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.

ON EUCLIDEAN G-MANIFOLDS WHICH HAVE TWO DIMENSIONAL ORBIT SPACES

2011 ◽  
Vol 22 (03) ◽  
pp. 399-406
Author(s):  
R. MIRZAIE
Keyword(s):  
Euclidean Space ◽  
Half Plane ◽  
Lie Group ◽  
Orbit Space ◽  
Two Dimensional ◽  
Orbit Spaces ◽  
Group Of Isometries ◽  
Connected Lie Group

We show that the orbit space of Euclidean space, under the action of a closed and connected Lie group of isometries is homeomorphic to a plane or closed half-plane, if the action is of cohomogeneity two.

scholarly journals Schubert presentation of the cohomology ring of flag manifolds

2015 ◽  
Vol 18 (1) ◽  
pp. 489-506 ◽  
Author(s):  
Haibao Duan ◽  
Xuezhi Zhao
Keyword(s):  
Lie Group ◽  
Maximal Torus ◽  
Cohomology Ring ◽  
Minimal System ◽  
Schubert Calculus ◽  
Flag Manifolds ◽  
Generators And Relations ◽  
Integral Cohomology ◽  
Minimal System Of Generators ◽  
Connected Lie Group

Let $G$ be a compact connected Lie group with a maximal torus $T$. In the context of Schubert calculus we present the integral cohomology $H^{\ast }(G/T)$ by a minimal system of generators and relations.

MAXIMAL TORUS ACTIONS ON SOLVMANIFOLDS AND DOUBLE COSET SPACES

1991 ◽  
Vol 02 (01) ◽  
pp. 67-76
Author(s):  
KYUNG BAI LEE ◽  
FRANK RAYMOND
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Double Coset ◽  
Torus Action ◽  
Torus Actions ◽  
Coset Spaces ◽  
Aspherical Manifold ◽  
Aspherical Manifolds ◽  
Connected Lie Group

Any compact, connected Lie group which acts effectively on a closed aspherical manifold is a torus Tk with k ≤ rank of [Formula: see text], the center of π1 (M). When [Formula: see text], the torus action is called a maximal torus action. The authors have previously shown that many closed aspherical manifolds admit maximal torus actions. In this paper, a smooth maximal torus action is constructed on each solvmanifold. They also construct smooth maximal torus actions on some double coset spaces of general Lie groups as applications.

scholarly journals Observability and Symmetries of Linear Control Systems

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 953
Author(s):  
Víctor Ayala ◽  
Heriberto Román-Flores ◽  
María Torreblanca Todco ◽  
Erika Zapana
Keyword(s):  
Euclidean Space ◽  
Control Systems ◽  
Lie Groups ◽  
Lie Group ◽  
Linear Control ◽  
Linear Control Systems ◽  
Euclidean Spaces ◽  
General Setup ◽  
Connected Lie Group

The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space R n and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups.

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.

Basis states for equivalent electrons. III. States for the Lie group SO(2l + 1) × SU(2) using Sp(4l + 2) states

1981 ◽  
Vol 59 (2) ◽  
pp. 207-212
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Unitary Group ◽  
Symplectic Group ◽  
Irreducible Representations ◽  
Slater Basis

The Lie group SO(2l + 1) × EU(2) is a subgroup of the symplectic group Sp(4l + 2), which in turn is a subgroup of the unitary group U(4l + 2). The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of U(4l + 2). The basis states for the irreducible representations of SO(2l + 1) × SU(2) are expressed in terms of the states for irreducible representations of Sp(4l + 2). The basis states for SO(2l + 1) × SU(2) are also expressed in terms of the Slater basis states.

scholarly journals Central lacunary sets for Lie groups

1988 ◽  
Vol 45 (1) ◽  
pp. 30-45 ◽  
Author(s):  
A. H. Dooley
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Infinite Subset ◽  
Semisimple Lie Group ◽  
Infinite Sets ◽  
Connected Lie Group

AbstractIf G is a compact connected Lie group every infinite subset of Ĝ contains an infinite central Λ(p) set, for p < 2 + 2 rank G/(dim G - rank G). A subset R of Ĝ is of type central Λ(2) if and only if the associated set of characters on the maximal torus is of type Λ(2). The dual of a compact connected semisimple Lie group contains infinite sets which are central p-Sidon for all p > 1. Every infinite subset of the dual of Su(2) contains such a set.

scholarly journals Divided difference operators in equivariant KK-theory

2014 ◽  
Vol 06 (02) ◽  
pp. 237-261
Author(s):  
Ho-Hon Leung
Keyword(s):  
Direct Summand ◽  
Lie Group ◽  
Maximal Torus ◽  
Divided Difference ◽  
Difference Operators ◽  
C Algebras ◽  
Work Done ◽  
Connected Lie Group

Let G be a compact connected Lie group with a maximal torus T. Let A, B be G-C*-algebras. We define certain divided difference operators on Kasparov's T-equivariant KK-group KKT(A, B) and show that KKG(A, B) is a direct summand of KKT(A, B). More precisely, a T-equivariant KK-class is G-equivariant if and only if it is annihilated by an ideal of divided difference operators. This result is a generalization of work done by Atiyah, Harada, Landweber and Sjamaar.

scholarly journals Weyl functions and the Ap condition on compact lie groups

1982 ◽  
Vol 33 (2) ◽  
pp. 185-192 ◽  
Author(s):  
Giancarlo Travaglini
Keyword(s):  
Trigonometric Polynomial ◽  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Weyl Function ◽  
Compact Lie Groups ◽  
Simply Connected ◽  
Weyl Functions ◽  
Uniformly Bounded ◽  
Connected Lie Group

AbstractLet G be a compact, simple, simply connected Lie group. The Lp-norm of a central trigonometric polynomial reduces naturally to a weighted Lp-norm of a trigonometric polynomial on a maximal torus T. The weight is | Δ |2-p, where Δ is the usual Weyl function. If p ≥ 2, we prove that | Δ |2-p satisfies Muckenhoupt's Ap condition if and only if the Lp-norms of the irreducible characters of G are uniformly bounded.

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.

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