# Questions tagged [tensor-products]

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245
questions

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### Computing a cone in a $\otimes$-triangulated category

I have a $\otimes$-triangulated category $\mathcal T$ and two triangles in $\mathcal T$:
$$
x_0\to x_1\to c_x\to \Sigma x\ \ \ \text{and}\ \ \ y_0\to y_1\to c_y\to \Sigma y.
$$
Consider the following ...

**1**

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28 views

### Sobolev tensor spaces and finite ranks

Let $W^{2,2}(\Omega_i)$, $\Omega_i = [-1,1]$, $i = 1,\ldots,d$ be the Sobolev spaces of twice weakly differentiable, square integrable functions. Let further $\otimes_a$ denote the algebraic tensor ...

**0**

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**1**answer

33 views

### Concentration tensor product with a rank-1 random tensor with sub-Gaussian elements

Suppose $A\in\mathbb{R}^{n^k}$ is a $k$-dimensional tensor with $n$ elements along each dimension. Morover suppose $u_1,u_2,\dots,u_k\sim\text{Unif}(\pm1)^n$ are $n$ dimensional vectors with each of ...

**4**

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89 views

### Linear maps preserved by algebra automorphisms

Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field.
Let $s_A:A\to F$ be an $F$-linear map.
Now consider an arbitrary field extension $K/F$, and define $s_{A\...

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109 views

### An explicit formula for characteristic polynomial of matrix tensor product [closed]

Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of ...

**4**

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**1**answer

309 views

### Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

Edit: According to the comment of Todd Trimble, I revise the question.
What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...

**5**

votes

**2**answers

420 views

### Zero tensor product over a complex algebra?

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space.
Q1: Can this ...

**3**

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**1**answer

157 views

### Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$

Every matrix $A\in M_4(\mathbb{R})$
can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$.
What is the least uniform upper bound $M$ for such $n(A)$...

**2**

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35 views

### 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\...

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116 views

### 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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40 views

### 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 ...

**3**

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**2**answers

194 views

### 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 ...

**17**

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**1**answer

178 views

### 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 ...

**2**

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38 views

### 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}(...

**2**

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**1**answer

122 views

### 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 ...