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2 votes
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Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules

I'm not quite sure this question rises to the level of MathOverflow, which is why I initially posted only a comment, but at the request of the question-asker I am converting my comment to an answer. ...
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6 votes
Accepted

Number of representations of a semisimple Lie algebra of any given dimension

For $\mathfrak{sl}_2\times \mathfrak{sl}_2$, the number of irreps of dimension $n$ is the number of factorizations $n=n_1n_2$ (you tensor the irreps of the two $\mathfrak{sl}_2$'s), so there's no ...
  • 42.4k
2 votes

$\varphi ( \mathbb{Z}^2)$ is not discrete in $G$

Unsure if it is truely research level. Anyways, the Abeliean subgroups of $\text{Aff}(\mathbb{R})^{o}$ are either the embedded torus $G_{m}(\mathbb{R})^{o} = GL_{1}(\mathbb{R})^{o} = \mathbb{R}_{>0}...
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3 votes

Topological vector spaces in direct sum

Let $H$ be a separable infinite-dimensional Hilbert space and $(e_i)_{i\in\mathbb N}$ an Orthonormal basis. Pick a vector $v_0$ not in the span of the $e_i$. Then the one-dimensional span $W$ of $v_0$ ...
  • 757
2 votes

What do the Pauli matrices say about the Threefold Way?

Q: What do the Pauli matrices say about the Threefold Way? In the context of Dyson's threefold way, the Pauli matrices produce two of the three ensembles of random Hamiltonians. A Hermitian matrix $H$ ...
1 vote
Accepted

What does the boundary of convex hulls look like in matrix Lie groups?

I guess you wanted to say smallest geodesic polytope (not polygon). It is unclear what is polytope in a the matrix Lie group, but it seems to require geodesic hypersurfaces. They do not exist in most ...
0 votes

Problem in understanding the coadjoint action of $\mathfrak {g}^{\ast}$ on $\mathfrak {g}$

As @Michael Barz pointed out in his comment above it turns out that $$\text {ad}_{b'}^{\ast} (\xi) = \varphi^{-1} \circ \left (\text {ad}_{b'} (-\xi) \right )^{\ast} \circ \varphi$$ where $\varphi : \...
3 votes

Complete representation theory of $\mathrm{SL}(2,\mathbb R)$

Yes, this is all completely settled and for the finite dimensional case it is the basic underpinning of much of Lie theory. The irreducible finite-dimensional representations of $\operatorname{SL}(2,\...
  • 777
7 votes
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Is a Lie subgroup whose center is closed, a closed subgroup itself?

No. Consider a group of the form $G=V\rtimes K$ where $V$ is a Euclidean group and $K$ is a compact 2-torus and $K$ acting faithfully on $D$, with no nonzero invariant vector. (For instance $G=(\...
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2 votes
Accepted

Sum of weights of an irreducible representation of $U(N)$

As discussed in the comments, your sum is a Weyl-fixed character, so trivial for $G = \operatorname{SU}(N)$ and a multiple of $\det = (1, \dotsc, 1)$ for $\operatorname U(N)$. To be concrete, as I ...
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5 votes
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Classification of Lie group structures on $\mathbb{R}^n$

YCor claims here that the contractible Lie groups (this is equivalent to being diffeomorphic to $\mathbb{R}^n$, since a connected Lie group is diffeomorphic to the product of a Euclidean space times ...
1 vote

Complete representation theory of $\mathrm{SL}(2,\mathbb R)$

For unitary infinite dimensional representations of $SL(2, \mathbb{R})$ and $SL(2, \mathbb{C})$, I would look at A. Knapp, Representation theory of semisimple groups, 1986 R. Takahashi, $SL(2, \...
2 votes
Accepted

Unitary dual of universal cover

Yes. (a) This is equivalent to ask about the existence of an extremal normalized positive-definite function $\phi$ on $G$ that is "faithful" on its infinite cyclic center $Z$, that is, $\phi^...
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10 votes
Accepted

Faithful locally free circle actions on a torus must be free?

This follows from Theorem 9.3 (page 216) of the book "Compact tranformation groups" by Bredon (note that it is freely available online). More is true: Any effective group action of a torus ...
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