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# Questions tagged [tensor-products]

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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 ...
0answers
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### 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 ...
1answer
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 ...
1answer
89 views

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 $s_{A\... 0answers 109 views ### An explicit formula for characteristic polynomial of matrix tensor product [closed] Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of ... 1answer 309 views ### Functors on the category of abelian groups which satisfy$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 $$F(G\times H)\cong ... 2answers 420 views ### Zero tensor product over a complex algebra? Let A be an algebra over \mathbb{C}. Let M be a left A-module, let N be a right A-module and consider the tensor product N \otimes_A M, which is a complex vector space. Q1: Can this ... 1answer 157 views ### Representation of 4\times4 matrices in the form of \sum B_i\otimes C_i Every matrix A\in M_4(\mathbb{R}) can be represented in the form of$$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$n(A)$... 0answers 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\...
0answers
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 ...
0answers
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 ...
2answers
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 ...
1answer
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 ...
0answers
38 views

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 $U_q(\mathcal{L}(... 1answer 122 views ### Simple modules for direct sum of simple Lie algebras I think that the following statement is true, but I do not know how to prove it. Let$\mathfrak{g}_1$and$\mathfrak{g}_2$be two real simple Lie algebras. If$M\$ is a (infinite dimensional) complex ...

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