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### $SO_4(\mathbb{R})$ and $SU_2 \otimes SU_2$ subgroups of $SU_4$

Yes, $SU(2)\otimes SU(2)$ is conjugate to the usual $SO(4)$ in $SU(4)$. There's probably a direct way to see it, but here's some theory. Suppose $G = G_1\times G_2$ is a product of compact Lie groups....
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### Equivalent definitions of the tensor product.

Your suspicion is correct: the definition given in your differential geometry class works for finitely generated free modules but not in general. There is a more general definition found in many ...

### Definitions - Tensor Products and Tensors

Let me try to answer question (ii). If $V$ is a vector space over a field $F$, then the space of $(p,q)$-tensors over $V$ is the space $$T^p_q(V)=V^{\otimes p} \otimes (V^*)^{\otimes q}.$$ A $(p,q)$-...
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### Confusion regarding the definition of tensor product of modules.

Based on your post, I'm going to assume that $R$ is commutative. A tensor product of the $R$-modules $M$ and $N$ is a pair $(T,\varphi)$ where $T$ is an $R$-module and $\varphi:M\times N\to T$ is a ...
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### Notation in defining the abstract tensor product

To illustrate with a simple example, let's take $n=2$ and $V_1=V_2=\mathbb{R}$. Then $F=\mathcal{F}(\mathbb{R}\times\mathbb{R})$ is the space of formal finite linear combinations of pairs of real ...
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For any $A$-module $M$ and any ideal $I$ of $A$, the natural map $M \otimes_{A} A/I \to M/IM$ sending the simple tensor $m \otimes [a]$ to $[am]$ is an isomorphism of $A$-modules. If $\rho \colon A \... • 19.5k 1 vote Accepted ### Product of projection maps It is not exactly tensor product, but composition of tensor product and diagonal map. In general for$f:X\to Y$and$g:A\to B$we define $$f\times g : X\times A \to Y\times B,\quad (f\times g)(x, a) =... • 1,090 1 vote Accepted ### A scalar field satisfying the Laplace's Equation is zero everywhere I'm going to assume that S is smooth closed surface that is embedded into \mathbb R^3. The smoothness assumption can be relaxed to C^1. Also, I think the assumption that S is embedded is ... • 4,296 1 vote ### Is this how to prove the two-sided ideal is spanned by v_1\otimes\dots\otimes v_k where the v_1,\dots,v_k are linearly dependent? Let I_n be the subspace of V^{\otimes n} generated by elementary tensors v_1\otimes\cdots\otimes v_n in which two consecutive factors are equal, and let f:V\times\cdots\times V\to V^{\otimes n}/... 1 vote Accepted ### Decomposition of an operator x\in B(H\otimes K) as a sum x=\sum_{i,j} x_{ij}\otimes E_{ij}. You have the identity\def\e{\epsilon}$$\tag1 \sum_i\pi_i(\xi\otimes\e_j)\otimes \e_i=\xi\otimes\e_j.$$Let$\tilde y=\sum_{\ell\in G_0} \eta_\ell\otimes\e_\ell$, the sum over some finite set$G_0\$....
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