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      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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      49 views

      A geometric property of certain Lie groups

      What is an example of a Lie group $G$ not ismorphic to the Poincare upper half space $H^n$ but satisfy the following: For every left invariant metric $g$ we draw all geodesics. Then we CAN rescale $...
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      33 views

      Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$

      Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\Delta = \{e_1 - e_2, ... , e_{n-1} - e_n, e_n\}$$ be a set of simple roots of $T$ corresponding to a ...
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      53 views

      Structure of extensions arising in Lie approximation of connected groups

      My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known: Let $G$ be a connected, locally compact, Hausdorff group, ...
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      59 views

      Finding the closest special orthogonal matrix in Frobenius norm sense

      Given a $3\times3$ matrix $M$, if we would like to get the closest $\mathrm{SO}(3)$ matrix $R$ that minimizes \begin{equation} \|R-M\|_F \end{equation} then $R$ = $UV^{T}$ where $U$ and $V^{T}$ are ...
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      49 views

      Nontrivial relations of the irreducible root systems

      For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$, we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,...
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      82 views

      Lattices in Lie groups

      In the literature, people seem to predominantly look at lattices in nilpotent or reductive groups. Is there a result that gives a general description of a lattice in an arbitrary Lie group? Something ...
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      1answer
      100 views

      Analog of the Lie Product formula for commutators

      Let $X, Y$ be elements of a Lie algebra. Consider the group $G$ generated by (limits of) arbitrary products of the elements $$ G = \langle{e^{tX},e^{sY}\rangle}$$ for all $t,s$. The Lie product ...
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      51 views

      Fierz identity for symplectic group

      It is known that for the fundamental matrix representation of SU(N), with normalization given by $$ {\rm Tr}(T^iT^j)=\frac{1}{2}\delta^{ij} $$ there is a Fierz identity: $$ \sum_{i=1}^{N^2-1}T^i_{ab}T^...
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      79 views

      Almost complex structures on real Lie groups

      Let $G$ be an even-dimensional Lie group, and let $\mathfrak{g}$ be its Lie algebra. I want to explore when a complex structure on $\mathfrak{g}$ induces a complex structure on $G$ making it into a ...
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      109 views

      On the diameter of left-invariant sub-Riemannian structures on a compact Lie group

      Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$. We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$. We may assume (in case is necessary) ...
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      51 views

      Haar measure of the zero set of a nonconstant analytic function on a connected Lie group

      Let $G$ be a connected Lie group equipped with its unique real analytic structure, $f : G \to \mathbb{R}$ a nonconstant real analytic function on $G$. Is the closed set $Z_f = f^{-1}(0)$ always of ...
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      1answer
      160 views

      Integral of Schur functions over $SU(N)$ instead of $U(N)$

      Schur functions are irreducible characters of the unitary group $U(N)$. This implies $$ \int_{{U}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\delta_{\lambda\mu},$$ where the overline means complex ...
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      51 views

      The Lie subgroup corresponding to inner derivations

      Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra. We know that $Aut(\mathfrak{g})$ is a closed real or complex Lie subgroup of $GL(\mathfrak{g})$. We also know that the Lie ...
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      1answer
      91 views

      Coinvariant representative of homogeneous space cohomology

      Given a compact homogeneous space $M = K/L$, consider its de Rham complex $(\Omega^*,d)$. Will every cohomology class $[\omega] \in ker(d)/im(d)$ contain a representative $\nu$ which is invariant with ...
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      133 views

      Characterize an element of $\operatorname{SL}_n(\mathbb Z)$

      I'm trying to generalize a theorem on $\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$. In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where $$e_{1,n}(m)= ...

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