# Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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### Tensor Product is associative, distributive, not commutative.

Tensor Product is associative, distributive, not commutative. Here is my attempt to show tensor product is associative, is it legit? If $T$ is a $p$-tensor and $S$ a $q$ tensor, then $T \otimes S$ ...
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### Dual of a finite dimensional algebra is a coalgebra (ex. from Sweedler)

Let $(A, M, u)$ be a finite dimensional algebra where $M: A\otimes A \rightarrow A$ denotes multiplication and $u: k \rightarrow A$ denotes unit. I want to prove that $(A^*, \Delta, \varepsilon)$ ...
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### Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
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### Why do we want or need cross-norms on tensor products?

If $(E, \|\cdot\|_1),(F, \|\cdot\|_2)$ are Banach spaces, their (algebraic) tensor product $E \otimes F$ is a vector space looking forward to be normed. A norm on a tensor product of vector spaces is ...
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### Is the tensor product of two real numbers a real number?

I am on a physics course where they've introduced the tensor product as if it was something that I would have seen before, and they don't tell me any of its properties. However, I am trying to work ...
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### when is the map $m \mapsto 1 \otimes m$ injective?

I just started to study tensor products and have a question. Suppose A is a (commutative with unity) ring, B an A-algebra and M an A-module. Consider map $\alpha: M \rightarrow B \otimes_AM$, defined ...
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### Tensor product of $k$-algebras, center, isomorphism.

Let $A$, $B$ be two $k$-algebras of finite dimension, where $k$ is a field. Here, $A$ and $B$ are not necessarily commutative. Do we have that$$Z(A \otimes_k B) \cong Z(A) \otimes_k Z(B),$$where $Z(-)$...
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### Alternative introduction to tensor products of vector spaces

One of the main obstacles in understanding the tensor product is that, unlike many other algebraic structures, you cannot really get hold of its element structure. This confuses many beginners. The ...
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### Is there a relationship between the trace and the Clifford/geometric product?

In what follows, let $V=\mathbb{R}^n$ (although the following probably applies also to a larger number of finite-dimensional spaces). We assume throughout that we have made a choice for an inner ...
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### Extension of scalars functor essentially surjective.

Let $f:A\rightarrow B$ be a morphism of rings. When is every $B$-module $N$ of the form $M\otimes_A B$ for some $A$-module $M$?. What are sufficient and necessary conditions on $f$? I know for ...
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### Tensor of tensored categories

Given two $V$-categories $C$ and $D$ tensored over a symmetric monoidal category $V$, could I form the "tensor" of $C$ and $D$? More precisely, is there a $V$-category $T(C,D)$ such that $V$-functors ...
### Decomposition of order-$n$ tensors
If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...