10 votes
Accepted

A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$

WLOG, we may assume that $\phi$ is injective and identify $\mathfrak{g}$ with its image in $\mathrm{End}(V)$. Our goal is to construct a reductive algebraic subgroup of $\mathrm{GL}(V)$, whose Lie ...
  • 4,865
7 votes

What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{SO(2)} )?

Write $z = x + iy, \bar{z} = x - iy$ as usual, where $x, y \in \mathbb{C}[x, y]$ are regarded as complex-valued polynomial functions on the plane. The action of $SO(2)$ diagonalizes as $$z \mapsto e^{...
5 votes

How to define cohomology of algebraic structures?

There is a tremendous amount of abstract formalism answering this question in various levels of generality depending on what you want to do. I'll pick one in the middle: the machinery of derived ...
5 votes
Accepted

Does the isometry group determine the Riemannian metric?

I think that you are missing some hypotheses on the action of $G$, otherwise there are trivial counterexamples. For example, if $G\subset\mathrm{Iso}(M,g)$ is the trivial group, then one clearly ...
5 votes
Accepted

Particular reduced expression of the longest element of Weyl group

Whenever $\ell(wv)=\ell(w)+\ell(v)$, you can construct a reduced word for $wv$ by producing one for $w$ and one for $v$ and then concatenating them. So, if you know an algorithm for producing reduced ...
  • 41.8k
4 votes

A few reference questions about the Baker–Campbell–Hausdorff formula

$\DeclareMathOperator\ad{ad}$The identity $$\log(e^X e^Y) = X + \frac{\ad_X}{1 - e^{-\ad_X}}Y + O(Y^2)\tag{$*$}\label{star}$$ is due to Poincaré [1], who showed that $W(t)=\log (e^X e^{tY})$ solves ...
3 votes

Particular reduced expression of the longest element of Weyl group

As I mentioned in a comment, this is a special case of finding a distinguished representative for a coset of a parabolic subgroup. Let's inductively define elements $w_n$, with the convenient ...
  • 8,683
1 vote

3-dimensional principal subalgebra in the real case

If your definition is a subalgebra $\mathfrak{s}$ which complexifies to a principal TDS you immediately have two distinct types $\mathfrak{s} \cong \mathfrak{su}_2$ or $\mathfrak{s} \cong \mathfrak{sl}...
  • 647
1 vote
Accepted

On Euler angles decomposition of $\mathrm{SU}(N)$

Building on the paper Idel and Wolf - Sinkhorn normal form for unitary matrices Colin McQuillan suggested, it is easy to see that every $\operatorname{SU}(N)$ matrix $m$ can be decomposed as $$ m = a \...

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