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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Doubts on von Neumann's infinite tensor product of Hilbert spaces

I am reading the original paper by von Neumann on infinite tensor product of Hilbert spaces, and there are few things that are not fully clear to me. Given $K = \otimes_{i \in \mathbb{Z}} (\mathbb{H}...
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Associtivity of Tensor Product of Modules Over Algebras in a Tensor Category

I am attempting to prove that modules over a commutative algebra (monoid) $A$ in a fixed tensor category $\mathcal{T}$ form a tensor category $\mathcal{T}_A$. All of the references I have found say it ...
Dakota's Struggling's user avatar
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If $f:A\rightarrow B$ is a ring morphism, and $M$ is a flat $A$-module, then $M_B := B \otimes_A M$ is a flat $B$-module.

The statement is: If $f:A\rightarrow B$ is a ring morphism, and $M$ is a flat $A$-module, then $M_B := B \otimes_A M$ is a flat $B$-module. By a proposition in theory, proving that $M_B$ is a flat $B$-...
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Is $\mathbb{Q}/\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Q} = 0$?

Given the abelian group $\mathbb{Q}/\mathbb{Z}$ and the group $\mathbb{Q}$, is it true that their tensor product over $\mathbb{Z}$ is the trivial group, i.e., $\mathbb{Q}/\mathbb{Z} \otimes_\mathbb{Z} ...
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Kernel of a modules aplication

Let $A$ be a ring, $I$ an $A$-ideal. We define the aplication $A/I\otimes _AM\longrightarrow M/IM$, $[a]\otimes x\longmapsto [ax]$. I want to see it is injective. My try is the following: let $[a]\in ...
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Equivalent Forms of Schur-Weyl Duality?

I'm trying to understand Schur-Weyl Duality, and in doing so have encountered two statements which various sources refer to as "Schur-Weyl Duality". The first I encountered was the statement ...
Jude Horsley's user avatar
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Tensor product Ore division ring of fractions

In this article, Lemma 4.2, it seems to be saying that $$K\otimes_{\mathscr{U}(Q)}\mathscr{F}(Q) = 0$$ where $\mathscr{U}(Q)$ is the universal enveloping algebra of the finitely generated Lie algebra $...
Jolia's user avatar
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Lattice and extension of scalar of vector space over valued field

Let $(K_1,w)\subseteq (K_2,v)$ be a valued field extension, where $K_1$ is a local field. Let $V$ be a finite dimension $K_1$-vector space and $L$ be a $\mathcal{O}_{K_1}$-lattice in $V$. Let $a,b\in ...
Yijun Yuan's user avatar
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$n$th symmetric power of a superspace

Given a vector space $V$, we consider the (trivial) associated even superspace $V\oplus 0$ and odd superspace $0\oplus V$. For any (super) vector space $W$ we define the $n$th symmetric power as $$ \...
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Prove that (Q · v) × (Q · w) = (det Q)Q · (v × w) with Levi-Civita Symbols [duplicate]

Assume that $Q$ is an Orthogonal Tensor and $v, w$ are two vectors. Is it true that: $$ (Q · v) × (Q · w) = (\det Q)Q · (v × w) $$ I got a little bit stuck by the proof with Levi-Civita Symbols: \...
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Proof that the derivative of a second order tensor w.r.t. a second order tensor is a fourth order tensor

We know that, since a linear map $T$ from a vector space $V$ to a vector space $W$ can be represented by a matrix, and, the derivative of a vector function $f:V \rightarrow W$ at $a \in V$ is the ...
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Building the tensor product of multiple algebras in sage?

I want to build $\Lambda\otimes\Lambda$ in Sage, where $\Lambda$ is the algebra of symmetric functions. You can build the algebra of symmetric functions in the Schur bases with SymmetricFunctions(QQ)....
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How to understand tensor product?

Can I conceptualize tensor product as a product between matrices of matrices? So could I apply the regular Matrix multiplication between rows and columns of matrices (made of matrices instead of ...
CoffeDeveloper's user avatar
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Dimensionality of Tensor Product in Bimodules with Distinct Actions

Failing Example Show Consider $Z=(Z\setminus\{0\}, \cdot, [1])$ as the multiplicative monoid of integers, denoted by elements $[1], [2], \dots$, etc. Denote the free $\mathbb{Z}$-ring $\mathbb{Z}[Z]$ ...
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If $T \in \mathscr L(V, W)$, then there exists a map $T^∗: \tau^k(W) \to \tau ^k(V ).$

$\mathscr L(V, W):=$ space of all linear transformations from $V$ to $W.$ $\tau^k(W):=$ Space of all $k-$linear transformation from $W\times W ...\times W(k-\text{times})\to \mathbb R.$ Similarly, $\...
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Rewriting complex matrix as a tensor product

Consider the following matrix: $$\tilde A=\begin{pmatrix}A_{11}^* & 0 & A_{12}^* & 0 & \dots & A_{1n}^* & 0 \\ 0 & A_{11} & 0 & A_{12} & \dots & 0 & A_{...
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A subtle point about vector space isomorphisms

