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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Tensor formula in SU(3) representations

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
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1answer
25 views

Tensor product of two polynomials

Given two polynomials $a$ and $b$ over some ring $R$, what is the explicit definition of their tensor product? If it's easier to be more concrete, take $R=\mathbb{Z}_2$.
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Isomorphism of Hecke algebra modules doesn't seem right

Define the affine Hecke algebra $H_n=H_n(q)$, as the algebra with generators $T_1, \dots, T_{n-1}, X_1^{\pm 1}, \dots, X_n^{\pm 1}$ that satisfy $$ (T_i + 1)(T_i − q) = 0$$ $$T_i T_j = T_j T_i \text{ ...
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42 views

tensor $k[x] \otimes_k k[x]$

I am thinking about tensor product $k[x] \otimes_k k[x]$. I know that $k[x] \otimes_k k[y] \cong k[x,y]$, but what about $k[x] \otimes_k k[x]$? As far as I understand I can't say that $k[x] ...
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16 views

Discrepancy between tensor product and Kronecker product?

I have tried evaluating $|\psi\rangle^2$, both using the tensor product notation and the Kronecker product. Why do the results look so different? Are they even right?
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24 views

How to prove tensor product of orthogonal projections is still orthogonal ?

Given $n$ finite dimensional Hilbert spaces, $H_1,\cdots, H_n$, and $n$ orthogonal projections $\mathcal{P_1},\cdots,\mathcal{P_n}$, For $i\in \{1,\cdots, n \}$, $\mathcal{P_i}$ is an orthogonal ...
3
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65 views

Understanding the tensor product

I know the definition of the tensor product, and I can somehow understand its importance, but among several constructions in mathematics, somehow I just never grasped the meaning of the tensor ...
4
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2answers
35 views

Tensorproduct of finite fields

I perfectly understand the tensor product of vector spaces over finite fields. But when I regard these vector spaces as finite fields I get confused. Let the vector spaces $\mathbb{F}_p^m$ and ...
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1answer
52 views

Spelling out uniqueness in a rigid category

As has been discussed in other posts, the dual of an object in a rigid category is unique up to unique isomorphism. As highlighted here, this does not mean that, for any two duals $(X^*,\epsilon,\nu)$ ...
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18 views

Surjectivity of tensor product map

This lemma can be found in "Noether's Problem in Galois Theory" by Richard Swan at this link: ...
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2answers
85 views

When do we use Tensor?

I'm carious to see applications of tensor product. Is there any set of things that if they happen or we encounter them then we use tensor product, tensor algebra, ...? I will be happy if it be ...
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16 views

Tensor products and decomposition of $SU(3)$ representations

For each finite irreducible representation of Lie algebra $su(3)$ one knows that it is characterized by highest weight $(\lambda_1, \lambda_2)$ with integral entries. In this notation, $(1,0)$ is ...
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1answer
42 views

Can someone help me understand the Euclidean metric?

A Euclidean metric is defined as: $g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$ Can someone explain the following: why do we use $dx^i$ instead of $x^i$ which ...
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1answer
30 views

Tensor Product of irreducible modules

Let $A$ be a $\mathbb C$ algebra. Let $S$ be an irreducible $A$ module? Then what $ S \otimes_A Hom_A(S,S)$? Is it equal to S? I know that $S \otimes_{\mathbb C} Hom_A(S,S)$ is isomorphic to $S$ as a ...
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19 views

Tensor of Relative Bases

Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of ...
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1answer
27 views

Tensor product of A-module M with itself equals M

Suppose that $A$ is a commutative ring and $M$ is an $A$-module such that $M \otimes_A M=M$. Is it true that $M=A$? If not, does the answer change if $M$ is a ring and $A \subset M$?
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76 views

Total quotient ring and localization

Let $R$ be a Noetherian normal ring, and let $\mathfrak p$ be a prime ideal of $R$, and let $Q(R)$ be the total quotient ring of $R$. For the tensor product $Q(R) \otimes_R R_{\mathfrak p}$, ...
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27 views

Composition of linear maps as a tensor product

$\mathbf{Question:}$ Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$. Fix $A,B \in \mathscr{L}(V)$. Consider the linear operator $T_{A,B} \in ...
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15 views

Reference on Tensor Product Over Semiring

I know that there has been research about tensor product not only of modules over rings, but also of semimodules over semirings. I would like to use it in my master thesis, so I'm looking for a ...
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1answer
38 views

Inverse Tensor map

Let $$\phi: M_n (\mathbb{C})\otimes M_m (\mathbb{C})\to M_{nm} (\mathbb{C})$$ $$ \phi({A}\otimes{B}) = \begin{bmatrix} a_{11} {B} & \cdots & a_{1n}{B} \\ \vdots & \ddots & \vdots \\ ...
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21 views

