Questions tagged [tensor-products]

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6
votes
0answers
72 views

Is this “semi-tensor product” something recently invented? Are there other usages of it?

The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
4
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0answers
92 views

Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$

It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
3
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3answers
151 views

Universal property of tensor products of bounded operators

Consider the tensor product of bounded operators. Does this tensor product satisfy the universal property of the tensor product, i.e., for any bilinear map $F: B(\mathcal H_1)\times B(\mathcal H_2)\to ...
3
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0answers
100 views

Spectral norm and “operator norm” for hypergraphs

Consider a $d$-regular, $k$-uniform hypergraph: the elements $S$ of its set $E$ of edges are subsets of $V$ of size $k$, and each vertex $v\in V$ is in $d$ edges. We can then define its adjacency ...
10
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1answer
342 views

Basis of invariant tensors of rank n in three dimensions

[This is a question motivated by theoretical physics, so apologies if the language is rough...] In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, ...
2
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1answer
91 views

Inclusion $M(A) \otimes M(B)\subseteq M(A\otimes B)$ of multiplier algebras

Consider the following definitions given in Timmerman's book "An invitation to quantum groups and duality": m Further in the book, it is claimed that if $A$ and $B$ are non-degenerate ...
6
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1answer
479 views

If $M\otimes_S T$ is an $A$-module, is $M$ an $A$-module?

Let $\mathbb{C}$ be the field of the complex numbers. Let $R=\mathbb{C}[x]$, $T=\mathbb{C}\langle x\rangle$ be the ring of entire series with convergence radius at least $1$, and let $S=\mathbb{C}\...
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0answers
62 views

Definition of tensor product of dense subspaces of Hilbert spaces

Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1}...
8
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1answer
213 views

A Hadamard product of binary (or ternary) matroids

I would like to know if anyone has studied the following ``Hadamard product" of binary (or ternary) matroids. (There is a notion of Hadamard product of matroids studied e.g. here but I think that ...
2
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1answer
91 views

Non-central tensor product of central algebras

This is an export of https://math.stackexchange.com/questions/4016545/non-central-tensor-product-of-central-algebras which despite a bounty has sadly attracted no answer. I repeat the question here: ...
1
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0answers
108 views

Why is this operator independent of the choice of basis

I asked this question in MSE but I received no answer https://math.stackexchange.com/questions/4009524/why-is-the-following-operator-independent-of-the-choice-of-basis/4013636#4013636 Let $G$ be a lie ...
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0answers
189 views

To show equivalence and full faithfulness of a functor PRESERVED under an action of a finite flat algebra

I have explained the two questions and then showed my effort on question $(1)$ as follows (Please at least check my effort below and suggest to make it perfect): Let $R, S,T$ be three commutative ...
2
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1answer
231 views

The maximal tensor product is a continuous functor

I am trying to prove continuity of the maximal tensor product functor. I have a problem in the proof that I cannot see how to handle; If anyone could give me a clue on how to go on from here, I would ...
2
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1answer
106 views

If $(\text{id}_A\otimes \text{ev}_x)(z)= 0$ for all $x \in X$. Do we have $z=0$?

Let $A$ be a $C^*$-algebra (not necessarily unital). Let $X$ be a compact Hausdorff space. We can consider the minimal $C^*$-tensor product $A \otimes C(X)$. On this space, we can consider the slice ...
0
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2answers
97 views

$(ST)_{[13]}= S_{[13]}T_{[13]}$ for $S,T \in B(\mathcal{H}\otimes \mathcal{H}).$

Let $T\in B(\mathcal{H} \otimes \mathcal{H})$ where $\mathcal{H}$ is a Hilbert space. We can define operators $$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$ and if $\Sigma: \mathcal{H} \otimes \...
7
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1answer
169 views

Function spaces satisfying $\mathcal{F}(M\times N)\simeq\mathcal{F}(M)\otimes\mathcal{F}(N)$

Let $M \mapsto\mathcal{F}(M)$ be a map associating topological vector spaces of some type (that I will call "function spaces") to geometric spaces $M$ of some type. For $M$, I'm mostly ...
8
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1answer
540 views

Origin of the symbol for the tensor product

I have recently realised that the Paleo-Hebrew (and Phoenician) graph for the Hebrew letter ט (Teth) is $\otimes$. This made me wonder if there is any relation between the choice of the symbol and the ...
1
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0answers
44 views

Is the product of two Banach algebras given by the injective cross-norm itself a Banach algebra?

I understand that you can take the tensor product of Banach spaces in many different ways by specifying different norms; of particular interest to me are the cross-norms. The projective and injective ...
3
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1answer
88 views

Show commutativity of a diagram involving multiplier $C^*$-algebras

Let me recall the following fact: If $A$ is a $C^*$-algebra and $\pi: A \to \mathcal{B}(\mathcal{H})$ is a faithful non-degenerate representation, then we can explicitely realise the multiplier ...
4
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1answer
127 views

Is the unit ball of $A \odot B$ strictly dense in that of $M(A \otimes B)$?

