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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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### 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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### 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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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: $$\sum_{i=1}^{N^2-1}T^i_{ab}T^... 0answers 79 views ### Almost complex structures on real Lie groups Let G be an even-dimensional Lie group, and let \mathfrak{g} be its Lie algebra. I want to explore when a complex structure on \mathfrak{g} induces a complex structure on G making it into a ... 0answers 109 views ### On the diameter of left-invariant sub-Riemannian structures on a compact Lie group Let G be a compact connected Lie group with Lie algebra \mathfrak g of dimension m. We fix an inner product \langle\cdot,\cdot\rangle on \mathfrak g. We may assume (in case is necessary) ... 0answers 51 views ### Haar measure of the zero set of a nonconstant analytic function on a connected Lie group Let G be a connected Lie group equipped with its unique real analytic structure, f : G \to \mathbb{R} a nonconstant real analytic function on G. Is the closed set Z_f = f^{-1}(0) always of ... 1answer 160 views ### Integral of Schur functions over SU(N) instead of U(N) Schur functions are irreducible characters of the unitary group U(N). This implies$$ \int_{{U}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\delta_{\lambda\mu},$$where the overline means complex ... 0answers 51 views ### The Lie subgroup corresponding to inner derivations Let \mathfrak{g} be a finite-dimensional real or complex Lie algebra. We know that Aut(\mathfrak{g}) is a closed real or complex Lie subgroup of GL(\mathfrak{g}). We also know that the Lie ... 1answer 91 views ### Coinvariant representative of homogeneous space cohomology Given a compact homogeneous space M = K/L, consider its de Rham complex (\Omega^*,d). Will every cohomology class [\omega] \in ker(d)/im(d) contain a representative \nu which is invariant with ... 0answers 133 views ### Characterize an element of \operatorname{SL}_n(\mathbb Z) I'm trying to generalize a theorem on \operatorname{SL}_n(\mathbb Z) to the Chevalley groups over \mathbb Z. In the theorem, there is a heavy use in the element e_{1,n}(1) where$$e_{1,n}(m)= ...

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