarding notation of a dual transformation

I'm reading Spivak's Calculus on Manifolds and in Chapter 4 he defines the dual transformation (although he doesn't call it that) as follows: If $f:V \rightarrow W$ is a linear transformation, a ...
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1answer
28 views

Does the tensor product distribute over a direct sum exactly if all the involved modules are bimodules?

Given a non-commutative ring $R$, a right module $M$, and a left module $N$, we can define the tensor product $M \otimes_R N$. I suspect that $$ \left(\bigoplus_{i=1}^n M_i\right) \otimes_R N \simeq ...
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1answer
43 views

Short exact sequences of quasi-coherent sheaves and closed subschemes

I am confused by this exercise in Ravi's notes: 16.3F (paraphrasing) Suppose $i : p \to A_k^1$ is the inclusion of the origin. Consider the associated short exact sequence of quasi-coherent modules: ...
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1answer
76 views

Tensor product of $\mathbb{Q}$ with an infinite product [closed]

How can I prove that the tensor $\mathbb{Q} \otimes \left( \prod_n \mathbb{Z}/n\mathbb{Z} \right)$, where the product is taken over all the positive integers $n$, is not trivial?
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1answer
42 views

Decomposing a tensor product space into direct sums

I'm trying to understand how to decompose certain symmetric and anti-symmetric tensor products of vector spaces into direct summands. Let $V$ be a complex finite dimensional vector space and denote ...
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1answer
38 views

Verify proof that if $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian

I have to prove that If $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian We let $S$ be a non-finitely generated submodule of $M\otimes_R ...
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1answer
105 views

Difference Between Tensoring and Wedging.

Let $V$ be a vector space and $\omega\in \otimes^k V$. There are $2$ ways (at least) of thinking about $\omega\otimes \omega$. 1) We may think of $\otimes^k V$ as a vector space $W$, and ...
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8 views

Adjoint of $SU(2)$ from tensor product of fundamental 2-dimensional representation

Given elements of the fundamental 2-dimensional representation of $SU(2)$, for example, $a=(1,0)$ and $b=(0,1)$, how can I multiply them correctly to yield an element of the adjoint? $$ a \otimes b ...
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1answer
15 views

Index notation of tensors and mnemonics

I've been trying to learn to manipulate tensors but I've got probably too comfortable with all the matrices in my Linear Algebra course, that it gets really difficult beyond rank-3 tensors. So, ...
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2answers
93 views

Applying the Yoneda-Lemma to prove the existence of Tensor-products

In class the professor said when he came to prove the existence of the tensor-product for $A$-modules ($A$ any ring) that the existence and properties of the tensor-product would be one-liners having ...