Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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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 $T \otimes S$ ...
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1answer
596 views

Dual of a finite dimensional algebra is a coalgebra (ex. from Sweedler)

Let $(A, M, u)$ be a finite dimensional algebra where $M: A\otimes A \rightarrow A$ denotes multiplication and $u: k \rightarrow A$ denotes unit. I want to prove that $(A^*, \Delta, \varepsilon) $ ...
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The “semi-symmetric” algebra of a vector space

If $V$ is a vector space over a field $K$ then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\in ...
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Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

$\newcommand{\Tr}{\operatorname{Tr}}$Let $\mathcal{S}(\mathbb{R}^k)$ denote the $k$-dimensional Schwartz space with the usual topology, and let $\mathcal{S}'(\mathbb{R}^{k}))$ denote its strong dual (...
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160 views

How do fractional tensor products work?

In this blog post, Terry Tao discusses the $n$-fold tensor product of a one-dimensional vector space $V^L$ ($L$ is just a non-numeric label, not an exponent). He claims that With a bit of ...
8
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340 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 product....
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149 views

Doubtful solution to an exercise on faithful flatness in Matsumura's Commutative Algebra

The exercise on page 30. It says that: Let $A, B$ be integral domain having the same field of fractions, $B \supseteq A$. Prove that $B$ is faithfully flat over $A$ only when $B = A$. My solution ...
8
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762 views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
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60 views

$M\otimes_A N\cong A$ implies $M$ is left $A$-projective?

Let $A$ be an algebra (possibly non-commutative). Let $M,N$ be $A-A$-bimodules. Suppose that $M\otimes_AN\cong A$. Can we conclude that $M$ is left $A$-projective? I tried many things but ultimately ...
7
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160 views

Is there a linear injection $ \Lambda^k V^* \otimes \Lambda^k V^* \to \Lambda^k (V^* \otimes V^*)$ which preserves decomposability?

Let $V$ be an $n$-dimensional real vector space, and let $2 \le k \le n-2$. Definitions We say an element $\omega \in \Lambda^k V$ is decomposable if $\omega=\alpha_1 \wedge \dots \wedge \alpha_k$, ...
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257 views

A conjecture on Schatten 1-norm

I have a conjecture on Schatten 1-norm. Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite ...
7
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220 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
7
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1answer
411 views

Symmetric kernel of tensor product

Let $V,W$ be two real vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with distinct kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\...
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176 views

Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
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898 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) \...
7
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216 views

Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
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238 views

Mnemonic device for relationships between Hom and Tensor

Probably this is a stupid question, but nevertheless... Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
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208 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
6
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1answer
89 views

If $a\otimes 1=1\otimes b$, must it be of the form $r\cdot (1\otimes 1)$?

Let $R$ be a reduced commutative ring, and $A,B$ be two faithfully flat $R$-algebras, note that any faithfully flat ring map is injective. (we can assume all of them are local ring and local maps or ...
6
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248 views

Isomorphism between tensor products (base extension)

Let $A,B$ be two $\Bbb Q$-algebras. Assume that $A \otimes_{\Bbb Q} \Bbb C \cong B \otimes_{\Bbb Q} \Bbb C$ as $\Bbb C$-algebras. Does it follows that $A \otimes_{\Bbb Q} \overline{\Bbb Q} \cong B \...
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400 views

Why do we want or need cross-norms on tensor products?

If $(E, \|\cdot\|_1),(F, \|\cdot\|_2)$ are Banach spaces, their (algebraic) tensor product $E \otimes F$ is a vector space looking forward to be normed. A norm on a tensor product of vector spaces is ...
6
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1answer
416 views

Is the tensor product of two real numbers a real number?

I am on a physics course where they've introduced the tensor product as if it was something that I would have seen before, and they don't tell me any of its properties. However, I am trying to work ...
6
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1answer
487 views

when is the map $m \mapsto 1 \otimes m$ injective?

I just started to study tensor products and have a question. Suppose A is a (commutative with unity) ring, B an A-algebra and M an A-module. Consider map $\alpha: M \rightarrow B \otimes_AM$, defined ...
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650 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 $Z(-)$...
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398 views

Alternative introduction to tensor products of vector spaces

One of the main obstacles in understanding the tensor product is that, unlike many other algebraic structures, you cannot really get hold of its element structure. This confuses many beginners. The ...
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197 views

Groupoid $C^*$ algebra of product groupoid

Let $G$ and $H$ be locally compact (Hausdorff, second countable) groupoids with Haar systems $\mu$ and $\nu$, respectively. Is it true then that the (full) groupoid $C^*$-algebras satisfy $$ C^*(G\...
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350 views

Invariant element in the tensor product of rectangular Specht modules?

Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module ...
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118 views

Does $\text{Tor}_1^R(M, M) = M$?