So I was studying tensor products from the book "An Introduction to Tensors and Group Theory for Physicists". After proving the fact that $ \{ e_i \otimes f_j\}_{i \in \mathcal{I}, \, j\in \...
Bilge K. Aksebzeci's user avatar
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Tensor product and product

If $A$ is an unital associative $\mathbb C-$algebra, and it admits a decomposition $$ A = U V \; , $$ where $U,V$ are subalgebras of $A$ and $U \cap V= \mathbb C \textbf 1$, then can we show that $$ A ...
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Extending scalars to get $\mathbb{C}\bigotimes_\mathbb{R} \mathbb{R}^{2n}\cong \mathbb{C}^{2n}$ as $\mathbb{C}$-modules

I am going through some lecture notes in commutative algebra. I am struggling with one basic example which I want to understand fully before going further. The example is the following: We take $\...
Haldot's user avatar
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Finding the eigenvalues of a tridiagonal block matrix of special form

Consider the following symmetric tridiagonal block matrix: $$\begin{bmatrix} 2I_{N \times N} & -I_{N \times N} & O & \dots & O &O \\ -I_{N \times N} & 2I_{N \times N} &...
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Determining mapping cone of free resolution

I am reading the concept of mapping cone of a resolution. I need help with the following. Let $I_1=\langle x_1^2-x_2 x_4, x_1 x_2-x_3 x_4, x_1 x_3-x_4^2,x_2^2-x_1 x_3, x_2 x_3-x_1 x_4,x_3^2-x_2 x_4 \...
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Flat base change preserves the non-degeneracy (Proposition 9.2 in Commutative Algebra, Matsumura)

Let $f : A \rightarrow B$ and $g : A \rightarrow C$ be homomorphisms of Noetherian rings.Suppose 1) $B \otimes_A C$ is Noetherian, 2) $f$ is flat and 3) $g$ is non-degenerate. Then $1_B \otimes g : B ...
RHspqr's user avatar
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How to define tensor product of operators on $B(H \otimes H)$

Let $H$ be a Hilbert space. Working on locally compact quantum groups, I have met with operator of the form $\iota \otimes \omega_{\xi, \eta}$ with $\xi,\eta \in H$ and $\omega_{\xi,\eta}(T) = (T\xi,\...
Valentin Massicot's user avatar
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Showing a tensor is non-zero without using universal property

The following question is from Vakil's The Rising Sea: Show that $Z/(12)⊗Z/(10)≅Z/(2).$ It is clear that all tensors are equal to either $1⊗1$ or $0⊗0$. My question is that how one can show $1⊗1\neq 0⊗...
gro's user avatar
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What is the empty tensor product of vector spaces?

The tensor product of a space with itself once is $V^{\otimes1}$, but what is $V^{\otimes0}$? Since it is an empty tensor product, it is - a fortiori - an empty product. So I'm looking for a "$1$&...
Hank's user avatar
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What's the definition of dual number at perspect of exterior algebra?

In Dual Number it said that "It may also be defined as the exterior algebra of a one-dimensional vector space with $\varepsilon$ as its basis element." But I can't find the detailed rigorous ...
Richard Mahler's user avatar
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Let $R$ be a ring $P$ a projective generator of right $R$-modules. If $M$ is a left $A$-module, then $M \cong \text{Hom}_R(P, P \otimes_R M)$

Suppose $R$ is a ring and $P$ is a projective generator of right $R$-modules. If $M$ is a left $A$-module, show that the natural map $M \to \text{Hom}_R(P, P \otimes_R M)$ that sends $m \mapsto (p \...
love and light's user avatar
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Shuffle product formula for coproduct

I'm studying the coproduct $\Delta$ defined on a tensor algebra $T(V)$ and its action on tensor products of elements from a vector space $V$. The coproduct is given by $\Delta(v) = v \boxtimes 1 + 1 \...
Martin Geller's user avatar
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2 answers
59 views

Confusion Over Distributive Property in Tensor and External Tensor Products

I've been delving into the properties of tensor ($\otimes$) and external tensor products ($\boxtimes$) within the context of coalgebra, particularly examining how the coproduct $\Delta$ applies to ...
Martin Geller's user avatar
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Module of type $FP_n$

I'm trying to understand the converse of theorem 1.3 in Robert's Bieri Homological dimension of discrete groups which says that a $\Lambda$-module $A$ is of type $FP_n$ if and only if for every direct ...
Jolia's user avatar
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Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
Brendan Murphy's user avatar
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What are the names of the conventions for defining the double dot product?