Tensor product of non-commutative algebras

Define the affine Hecke algebra, $H_n=H_n(q)$, as the algebra with generators $T_1, \dots, T_{n-1}, X_1^{\pm 1}, \dots, X_n^{\pm 1}$ that satisfy $$ (T_i + 1)(T_i − q) = 0$$ $$T_i T_j = T_j T_i \text{ ...
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1answer
33 views

A tensor product over a ring $R$ and over a domain $k$

Let $R=k[x,y]$, where $k$ is an integral domain, and let $\mathfrak m=xR+yR$. We can identify the $R$-module $\mathfrak m/\mathfrak m^2$ with $V$, where $V=k\bar x\oplus k\bar y$ is acted on trivially ...
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1answer
30 views

Dimension of the basis $\{x \otimes y + y \otimes x\}$

I'm trying to prove that the annihilator of $I = \left<x \otimes y - y \otimes x \right>$ is $\left<x \otimes y + y \otimes x \right>$. To do this I am trying to compare dimensions. So if ...
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1answer
36 views

Dimension of a tensor algebra quotient

For example, for arbitrary $V$ we can find a dimension of $ \bigwedge V = T(V) / (u \otimes u)$ by writing basis explicitly -- the dimension of $\bigwedge V$ is $2^{\dim{V}}$. Let's consider a tensor ...
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1answer
36 views

tensor product of modules over commutative ring

It is well-know that the tensor product of two finitely generated modules over a given ring $R$ is finitely generated. Now my question is Let $M$ and $N$ be $R$-modules where $R$ is a commutative ...
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2answers
33 views

tensor algebra $T(\mathbb{R})$

I would like to understand the product on the tensor algebra $T(V)$ by a concrete example. I know the general construction and the product is defined just by juxtaposition of the elements. So if I ...
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1answer
16 views

How does $R[Y\times_K X]\simeq RY\otimes_{RK} RX$?

Suppose a group $G$ acts on two finite sets $X$ and $Y$ by a left action on $X$, and a right action on $Y$. If $R$ is a commutative, unital ring, and $K$ a subgroup of $G$, and $R[Y\times_K X]$ is the ...
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1answer
50 views

Having Problem With Kronecker and Outer Product

I'm having an issue with some outer & Kronecker products where I am doing two different processes which should result in the same answer, but I'm getting a different answer for each. Can anyone ...
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1answer
53 views

Is $\operatorname{Hom}_{RK}(R,RX)\cong R\otimes_{RK} RX$?

Suppose a group $K$ acts on a finite set $X$, and let $RX$ be the permutation module of $X$ over a commutative unital ring $R$. Is it true that $$ \operatorname{Hom}_{RK}(R,RX)\cong R\otimes_{RK} RX? ...
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1answer
44 views

If $\mathfrak{g}\subseteq\mathfrak{so}(V)$, then $H^{0,2}(\mathfrak{g})=(\mathfrak{so}(V)/\mathfrak{g})\otimes V^*$

Problem I'm learning about $G$-structures and was assigned this exercise (Cartan for Beginners Exercise 8.3.6.1): Let $$ H^{0,2}(\mathfrak{g})=(V\otimes\Lambda^2V^*)/\delta (\mathfrak{g}\otimes ...
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60 views

Tensor product of $k$-algebras, center, isomorphism.

Let $A$, $B$ be two $k$-algebras of finite dimension, where $k$ is a field. Here, $A$ and $B$ are not necessarily commutative. Do we have that$$Z(A \otimes_k B) \cong Z(A) \otimes_k Z(B),$$where ...
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1answer
72 views

Is this 3 Qubit state Entangled?

Is this 3-Qubit state entangled? http://prntscr.com/8vsg0b $|X\rangle=\frac{1}{\sqrt 2}~~|000\rangle+ \frac{i}{\sqrt 2}|111\rangle$ I've worked with 2-Qubit states and you can turn them into ...
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28 views

Covariant derivative of a covariant vector

$$\mathbf{A} = A^{1}\mathbf{e_{1}}+A^{2}\mathbf{e_{2}}$$ $$A= \sum_{i=1}^{n}A^{i}\mathbf{e_{i}}$$ Taking the derivative wrt the tangent basis vector and dropping the summation by Einstein convention: ...
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1answer
59 views

Does the tensor product with a finite dimensional vector space gives you some kind of semi-basis?

Let $V$ be a (possibly infinite dimensional) vector space, and $W$ a finite dimensional one. Then, one can define the tensor product $V\otimes W$ as the free vector space on $V\times W$ modulo some ...
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1answer
20 views

Surjectiveness of convolution

Consider the convolution mapping $j^*: Hom(X, Y) \otimes X \to Y$, given by bilinear formula $(\phi, x) \mapsto \phi(x)$, in a category of coherent sheaves or, generally, in any abelian category. I ...
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How would you express $(\underline{a} \times \underline{b}) \times (\underline{a} \times \underline{c})$ in index notation?

