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        Questions tagged [tensor-products]

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        5
        votes
        1answer
        104 views

        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
        vote
        0answers
        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
        votes
        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 ...
        4
        votes
        1answer
        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\...
        1
        vote
        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 ...
        4
        votes
        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 ...
        5
        votes
        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 ...
        3
        votes
        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 $n(A)$...
        2
        votes
        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\...
        0
        votes
        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 ...
        2
        votes
        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 ...
        3
        votes
        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 ...
        17
        votes
        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 ...
        2
        votes
        0answers
        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
        votes
        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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