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.

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 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 Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals On the K-theory of the loop space of a Lie group

1974 ◽  
Vol 76 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Francis Clarke
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Simple Structure ◽  
Homology Theory ◽  
Compact Lie Group ◽  
Classifying Space ◽  
K Theory ◽  
Simply Connected ◽  
Torsion Free ◽  
Multiplicative Structures

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

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.

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 115-139 ◽  
Author(s):  
L. MAGNIN
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Normal Forms ◽  
Equivalence Classes ◽  
Complex Structures ◽  
Simply Connected ◽  
Connected Lie Group

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

scholarly journals Hardy-Littlewood maximal functions on some solvable Lie groups

1988 ◽  
Vol 45 (1) ◽  
pp. 78-82 ◽  
Author(s):  
G. Gaudry ◽  
S. Giulini ◽  
A. Hulanicki ◽  
A. M. Mantero
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Weak Type ◽  
Maximal Functions ◽  
Solvable Lie Groups ◽  
Littlewood Maximal Function ◽  
The Family ◽  
Fixed Power ◽  
Simply Connected

AbstractLet N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

scholarly journals The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces

2019 ◽  
Vol 71 (4) ◽  
pp. 843-889
Author(s):  
Katsuhiko Kuribayashi ◽  
Luc Menichi
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Group ◽  
Hochschild Cohomology ◽  
General Theorem ◽  
String Topology ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Conformal Field ◽  
Connected Lie Group

AbstractFor almost any compact connected Lie group$G$and any field$\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra$H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if$p$is odd or$p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology$HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over$\mathbb{F}_{2}$, such an isomorphism of Batalin–Vilkovisky algebras does not hold when$G=\text{SO}(3)$or$G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

Self-Maps of Low Rank Lie Groups at Odd Primes

2010 ◽  
Vol 62 (2) ◽  
pp. 284-304 ◽  
Author(s):  
Jelena Grbić ◽  
Stephen Theriault
Keyword(s):  
Lie Groups ◽  
Structural Properties ◽  
Lie Group ◽  
Low Rank ◽  
Rank Condition ◽  
Homotopy Classes ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a simple, compact, simply-connected Lie group localized at an odd prime p. We study the group of homotopy classes of self-maps [G, G] when the rank of G is low and in certain cases describe the set of homotopy classes ofmultiplicative self-maps H[G, G]. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.

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