# Questions tagged [lie-groups]

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
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### The largest abelian subgroups of a Lie group [duplicate]

Let $G$ be a semisimple Lie group. Denote by $d(G)$ the maximal integer $p$ such that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$, and let $c(G)$ denote the maximal integer $q$ such ...
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### Classification of Lie group structures on $\mathbb{R}^n$

Is it possible to describe, up to isomorphism, all Lie groups $G$ whose underlying manifold is diffeomorphic to $\mathbb{R}^n$ (with its standard smooth structure)? In fact, I haven't found any such ...
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### Complete representation theory of $\mathrm{SL}(2,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...
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### Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .$$ Basic ...
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### Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\$?

Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
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### The homotopy type of the space of symplectic structures

While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
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### Left translations respect the Schouten bracket

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $r \in \bigwedge^2 \mathfrak{g}$. For $x \in G$ let $\lambda_x$ denote the left multiplication by $x$. Let $[\cdot, \cdot]$ ...
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### Faithful locally free circle actions on a torus must be free?

Do we have an example of a smooth action $S^1 \curvearrowright T^n$ which is faithful, locally free but not free? I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$. Another ...
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### Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made. In this paper, $\mathfrak{g}$ is a ...
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### Positive radial function with compactly supported Fourier transform

Let $G$ be a non-compact semisimple Lie group with finite center (for example $SL_2(\mathbb{R})$). Choose a maximal compact subgroup $K$. A bi-$K$-invariant function is called radial. Let $A$ be a ...
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### On the center of the universal enveloping algebra of a Lie algebra

Consider a finite-dimensional Lie algebra over C. Sometimes (e.g. for any semisimple Lie algebra) the center of its universal enveloping algebra is isomorphic to a freely generated polynomial ring. ...
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### Symmetric spaces and spherical functions

Let $G$ be a connected compact Lie group, $\sigma$ an involutive automorphism and $K$ a subgroup such that $(G^\sigma)_0\subset K\subset G^\sigma$. Then $M=G/K$ is a Riemannian symmetric space. ...
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### Relation between projective representation and the representation of the universal cover of a Lie Group

I am reading this paper, in what says exactly: "Weare dealing with a ray representation os the conformal group AND THEREFORE with a representation of the universal covering group of the conformal ...
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### Injectivity of the cohomology map induced by some projection map

Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence $$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$ where $G_c$ is the normal subgroup which ...
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### Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds

I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...
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### How many diagrams interlace a given Young diagram?

For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff \lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...
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### Derived subgroups of 2-adic congruence subgroups of $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$. Explicit formulas with formal ...
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### Free $S^1$-action on compact homogeneous spaces

Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$. If $r(G) > r(K)$ (...
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Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Denote by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $r$ be an element of $\mathfrak{g}^*$ such that $G_r$ the stabilizer of ...
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### Generalization of $G/T \simeq G_\mathbb{C}/B$

Let $G$ be a compact Lie group and Let $G_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$. Consider $H$ to be a Lie subgroup of $G$ and denote ...
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Let $M$ be a compact connected manifold. The degree of symmetry of $M$, denoted $N(M)$, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $M$. ...