The double dot product of two matrices $A : B$ can be defined as either: $A : B = Tr(AB^T) = A_{ij}B_{ij}$ $A : B = Tr(AB) = A_{ij}B_{ji}$. I've seen the first convention called Frobenius or ...
Tomek Dobrzynski's user avatar
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Induced change of basis on a (p,q) tensor

I'm struggling to simplify the last step of a $(p,q)$ tensor and how its components change with a linear change of basis on the associated vector space. So far I have: Given a vector space $V$ over ...
Tyler Roche's user avatar
3 votes
1 answer
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Proving a criterion for flatness of modules

I am following Qing Liu's textbook "Algebraic Geometry and Arithmetic Curves," and have come upon the following statement (the truth of which is well-known): Theorem: Let $M$ be an $A$-...
Inbo Gottlieb-Fenves's user avatar
1 vote
1 answer
51 views

$\sigma$-weak continuity of $x \mapsto x \otimes 1$ from $B(H)$ to $B(H \otimes H)$

Let $H$ be a separable Hilbert space. Write $B(H)$ for the set of linear bounded operator on $H$ and $H \otimes H$ the tensor product of Hilbert space. For every $A,B \in B(H)$ we can define an ...
Valentin Massicot's user avatar
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Is the infinite tensor product of flat modules still flat? [closed]

Suppose $(M_i)_{i \in I}$ is a collection of flat $A$-modules ($A$ is a commutative ring with $1$). Is the tensor product $\bigotimes_i M_i$ still flat? This is obviously true by induction when $I$ is ...
Inbo Gottlieb-Fenves's user avatar
3 votes
2 answers
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Basis free proof of the Frobenius formula

Let $G$ be a finite group and $H<G$ a subgroup. Let $V$ be a representation of $H$ with character $\chi$. The Frobenius formula states that the character of the induced representation $\text{Ind}_H^...
Fungaria's user avatar
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1 answer
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tensor product of a graded vector space and an object in k-linear category

In the book "Fourier-Mukai Transforms in Algebraic Geometry", the author has been using the following terminology quite a few times in the first two chapters (Proof of Lemma 1.58, Definition ...
Ray's user avatar
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Intersection of Submodules inside a Tensor Product

Let $R \subset S$ a flat, injective extension of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M \otimes_S 1_R \subset N \otimes_R 1_S \subset N \otimes S$ as $R$-...
user267839's user avatar
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1 answer
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A proof of no-cloning theorem in the case of pures states in a qubit system

I am working on this problem Consider a qubit $\scr H =\Bbb C^2$ and pure states. Prove the no-clonning theorem ( hint: Use the linearity of the channel to arrive to a contradiction) I wonder if the ...
darkside's user avatar
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Tensor confusion: scalar $\otimes$ vector = OK?...

I am trying to understand tensors and one particular question have caused me a great deal of confusion. The particular example with the metric tensor below is an attempt to highlight where my ...
evolhart's user avatar
4 votes
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Combinations of simple tensors

This is a follow-up question to https://math.stackexchange.com/a/4872343 Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. Let $A(u)$ be the collection of all finite sets of ...
Lorenzo Guglielmi's user avatar
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A particular quotient in the study of tensor products of $\mathfrak{sl}_2$-modules

I’m studying Verma modules of $\mathfrak{sl}_{2} = \mathfrak{sl}_{2} (\mathbb{C})$. Let’s introduce standard notation. Elements $h,e,f\in \mathfrak{sl}_{2} $ form the basis of $\mathfrak{sl}_{2}$, ...
Matthew Willow's user avatar
6 votes
1 answer
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Inner product of signatures of piecewise linear paths

It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$...
Oscar's user avatar
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Spectral family/resolution for $A \otimes 1+ 1 \otimes B$

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$ with spectra $\sigma(A), \sigma(B)$. We know that : $\sigma(A \otimes 1 + 1 \otimes B) = \overline{\...
Hugo's user avatar
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5 votes
2 answers
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The projective tensor norm on tensor product of Banach spaces implies the inner product on tensor product of Hilbert spaces?

As presented in the answer of this post, the projective tensor norm on the algebraic tensor product of two Banach spaces $X$ and $Y$ is given by \[ \Vert \omega\Vert_{\pi} = \inf\left\{\sum \lVert x_{...
Keith's user avatar
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3 votes
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Double dual and tensor product for infinite dimensional spaces

It is well known that finite dimensional vector spaces (over any field $K$) canonically satisfy $V''\otimes W''\simeq (V\otimes W)''$. My question is, roughly, if this canonical map can somehow be ...
Tim Seifert's user avatar
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Discrete spectrum of $A \otimes 1+ 1 \otimes B$

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...
Hugo's user avatar
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Natural isomorphism between tensor product and exterior product

I am requesting help with the following problem. Below all rings are commutative with unit, and for a ring $R$, we define an $R$-algebra to be a ring $R'$ with ring homomorphism $f: R \to R'$. This ...
Abced Decba's user avatar
1 vote
1 answer
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Derivatives of a polynomials on a Banach space as a multilinear maps

I'm reading "Differential Equations Driven by Rough Paths" by Lyon, Caruana and Lévy and I can't wrap my head around the following part (beginning of Section 1.4.2). Let $V$ be a finite-...
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