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Tagged with lie-groups matrix-exponential
5 questions
5
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146
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Is every linear Lie group of bounded geometry?
$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...
4
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277
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What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
3
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Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators
Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...
2
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Does restricting the eigenvalues of Hermitian matrices to the interval $[0,2\pi)$ make the exponential map to the unitary group bijective?
Let $U \in U(n)$ be a generic unitary matrix. Since the unitary group $U(n)$ is compact and connected, I know that the exponential map is surjective, i.e. that every $U \in U(n)$ has the form $U = e^{...
1
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Possible compatible ""structure" on the Lie algebra associated with a Lie group with respect to group structure
For simplicity, let us work on an example.
Regard $GL_r(\mathbb C)$ as a Lie group with associated Lie algebra $M_r(\mathbb C)$, then there exists a canonical so called exponential map:
$$\exp: M_r(\...