# Questions tagged [tensor-products]

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### Locality of a tensor product of two fields

Assume that $K$ and $L$ are two field extensions of a field $k$. Is it known when a tensor product $K \otimes_k L$ is local? (A unital commutative ring is said to be local if it contains a unique ...

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### What is the suitable tensor product for Holder spaces

We know that for $X\subset\mathbb R^m,Y\subset\mathbb R^n$ open, then $C^0(\bar X\times\bar Y)=C^0(\bar X)\hat\otimes_\varepsilon C^0(\bar Y)$ where $V\hat\otimes_\varepsilon W$ is the injective ...

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### Ideal structure of a tensor product of certain algebras

I would be grateful if anyone could give me a reference regarding the following question.
Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar ...

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### Interpretations for higher Tor functors

Let's work in the category $R$-${\sf mod}$, for concreteness. I know that one can see the modules ${\rm Ext}^n_R(M,N)$ as modules of equivalence classes of $n$-extensions of $M$ by $N$ (Yoneda ...

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### Highest-$\ell$-weight tensor products and diagram subalgebras

Let $U_q(\mathcal{L}({\mathfrak{g}}))$ be a quantum loop algebra and $I$ the set of indexes of Dynking diagram of $\mathfrak{g}$. Consider $J\subset I$ a connected subdiagram, so that $U_q(\mathcal{L}(...

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### Simple modules for direct sum of simple Lie algebras

I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...

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### On diagonal part of tensor product of $C^*$-algebras

Suppose we have a $C^*$-algebra $\mathcal{U}$, Consider the $C^*$-subalgebra generated by elements of the form $a\otimes a$, what is it isomorphic to? Is it isomorphic to $\mathcal{U}$ itself?

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### Non-tensor-representable ultrafilters on $\omega$

If ${\cal U}$ and ${\cal V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\cal U}\otimes{\cal V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\...

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### Axiom of choice and algebraic tensor product

The first part of the question was asked on Math-stackexchange.
Let $V$, and $W$ be vector spaces. By the universal property of the tensor product,
there is a canonical map from $V^*\otimes W^*$ ...

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### If C is a cocomplete coalgebra, then $\psi:C\rightarrow B\Omega C$ is a filtered quasi-isomorphism

I am reading the PhD thesis thesis of Kenji Lefèvre-Hasegawa and the corresponding errata by Bernhard Keller, my question is about the first error found in the thesis. Lemma 1.3.2.3 c states 'the ...

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### Tensor square of duals over a domain

The title is motivated by my needs ($M=N$ in the sequel).
Linked to the question here and there (in the case of products) is the following.
Let $M,N$ $k$-modules ($k$ a commutative ring), then we ...

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### The symmetric power of a tensor product

In the representation theory, if $S^{\lambda}(V)$ is the irreductible representation of $\text{GL}(V)$ associated to a partition $\lambda \vdash n$ (in perticular, $S^n(V)$ is the $n^{\text{th}}$ ...

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### tensor stability of block-positive matrices

Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation)
$\langle \psi |...

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### How to derive a bound of distortion / error between two different tensor decompositions

Consider a tensor $\mathcal{X}\in\mathbb{R}^{I\times J\times K}$. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition:
$\mathcal{X}\approx \sum_{p=...

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### Global dimension of the tensor algebra

Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$.
Question 1: Is there a simple proof that $...