On the ‘strange formula’ of Freudenthal and de Vries

1989 ◽  
Vol 105 (2) ◽  
pp. 249-252 ◽  
Author(s):  
H. D. Fegan ◽  
B. Steer
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Weyl Chamber ◽  
Inner Product ◽  
Killing Form ◽  
De Vries ◽  
Positive Roots ◽  
Connected Lie Group

Suppose that G is a semi-simple, compact, connected Lie group. Endow g, its Lie algebra, with the inner product which is the negative of the Killing form. Choose a fundamental Weyl Chamber and let R+ denote the positive roots so determined.

scholarly journals Conformal Killing forms on 2-step nilpotent Riemannian Lie groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Viviana del Barco ◽  
Andrei Moroianu
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Conformal Killing ◽  
Nilpotent Lie Groups ◽  
Killing Form ◽  
Invariant Metric ◽  
Nilpotent Lie Algebra ◽  
Simply Connected ◽  
Connected Lie Group

Abstract We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.

scholarly journals Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

2020 ◽  
Author(s):  
Daniel Oeh
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Polar Decomposition ◽  
Complex Hilbert Space ◽  
Analytic Extension ◽  
Involutive Automorphism ◽  
Finite Dimensional ◽  
Continuous Unitary Representation ◽  
Continuous Representations ◽  
Connected Lie Group

Abstract Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak{g}}} = {{\mathfrak{h}}} \oplus{{\mathfrak{q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal{H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak{q}}},{{\mathfrak{q}}}] \oplus i{{\mathfrak{q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak{h}}} \oplus i{{\mathfrak{q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

scholarly journals A Note on Simple Anti-Commutative Algebras Obtained from Reductive Homogeneous Spaces

1968 ◽  
Vol 31 ◽  
pp. 105-124 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Commutative Algebras ◽  
Direct Sum ◽  
Reductive Homogeneous Spaces ◽  
Connected Lie Group

LetGbe a connected Lie group andHa closed subgroup, then the homogeneous spaceM = G/His calledreductiveif there exists a decomposition(subspace direct sum) withwhereg(resp.) is the Lie algebra ofG(resp.H); in this case the pair (g,) is called areductive pair.

scholarly journals Strictification of étale stacky Lie groups

2011 ◽  
Vol 148 (3) ◽  
pp. 807-834 ◽  
Author(s):  
Giorgio Trentinaglia ◽  
Chenchang Zhu
Keyword(s):  
Fundamental Group ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Crossed Module ◽  
Simply Connected ◽  
The Given ◽  
Connected Lie Group

AbstractWe define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and étale stacky Lie group is equivalent to a crossed module of the form (Γ,G) where Γ is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.

scholarly journals Some Homogeneous Einstein Manifolds

1970 ◽  
Vol 39 ◽  
pp. 81-106 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Closed Subgroup ◽  
Einstein Manifold ◽  
Homogeneous Spaces ◽  
Einstein Manifolds ◽  
Direct Sum ◽  
Reductive Homogeneous Spaces ◽  
Connected Lie Group

Let G be a connected Lie group and H a closed subgroup with Lie algebra such that in the Lie algebra g of G there exists a subspace m with (subspace direct sum) and In this case the corresponding manifold M = G/H is called a reductive homogeneous space and (g,) (or (G,H)) a reductive pair. In this paper we shall show how to construct invariant pseudo-Riemannian connections on suitable reductive homogeneous spaces M which make M into an Einstein manifold.

scholarly journals COMPACT ORBITS OF PARABOLIC SUBGROUPS

2021 ◽  
pp. 1-9
Author(s):  
LEONARDO BILIOTTI ◽  
OLUWAGBENGA JOSHUA WINDARE
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Reductive Group ◽  
Kähler Manifold ◽  
Parabolic Subgroups ◽  
Cartan Decomposition ◽  
Kahler Manifold ◽  
Real Submanifold ◽  
Connected Lie Group

Abstract We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra $\mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{\mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $G\subset U^{\mathbb {C}}$ is compatible, there exists a gradient map $\mu _{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$ . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $\mu _{\mathfrak p}$ .

Uniform Finite Generation of SU(2) and SL(2, R)

1972 ◽  
Vol 24 (4) ◽  
pp. 713-727 ◽  
Author(s):  
Franklin Lowenthal
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Lie Group ◽  
Finitely Generated ◽  
Finite Product ◽  
Finite Generation ◽  
Arcwise Connected ◽  
Finite Products ◽  
Infinitesimal Transformations ◽  
Connected Lie Group

A connected Lie group H is generated by a pair of one-parameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups, i.e., if and only if the subalgebra generated by the corresponding pair of infinitesimal transformations is equal to the whole Lie algebra h of H (observe that the subgroup of all finite products is arcwise connected and hence, by Yamabe's theorem [5], is a sub-Lie group). If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n ; otherwise define it as infinity.

Equivariant prequantization and the moment map

2021 ◽  
Vol 33 (3) ◽  
pp. 593-600
Author(s):  
Roberto Ferreiro Pérez
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Moment Map ◽  
The Moment ◽  
Necessary And Sufficient ◽  
Connected Lie Group

Abstract If ω is a closed G-invariant 2-form and μ is a moment map, we obtain necessary and sufficient conditions for equivariant prequantizability that can be computed in terms of the moment map μ. Our main result is that G-equivariant prequantizability is related to the fact that the moment map μ should be quantized for certain vectors on the Lie algebra of G. We also compute the obstructions to lift the action of G to a prequantization bundle of ω. Our results are valid for any compact and connected Lie group G.

scholarly journals Integrating -algebras

2008 ◽  
Vol 144 (4) ◽  
pp. 1017-1045 ◽  
Author(s):  
André Henriques
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Explicit Construction ◽  
Classifying Space ◽  
Simply Connected ◽  
Work Done ◽  
Connected Lie Group

AbstractGiven a Lie n-algebra, we provide an explicit construction of its integrating Lie n-group. This extends work done by Getzler in the case of nilpotent $L_\infty $-algebras. When applied to an ordinary Lie algebra, our construction yields the simplicial classifying space of the corresponding simply connected Lie group. In the case of the string Lie 2-algebra of Baez and Crans, we obtain the simplicial nerve of their model of the string group.

Sign in / Sign up

Export Citation Format

Share Document