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        Questions tagged [tensor-products]

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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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        1answer
        138 views

        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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        184 views

        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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        198 views

        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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        38 views

        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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        134 views

        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 $...
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        When are “square spans” not transversal?

        Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...
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        Jacobson radical of a tensor product

        Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
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        How do fractional tensor products work?

        [I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...
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        214 views

        Direct proof of “Nuclear implies $C_{red}^*(G) \cong C^*(G)$”

        It is well-known that for a discrete group $G$ the following statements are equivalent: $C_{red}^*(G)$ nuclear $C_{red}^*(G) \cong C^*(G)$ canonically i.e. there exists an *-isomorphism between the ...

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