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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
20 views

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}...
1 vote
1 answer
220 views

One-to-one correspondence between the prime ideals lying over $\mathfrak{p}$ and the primes of $B\otimes_A\kappa(\mathfrak{p})$

Let $A\to B$ be a ring homomorphism and let $\mathfrak{p}$ be a prime ideal of $A$. Then the prime ideals of $B$ lying over $\mathfrak{p}$ are in one-to-one correspondence with the prime ideals of $B\...
0 votes
1 answer
25 views

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 $...
1 vote
1 answer
24 views

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 $B$-...
1 vote
0 answers
19 views

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 ...
2 votes
1 answer
2k 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 ...
0 votes
0 answers
37 views

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 ...
2 votes
1 answer
73 views

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} ...
2 votes
2 answers
220 views

Compute the tensor product $\mathbb{Z}/726\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}[\frac{1}{77}]$

I am computing the cardinality of the tensor product $\mathbb{Z}/726\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}[\frac{1}{77}]$. Following are my several attempts: Let $R=\mathbb{Z}$, $I=726\mathbb{Z}$ ...
0 votes
1 answer
36 views

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 ...
2 votes
1 answer
62 views

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 \...
0 votes
1 answer
17 views

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 ...
0 votes
1 answer
82 views

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: \...
1 vote
2 answers
133 views

Minimal tensor product of $B(H)$ and $C(G)$

Let $H$ be a finite dimensional vector space, and $G$ be a compact group. Let $B(H)$ be the bounded operators on $H$, let $C(G)$ be the complex valued continuous functions on $G$, and let $C(G;B(H))$ ...
2 votes
1 answer
62 views

$n$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 $$ \...
0 votes
0 answers
12 views

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 ...
3 votes
1 answer
44 views

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)....
0 votes
1 answer
43 views

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 $\mathbb{Z}[Z]$ ...
0 votes
0 answers
26 views

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 ...
1 vote
0 answers
29 views

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, $\...
2 votes
1 answer
1k views

Associated Tensors.

I came across this question which asked us to come up with the "second order antisymmetric tensor associated with a vector" which was given in the problem. The components of the vector were ...
0 votes
0 answers
29 views

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_{...
3 votes
1 answer
3k views

Kronecker Delta as a product of partial derivatives

So, I'm currently teaching myself the basics of tensors, and one of the definitions I continually run into for Contravariant and Covariant tensors is that they transform according to $\bar A^i = \frac{...
0 votes
1 answer
50 views

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 \...
0 votes
0 answers
44 views

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 ...
1 vote
1 answer
45 views

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 $\...
0 votes
0 answers
33 views

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} &...
2 votes
1 answer
1k views

Universal property of the Tensor Algebra

Let M be an A-module over a commutative ring A. For any A-algebra N and A-module homomorphism $\phi : M \rightarrow N$ there is a unique A-algebra homomorphism $\Phi : T(M) \rightarrow N$ (where T(M) ...
1 vote
0 answers
28 views

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 ...
1 vote
1 answer
33 views

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,\...
4 votes
3 answers
508 views

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 "$1$&...
1 vote
0 answers
38 views

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 \...
0 votes
0 answers
20 views

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⊗...
6 votes
1 answer
50 views

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 $\Delta x$...
2 votes
1 answer
36 views

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 ...
2 votes
0 answers
21 views

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 \...
1 vote
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 ...
1 vote
1 answer
54 views

Let $R$ be a commutative ring and $A,B,C$ be $R$-modules, $A$ finitely presented, $C$ flat. Then Hom$(A,B\otimes C)\cong\text{Hom}(A,B)\otimes C$

Let $R$ be a commutative ring with $A$, $B$, and $C$ all $R$--modules. Suppose that $A$ is finitely presented and $C$ is flat (that is, the functor ${-} \otimes C$ preserves short exact sequences). ...
1 vote
1 answer
38 views

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 ...
0 votes
0 answers
14 views

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 ...
0 votes
0 answers
31 views

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 ...
7 votes
3 answers
194 views

Map is almost isomorphism iff tensoring with $\widetilde{\mathfrak{m}} = \mathfrak{m}\otimes_R \mathfrak{m}$ yields isomorphism

Let $R$ be a commutative ring with ideal $\mathfrak m$ satisfying $\mathfrak m^2 = \mathfrak m$, and let $f:M\to N$ be a map of $R$-modules. I want to show that the kernel and cokernel of $f$ are ...
0 votes
1 answer
60 views

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 ...
3 votes
1 answer
47 views

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 $A$-...
3 votes
2 answers
78 views

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^...
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 ...
1 vote
0 answers
88 views

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 ...
2 votes
1 answer
48 views

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 ...
2 votes
1 answer
43 views

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 $R$-...
2 votes
1 answer
34 views

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 ...

1
2 3 4 5
91