At a guess, I would have said that the answer is $\varepsilon_{ijk}\varepsilon_{jlm}a_{l}b_{m}\varepsilon_{kpq}a_{p}c_{q}$, but I'm guessing that this is incorrect. What is the corrdct expression? ...
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1answer
23 views

How would you use index notation to show that these vector equations are equal?

How would you use index notation to show that $(\underline{a} \times \underline{b}) \cdot (\underline{a} \times \underline{b}) = |\underline{a}|^{2} |\underline{b}|^{2}-(\underline{a} \cdot ...
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1answer
44 views

Show that $R \otimes_R M\cong M$ is isomorphic to $M$, for every left $R$-module $M$, $R$ a ring

My professor says it is trivial, but I cannot still see the triviality :( Let $R$ be a ring. Show that $R \otimes_R M\cong M$ for a left $R$-module $M$. Any ideas how to address this?
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1answer
27 views

$f: A\longrightarrow B$, $I \subset A$, then $B\otimes_AA/I=B/IB$?

A ring homomorphism $f: A\longrightarrow B$ and $I$ is an ideal of $A$, then do we have $B\otimes_AA/I=B/IB$ ? I am trying to prove that if $f:X\longrightarrow Y$ is a morphism between two schemes, ...
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23 views

Finite-Dimensional Representations of The Classical Groups in Tensor Spaces: Invariant Theory

I. When we study finite-dimensional irreducible representations over the space of general tensors (e.g.,Chapter 13, Group Theory in Physics by Wu-Ki Tung), is it enough to obtain all the ...
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1answer
98 views

Why is $\mathrm{id}\otimes\varphi:\mathbb{Q}\otimes_{\mathbb{Z}}A'\to\mathbb{Q}\otimes_{\mathbb{Z}} A$ a monomorphism, if $\varphi$ is a monomorphism?

Let $\varphi:A'\to A$ a monomorphism of abelian groups. Prove that $$\mathrm{id}\otimes \varphi:\mathbb{Q}\otimes_{\mathbb{Z}}A'\to \mathbb{Q}\otimes_{\mathbb{Z}} A$$is a monomorphism. I'm stuck ...
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28 views

The explicit expression of symmetric tensors, and the Symmetrize gradient of symmetric tensors

This question has been asked here long time ago but I am still confused... Let me repeat some definitions. In general we define on $\mathbb R^2$ that $$ \mathcal T^k(\mathbb R^2):=\{\xi:\,\mathbb ...
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1answer
29 views

Corresponding matrix field basis

Hi people, I'm reviewing my notes for an exams and this is a question which I was unable to wrap my head around for many months. It should be fairly simple but I might be lacking a crucial piece of ...
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15 views

Is there any clue for the supremum of projective norm in an unit ball of injective tensor product space

Is there any information about the supremum, that is in a d-order tensor space $\mathcal{T}=\mathcal{R}^{n_1}\times\mathcal{R}^{n_2}\times\cdots\times\mathcal{R}^{n_d}$, $x\in \mathcal{T}$ what is ...
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1answer
33 views

Question about the metric tensor

From wikipedia: The metric can be written in the form $g=g_{ij}dx^i \otimes dx^j$. The metric is thus a linear combination of tensor products of one form gradients of coordinates. If we denote the ...
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14 views

Quotients of T(V)

I want to prove that any finitely generated algebra can be written as a quotient algebra of $T(V)$, the tensor algebra where $V$ is a vector space. I was hoping to use the universal property to do ...
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1answer
50 views

$0^{th}$ tensor power, $V^{\otimes 0} = \Bbb F$, definition, or mathematical construction?

95% sure I will be told it's just a definition, move on etc: Is there mathematical reason why the $0^{th}$ tensor power is defined as: $$V^{\otimes 0} = \Bbb F$$ Where $\Bbb F$ is the field ...
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1answer
27 views

Base change and power series rings

I am confused about base change for power series rings. For a concrete example, take the $\mathbb{Z}_p$-algebra $R = \mathbb{Z}_p[[X]]$. Do we have $R \otimes \mathbb{F}_p = \mathbb{F}_p[[X]]$ ? Or do ...
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3answers
186 views

How to simplify $\frac{\partial^m}{\partial y_i^m}\mathrm{div }(A\nabla u({\bf x}({\bf y})))$

Let $\bf{x}\in\mathbb{R}^n$ (interesting in $n\in\{2,3\}$) and let $A=A_{n\times n}=\mathrm{diag}(a_1({\bf x}),\dots,a_n({\bf x}))$, that is $$A_{2\times2}=\begin{bmatrix} ...
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1answer
71 views

Natural Isomorphism $(V\otimes W)^*\cong V^*\otimes W^*$

According to Wikipedia, there exists a natural isomorphism $(V\otimes W)^*\cong V^*\otimes W^*$ for finite-dimensional vector spaces $V$ and $W$. I can show that given bases $v_i$, $i\in I$ and $w_j$, ...