Let $k$ be a field and let $R = k[x]$. Consider the $R$-module $M := \frac{k[x, x^{-1}]}{x \cdot k[x]}$ (i.e. so typical elements are Laurent polynomials with no positive powers). I have computed $\...
5
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1answer
136 views

Symmetric Rank-1 Decomposition for Density Matrices

Let $(H,\langle\cdot,\cdot\rangle)$ be an $n$-dimensional complex Hilbert space. For concreteness, you can just take $H=\mathbb{C}^n$ with standard inner product. Note that we will use the physicist's ...
5
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38 views

No canonical correspondance between bilinear forms $b:V\times V \rightarrow \mathbb{R}$ and linear forms $\hat{b}:V\otimes V \rightarrow \mathbb{R}$?

In the wiki on bilinear forms, the universal product says that to each bilnear map $b : V\times V \rightarrow \mathbb{R}$ we can associate a linear map $\hat{b} : V\otimes V \rightarrow \mathbb{R}$, ...
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95 views

Problem with equivalent definition of a integrable $G$-structure

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 2. It this proposition: My problem is that I don't understand the converse of this ...
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302 views

Kernel of the tensor product of maps

A theorem about the tensor product of modules is that If $\varphi: B \to C$ and $\psi: B' \to C'$ are surjective, then $$\llap{\text{the kernel of} \quad} \varphi \otimes \psi: B \otimes B' \to ...
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144 views

Is there a relationship between the trace and the Clifford/geometric product?

In what follows, let $V=\mathbb{R}^n$ (although the following probably applies also to a larger number of finite-dimensional spaces). We assume throughout that we have made a choice for an inner ...
5
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104 views

Extension of scalars functor essentially surjective.

Let $f:A\rightarrow B$ be a morphism of rings. When is every $B$-module $N$ of the form $M\otimes_A B$ for some $A$-module $M$?. What are sufficient and necessary conditions on $f$? I know for ...
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236 views

Explicit examples of tensor products

Let $R$ be a commutative ring, I'm looking for sufficiently general and explicit examples of tensor products of modules over $R$. Here are some examples that I already know: If $R$ is a domain and $...
5
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280 views

Hyperdeterminant of 4x4x4 hypermatrix

If given the hypermatrix (which I've written here in bracket notation since I'm not all too sure how to display this) { {{1,1,1,1},{1,1,-1,-1},{1,-1,-1,1},{1,-1,1,-1}}, {{1,1,1,1},{1,1,-1,-1},{1,-1,1,-...
5
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1answer
143 views

Defining multiplication on the tensor product of $R$-algebras.

If $M$ and $N$ are $R$-algebras, then one can define a multiplication of elementary tensors as follows; $(m \otimes n) \cdot (m' \otimes n') = mm' \otimes nn'$. My question is how can we show, using ...
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Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let $...
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107 views

Tensor of tensored categories

Given two $V$-categories $C$ and $D$ tensored over a symmetric monoidal category $V$, could I form the "tensor" of $C$ and $D$? More precisely, is there a $V$-category $T(C,D)$ such that $V$-functors ...
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171 views

Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
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651 views

When is the canonical extension of scalars map $M\to S\otimes_RM$ injective?

Let $\alpha:R\to S$ be a map of unital rings, and let $M$ be an $R$-module. We have a canonical map of $R$-modules: $$\begin{array}{rcl}i:M&\longrightarrow&S\otimes_RM\\[.05in]m&\...
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141 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 ...
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42 views

Exterior power of symmetric product of GL(2,R) tensor representations

A paper I am reading uses an identification of $GL(2,\mathbb{R})$ representations: \begin{align} \Lambda^2(\text{Sym}^3\mathbb{R}^2) = (\text{Sym}^4\mathbb{R}^2 \otimes \Lambda^2\mathbb{R}^2 ) \...
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59 views

Conditions to Make $\text{Hom}(M, N) \cong M^* \otimes N$.

Let $R$ be a commutative ring. Let $M$, $N$, and $L$ be $R$-modules. When do we have $\text{Hom}(M, N) \cong \text{Hom}(M, R) \otimes N$? I am looking for the most general condition you can think of. ...
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102 views

Kronecker Product Interpretation

The algebraic expression for a Kronecker product is simple enough. Is there some way to understand what this product is? The expression for matrix-vector multiplication is easy enough to understand. ...
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106 views

Natural transformations from coinvariants to invariants

Consider a group $G$ and a ring $R$. Write $R[G]\overset{\varepsilon}{\longrightarrow}R$ for the counit map. Write $(-)_G\dashv \varepsilon ^\ast \dashv (-)^G$ for the adjoint triple on modules ...
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112 views

Writing a representation as an iterated symmetric tensor product

I would like to go in the reverse direction than the usual one. Usually, one asks how does one decompose a tensor product as a direct sum of irreducible representations, but I would like to go in the ...
4
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260 views

Jordan Block of Kronecker Product

Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\...
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327 views

What are the units of a tensor product of rings?

Let $R$ and $S$ be commutative $k$-algebras, for some commutative ring $k$. All rings are assumed to have the identity. Let $U(R)$ denote the group of invertible elements of $R$. What can we say ...
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218 views

Understanding the construction of Exterior Algebra

Background The tensor space of type $(r,s)$ associated with $V$ is the vector space $$\underbrace{V\otimes \ldots \otimes V}_{\text{r copies}} \otimes \underbrace{V^* \otimes \ldots \otimes V^*}_{\...

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