3
votes
Accepted
Spanning set of symmetric invariants of tensor powers
Here is a counterexample. Let $k$ be a field of characteristic $2$, let $R=k[x,y,z]$, and let $M$ be the ideal $(x,y,z)\subset R$. Note that $xy\otimes z=x\otimes yz=xz\otimes y=z\otimes xy$ in $M^{\...
- 312k
3
votes
Accepted
Intuition behind canonical definition of inner product structure of tensors
Canonical is an imprecise word, but here is a justification. An inner product on $V$ is a map $V \otimes V \to k$ with certain properties. So having inner products on $V$ and $W$ gives a map
$$(V \...
- 7,781
3
votes
Accepted
Is $SU(2)\otimes SU(3)$ a subgroup of $SU(4)$?
I think that usually $\mathrm{SU}(2)\times\mathrm{SU}(3)$ is the direct product, where we can interpret its elements concretely as block-diagonal matrices with blocks from $\mathrm{SU}(2)$ and $\...
- 19.4k
2
votes
$E^{*} \otimes E^{*}$ is canonically isomorphic to $Hom^2(E × E , \mathbb{R})$
I would solve part II by using part I. I assume your answer to part I was something like this:
The tensor product $T:=E\otimes E$ is uniquely determied by the following property:
There exists a ...
- 611
2
votes
Accepted
$E^{*} \otimes E^{*}$ is canonically isomorphic to $Hom^2(E × E , \mathbb{R})$
Take the map
$f\colon E^* \times E^* \to \mathrm{Hom}(E\times E;\mathbb R)$
mapping $(\alpha,\beta)$ to a homomorphism $f(\alpha,\beta)$ acting on $(v,w) \in E \times E$ as $f(\alpha,\beta)(v,w) := \...
- 7,854
2
votes
Accepted
Matrix of operator on space of symmetrical matrices
I am going to reformulate the problem somewhat. It suffices to only consider this problem when working over $\mathbb R$. We can easily abstract to arbitrary fields (see end of post).
Let $V:=M_n(\...
- 8,725
2
votes
Accepted
Tensor Products of R-Algebras from Atiyah and Macdonald
That is an incorrect internal reference. It should say (2.12) instead of (2.11).
Anyway, setting $M = S \otimes_R T$, you are asking how a linear map $\psi \colon M \otimes_R M \to M$ corresponds to ...
- 37.2k
2
votes
Accepted
On the ideal of entries of a morphism between free modules
Yes, they are equivalent.
Write down the matrices representing the maps $f \otimes_R M : M^{\oplus a} \to M^{\oplus b}$ and $\operatorname{Hom}_R(f,M) : M^{\oplus b} \to M^{\oplus a}$. The former is ...
- 12.1k
2
votes
Induced filtration on polynomial ring with coefficients in a filtered associative algebra
It depends.
If you really consider $t$ to have degree $0$ and therefore to have $k[t]$ unfiltered (or rather with trivial filtration), then the induced filtration on $A[t]$ should be the first one.
If ...
- 23.7k
1
vote
Accepted
Deducing a $\frac{Dv}{Dt}$=$\frac{∂v}{∂t}$+$\frac{1}{2}$∇ (|v|$^2$) for an irrotational flow.
The solution given by ryaron is clever but overly complicated. The result can be very easily obtained using the product rule and the symmetry of the deformation tensor.
Let $\mathbf D=\nabla\...
- 10.6k
1
vote
Accepted
If $B$ is a flat $A$-algebra, $M$ an $A$-module, and $x\neq 0\in M$, why does $Ax \cong A/\mathfrak a$ imply $B(1\otimes x) \cong B/\mathfrak{a}^e$
There is an exact sequence $$0\to\mathfrak{a}\to A\stackrel{x}\to M.$$ Since $B$ is flat, this exact sequence remains exact after tensoring with $B$, giving an exact sequence $$0\to\mathfrak{a}\otimes ...
- 312k
1
vote
Tensor product over several modules
For each $p \in P$, the map
\begin{align}
M \times N &\to M \otimes N \otimes P \\
(m,n) & \mapsto m \otimes n \otimes p
\end{align}
is bilinear, and so there is a unique linear map
$$
f_p \...
- 19.6k
1
vote
Accepted
Why are the last 2 expressions in the following diagram are equal?
Fix $n$. We have:
$$
\begin{aligned}
\sum_{\substack{p,q\ge 0\\p+q=n}}
\sum_{\substack{k,l\ge 0\\k+l=p}}
t^k\otimes t^l\otimes t^q
&=
\sum_{\substack{q,k,l\ge 0\\k+l+q=n}}t^k\otimes t^l\otimes t^q\...
- 28.2k
1
vote
Show two nonzero vectors $v$ and $w$ have the property $v\otimes w=w\otimes v$ iff there is $\lambda$ such that $v=\lambda w$
I just found a way with the universal property that needs not you trust that $e_i \otimes e_j$ form a basis. (As Olivier showed, if you trust that, then the statement is trivial since it is just ...
- 299
1
vote
Proving the uniqueness of a map
It seems you use the cocommutativity of $D$ while proving $g = (f \otimes f' ) \circ \Delta_D$ is a coalgebra morphism. Here $\Delta_D$ is a coalgebra morphism if and only if $D$ is cocommutative, and ...
- 3,160
1
vote
Accepted
Help Understanding the Tensor Algebra
The first question was already answered in a comment. It is true that if $A, B \in T^2(V)$, then $A \otimes B$ is in general not in $T^2(V)$.
The interpretation of $S(X)$ is as in the answer in the ...
- 7,854
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