# Questions tagged [tensor-powers]

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16
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### Is there an element in an infinite tensor power of a C*-algebra that is invariant under finite permutations?

Let $A$ be a C*-algebra, and consider the infinite tensor power $A^{\otimes {J}}$, where $J$ is infinite (we consider the minimal or maximal tensor product). To any finite permutation, which is a ...

9
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### Characterising natural transformations between tensor functors

I would like to know if the following conjecture is correct and if so what's a good citation for its proof.
Let $\mathsf{E}$ be the category of euclidean vector spaces, i.e. objects are finite-...

7
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### Is every basis for $\bigwedge^kV$ satisfying a "complementary" property a rescaling of a "standard" basis?

This is a cross-post.
Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$
be a basis for $\bigwedge^2V$, where each $\omega_{i_1,i_2}$ is decomposable. Suppose that for every $\...

4
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### Symmetric power contained in tensor power?

Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be ...

2
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0
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### The first non-trivial Schur functor [closed]

I am trying to understand the Schur functor $S^{(2,1)}$. Let's try on a vector space $V$ of dimension 3. The general definition is :
$S^{\lambda}V = V^{\otimes n} \otimes_{S_n} V^{\lambda}$
where $V^...

19
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2
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### Exponentiation of vector spaces?

This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...

3
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0
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357
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### Symmetric power of an algebra

Given an algebra $A$ over $k$ with characteristic zero and a positive integer $n$, the subspace of $A^{\otimes n}$ consisting of all tensors invariant under the action of all permutations $\sigma\in\...

0
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2
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### Tensor powers of an algebra all isomorphic

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume ...

4
votes

2
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448
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### Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers

Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map
$$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...

2
votes

1
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### Iterated Reduced Tensor Power of Graded Vector spaces

This might be inappropriate for the MO-level. If so I'll delete it...
Suppose $V$ is a $\mathbb{Z}$-graded vector space and
$\overline{T}(V):=V \oplus V\otimes V \oplus \otimes^3 V \ldots$
is the '...

6
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2
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### Alternating multilinear invariants of GL(n) on End (k^n)

Introduction. Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus ...

6
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3
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### Torsion-free tensor powers

Let $R$ be an integral domain. If $M$ is an $R$-module such that every tensor power of $M$ over $R$ is $R$-torsion-free, then is $M$ necessarily flat as an $R$-module? If not, then does this ...

1
vote

0
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### Decomposition of product of exterior products

Suppose $V$ is a finite dimensional vector space of dimension n.
What is the kernel of the map
$$\bigwedge^p V \otimes \bigwedge^q V ----> \bigwedge^{p+q} V$$ ?
(here $p+q< n$)
Thanks..
...

4
votes

1
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### Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?

Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...

2
votes

1
answer

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### Restricted universal enveloping algebra of Abelian p-Lie algebra

Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...

16
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### What do gerbes and complex powers of line bundles have to do with each other?

We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but ...