The Kernel of the Adjoint Representation of ap-Adic Lie Group Need Not Have an Abelian Open Normal Subgroup

2016 ◽  
Vol 44 (7) ◽  
pp. 2981-2988
Author(s):  
Helge Glöckner
Keyword(s):  
Normal Subgroup ◽  
Lie Group ◽  
Adjoint Representation

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 Lie Supergroups Obtained from 3-Dimensional Lie Superalgebras Associated to the Adjoint Representation and Having a 2-Dimensional Derived Ideal

2008 ◽  
Vol 2008 ◽  
pp. 1-9
Author(s):  
I. Hernández ◽  
R. Peniche
Keyword(s):  
Lie Group ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Manifold ◽  
3 Dimensional ◽  
Lie Supergroups

We give the explicit multiplication law of the Lie supergroups for which the base manifold is a 3-dimensional Lie group and whose underlying Lie superalgebrag=g0⊕g1which satisfiesg1=g0,g0acts ong1via the adjoint representation andg0has a 2-dimensional derived ideal.

scholarly journals Low codimensional embeddings of Sp(n) and Su(n)

1984 ◽  
Vol 27 (1) ◽  
pp. 25-29 ◽  
Author(s):  
G. Walker ◽  
R. M. W. Wood
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Maximal Torus ◽  
Unitary Group ◽  
Normal Bundle ◽  
Symplectic Group ◽  
Adjoint Representation ◽  
General Technique ◽  
Special Cases ◽  
Connected Lie Group

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

scholarly journals Expanding Cone and Applications to Homogeneous Dynamics

2019 ◽  
Author(s):  
Ronggang Shi
Keyword(s):  
Normal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Error Rate ◽  
Spectral Gap ◽  
General Setting ◽  
Semisimple Lie Group ◽  
Homogeneous Dynamics ◽  
Maximal Split ◽  
Split Torus

Abstract Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

Lifting homotopies through fixed points II

1984 ◽  
Vol 96 (3-4) ◽  
pp. 201-205 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Normal Subgroup ◽  
Fixed Points ◽  
Lie Group ◽  
Hausdorff Space ◽  
Final Section ◽  
Invariant Subgroup ◽  
Compact Lie Group ◽  
Discontinuous Group ◽  
Group Of Homeomorphisms ◽  
Path Connected

SynopsisThis note complements an earlier paper of the same title. Let G be a discontinuous group of homeomorphisms of a connected, locally path connected, Hausdorff space X, and let ∏:X → X/G denote the associated projection. We work relative to a G-invariant subgroup H of the fundamental group of X and investigate the quotient group ∏1(X/G)/∏*(H). By choosing H appropriately, we can calculate ∏1(X/G) and show that ∏1(X/G)/∏*(∏1(X)) is isomorphic to G/F, where F is the normal subgroup of G generated by those elements which have fixed points. In a final section, we give analogous results for actions of a compact Lie group.

scholarly journals FIELD OF INVARIANTS OF BORELEAN GROUP OF ADJOINT REPRESENTATION OF GL(n,K)

2017 ◽  
Vol 20 (3) ◽  
pp. 34-40
Author(s):  
K.A. Vyatkina
Keyword(s):  
Explicit Form ◽  
Lie Group ◽  
Invariant Theory ◽  
Algebraic Independence ◽  
Adjoint Representation ◽  
Unitriangular Group ◽  
Joint Representation

The paper is devoted to invariant theory problems, in particular to the problem of finding generators of invariant fields in an explicit form. The set of generators is given for invariant field of unitriangular group concerning the ad-joint representation of GL(n, K) group. Moreover, the set of generators of Borel group for the field of invariants is constructed and their algebraic independence is proved. Lie group;adjoint representation;field of invariant;generators of the field of invariants;Borel group;

A homotopy construction of the adjoint representation for Lie groups

2002 ◽  
Vol 133 (3) ◽  
pp. 399-409 ◽  
Author(s):  
NATÀLIA CASTELLANA ◽  
NITU KITCHLOO
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Maximal Rank ◽  
Adjoint Representation ◽  
Simply Connected

Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories [Cscr ], hocolim[Cscr ]BGI where GI are certain connected maximal rank subgroups of G.

scholarly journals TOPOLOGICAL LOOPS HAVING SOLVABLE INDECOMPOSABLE LIE GROUPS AS THEIR MULTIPLICATION GROUPS

2020 ◽  
Author(s):  
Á. FIGULA ◽  
A. AL-ABAYECHI
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Abelian Extension ◽  
Solvable Lie Groups ◽  
Multiplication Group ◽  
Simply Connected

Abstract We prove that the solvability of the multiplication group Mult(L) of a connected simply connected topological loop L of dimension three forces that L is classically solvable. Moreover, L is congruence solvable if and only if either L has a non-discrete centre or L is an abelian extension of a normal subgroup ℝ by the 2-dimensional nonabelian Lie group or by an elementary filiform loop. We determine the structure of indecomposable solvable Lie groups which are multiplication groups of three-dimensional topological loops. We find that among the six-dimensional indecomposable solvable Lie groups having a four-dimensional nilradical there are two one-parameter families and a single Lie group which consist of the multiplication groups of the loops L. We prove that the corresponding loops are centrally nilpotent of class 2.

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