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Journal of Lie Theory, 1995
A connected Lie group \(G\) is called spacious if there exists an open subset \(U \subseteq G\) such that \(U^n \cap U^{n + 1} = \emptyset\) for all \(n \in N\). This property is closely related to the behaviour of the exponential function \(\text{exp} : g \to G\) because, according to a result of Jaworski, \(G\) is spacious if and only if \(\text{exp }
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A connected Lie group \(G\) is called spacious if there exists an open subset \(U \subseteq G\) such that \(U^n \cap U^{n + 1} = \emptyset\) for all \(n \in N\). This property is closely related to the behaviour of the exponential function \(\text{exp} : g \to G\) because, according to a result of Jaworski, \(G\) is spacious if and only if \(\text{exp }
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The Annals of Mathematics, 1947
where i a =-(2(ai)2)1/2 and where F satisfies the sole condition that F -O 0 as a -O 0, b 0. The coordinate system is right-regular if I b I replaces I a I in (1.1). A coordinate system a, *., at is analytic if the coordinates (ab)t of ab are expressible as power series in a', * , a, bl, * , bT which converge for some domain: I a I < 6, I b I < 6.
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where i a =-(2(ai)2)1/2 and where F satisfies the sole condition that F -O 0 as a -O 0, b 0. The coordinate system is right-regular if I b I replaces I a I in (1.1). A coordinate system a, *., at is analytic if the coordinates (ab)t of ab are expressible as power series in a', * , a, bl, * , bT which converge for some domain: I a I < 6, I b I < 6.
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On the tangent Lie group of a symplectic Lie group
Ricerche di Matematica, 2019zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Journal of Lie Theory, 1995
Various notions of approximation of locally compact groups by Lie groups have been studied in the literature, and there are indications that there is some danger of confusion. Therefore, the author undertakes a systematic comparison. A Lie-normal family in a locally compact Hausdorff group \(G\) is defined as a set \({\mathcal N}\) of normal subgroups ...
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Various notions of approximation of locally compact groups by Lie groups have been studied in the literature, and there are indications that there is some danger of confusion. Therefore, the author undertakes a systematic comparison. A Lie-normal family in a locally compact Hausdorff group \(G\) is defined as a set \({\mathcal N}\) of normal subgroups ...
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Canadian Journal of Mathematics, 1994
AbstractThe heat kernel of an amenable Lie group satisfies either pt ~ exp(—ct1/3) or pt ~ t-a as t → ∞. We give a condition on the Lie algebra which characterizes the two cases.
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AbstractThe heat kernel of an amenable Lie group satisfies either pt ~ exp(—ct1/3) or pt ~ t-a as t → ∞. We give a condition on the Lie algebra which characterizes the two cases.
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2020
In this chapter, we recall some well-known results on Lie groups and Lie algebras. In particular, we discuss the third Lie theorem, the Ado theorem, and the Cartan semisimplicity criterion. Some important types of Lie algebras and Lie groups together with their important ideals and normal subgroups are discussed.
Valerii Berestovskii, Yurii Nikonorov
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In this chapter, we recall some well-known results on Lie groups and Lie algebras. In particular, we discuss the third Lie theorem, the Ado theorem, and the Cartan semisimplicity criterion. Some important types of Lie algebras and Lie groups together with their important ideals and normal subgroups are discussed.
Valerii Berestovskii, Yurii Nikonorov
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2004
In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion ...
William Fulton, Joe Harris
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In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion ...
William Fulton, Joe Harris
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2014
The relationship between Lie algebras and Lie groups is of great importance. Let the Lie algebra be g and the corresponding Lie group G. The relation is $$\displaystyle{ \text{Lie algebra}\qquad g \ni X_{i}\mathrm{\ \ }(i = 1,\ldots,r) }$$ (4.1) $$\displaystyle{ \text{Lie group}\qquad G \ni \exp \left (\sum _{i=1}^{r}\alpha _{ i}X_{i ...
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The relationship between Lie algebras and Lie groups is of great importance. Let the Lie algebra be g and the corresponding Lie group G. The relation is $$\displaystyle{ \text{Lie algebra}\qquad g \ni X_{i}\mathrm{\ \ }(i = 1,\ldots,r) }$$ (4.1) $$\displaystyle{ \text{Lie group}\qquad G \ni \exp \left (\sum _{i=1}^{r}\alpha _{ i}X_{i ...
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1976
As we pointed out in 6.2, there are exactly two simple real Lie algebras of dimension 3. These are: the algebra \( {{\mathfrak{g}}_{1}} = \mathfrak{s}\mathfrak{l}\left( {2,R} \right) \) of real matrices of the second order with zero trace and the algebra \( {{\mathfrak{g}}_{2}} = \mathfrak{s}\mathfrak{o} = \left( {3,R} \right) \) of real skew-symmetric
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As we pointed out in 6.2, there are exactly two simple real Lie algebras of dimension 3. These are: the algebra \( {{\mathfrak{g}}_{1}} = \mathfrak{s}\mathfrak{l}\left( {2,R} \right) \) of real matrices of the second order with zero trace and the algebra \( {{\mathfrak{g}}_{2}} = \mathfrak{s}\mathfrak{o} = \left( {3,R} \right) \) of real skew-symmetric
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