Let $A$ and $B$ be $C^*$-algebras and let $A \otimes B$ their minimal tensor product and $M(A \otimes B)$ the associated multiplier algebra. On $M(A \otimes B)$, we consider the strict topology which ...
4
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0answers
96 views

non-abelian tensor products of several groups

R. Brown and J-L. Loday had defined the tensor product of two arbitrary groups acting on each other. Let $G,H$ be groups with actions on each other on the right. each group act on itself by ...
3
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2answers
208 views

Convolution of functionals on compact quantum group

Let $\mathbb{G}= (A, \Delta)$ be a ($C^*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^*= B(A, \mathbb{C})$ obtains a product $$\omega_1*\omega_2:= (\omega_1\otimes \omega_2) ...
3
votes
1answer
165 views

Completed tensor product and power series rings

I want to know if the notion of completed tensor product in Stacks Project Tag 0AMU is the one that yields $$k[[x]] \widehat{\otimes} k[[y]]≅k[[x,y]].$$ Here I should be considering the inverse limit ...
3
votes
1answer
110 views

Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$

It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$. For ...
5
votes
1answer
160 views

Simple quotients of a triple tensor product

Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let also $V_1, V_2, V_3$ finite-dimensional simple modules over $\mathcal{H}$ and $Q$ be a simple quotient of $V_1\otimes V_2\otimes V_3$. Is it ...
3
votes
1answer
125 views

A different notion of a decomposable symmetric tensor (besides Veronese)

$\DeclareMathOperator{\complex}{\mathbb{C}}$ Let $\bigvee^m(\complex^n)\subseteq (\complex^n)^{\otimes m}$ denote the space of symmetric tensors, i.e. the set of $x \in (\complex^n)^{\otimes m}$ that ...
12
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1answer
489 views

Infinite dimensional irreducible representations of a tensor product

The second part of Theorem 3.10.2 of "Introduction to representation theory" by Etingof, Golberg, Hensel, Liu, Schwender, Vaintrob and Yudovina states that if $A$ and $B$ are $k$-algebras ($...
2
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1answer
149 views

Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras?

Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\...
0
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1answer
95 views

Best-approximation with tensors of rank $\ge2$

Let $k\in\mathbb N$, $H_i$ be a (finite-dimensional, if necessary) $\mathbb R$-Hilbert space for $i\in I:=\{1,\ldots,k\}$, $H:=\bigotimes_{i\in I}H_i$ denote the tensor product$^1$ of $(H_i)_{i\in I}$ ...
4
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1answer
130 views

Concrete example of non-nuclear operator $E \to F$ and isometry $F \hookrightarrow G$ so that the composition $E \to F \hookrightarrow G$ is nuclear

DISCLAIMER: I posted the same question a week ago on Mathematics Stack Exchange. We know by an abstract argument that there exist Banach spaces $E$, $F$, $G$ and maps $E \to F \hookrightarrow G$ such ...
3
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1answer
219 views

Exactness of injective tensor products

For (algebraic) tensor products, it is well-known that the functor $A\otimes_R \cdot:Mod_R\rightarrow Mod_R$ is only (left-) exact when $A$ is a flat $R$-module. In particular, all vector spaces are ...
1
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1answer
66 views

Has the covariant Hom-functor of the category of additive categories a left adjoint?

Let $\mathsf{Add}$ denote the (strict) 2-category of small additive categories and additive functors. Because categories of additive functors are itself additive, we have for each additive category $\...
0
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0answers
45 views

Questions about a structure related to simplicial complexes

While researching some superficially unrelated theory, a structure similar to the one described below presented itself to me. I'm having trouble with identifying what's the structure name. It seems to ...
1
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0answers
41 views

What is the maximal weight submodule of $\text{Hom}_{\mathfrak{g}}(M,N)$?

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. For ...
2
votes
1answer
174 views

The local flatness criterion

I am self studying the book "Commutative Ring Theory" by H. Matsumura. The main theorem of section 22 is the theorem 22.3, which characterizes flatness of a module $M$ over any ring $A$. The (part of ...
5
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1answer
379 views

Cohomology of derived tensor product of complexes and Künneth spectral sequence

Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
5
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0answers
95 views

A variant of the capset problem

Let $p > 2$ be a prime of bounded size. Suppose $A$ is a subset of $G = \mathbf{F}_p^n$ with only degenerate solutions to $$x + y = 2a,\\ x+z = 2b,\\ y + z = 2c,$$ where a solution is considered ...
0
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1answer
130 views

Reverse the construction of a basis for a tensor product of vector spaces [closed]

If $V,W$ are infinite-dimensional vector spaces with basis {${v_i}$} and {${w_j}$} respectively it holds that $V\otimes W$ has as basis {${v_i⊗w_j}$}. What about the reciprocal? That is: if {${v_i}$} ...
1
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0answers
26 views

Compute Frobenius inner product of two tensor-trains in terms of tensor contractions

Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...
20
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1answer
543 views

Semisimplicity for tensor products of representations of finite groups

Let $G$ be a group and $k$ a field of characteristic $p>0$. Let $$\rho_i: G\to GL(V_i),~ i=1,2$$ be two finite-dimensional semisimple $k$-representations of $G$, with $\dim(V_1)+\dim(V_2)<p+2.$ ...
2
votes
1answer
426 views

Varieties with everywhere good reduction that are isomorphic over every completion have isomorphic generic fibers

Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the completion of the localization at $\...
3
votes
1answer
136 views

Generalized tensor-train decomposition

If $U\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ and $U^{(k)}\in\mathbb R^{r_{k-1}}\otimes\mathbb R^{n_k}\otimes\mathbb R^{r_k}$ with$^1$ $$U_{i_1,\:\ldots\:,i_p}=\sum_{j_0=1}^{r_0}\cdots\sum_{j_p=1}^{r_p}U^...
0
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0answers
149 views

Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...
1
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0answers
15 views

Show that a tensor-train is contained in a recursive sequence of subspaces

Let $p\in\mathbb N$; $n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$; $u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
0
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0answers
32 views

How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?

Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...
0
votes
1answer
56 views

Cores in the tensor-train decomposition

Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...
-1
votes
1answer
89 views

Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
9
votes
2answers
764 views

Is the tensor product of chain complexes a Day convolution?

Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...
0
votes
0answers
59 views

Does “tensoring” with a fixed field preserve Galois extensions of finite fields?

Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...
8
votes
1answer
162 views

A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm

Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide? A somewhat similar question is discussed here.

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