scholarly journals The openness condition for a coadjoint orbit projection of the semidirect product Lie group M((n, p), ℝ) ⋊ GL (n, ℝ)

2021 ◽  
Vol 1872 (1) ◽  
pp. 012017
Author(s):  
E Kurniadi
Keyword(s):  
Semidirect Product ◽  
Lie Group ◽  
Coadjoint Orbit

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

DISCRETE DECOMPOSABLE BRANCHING LAWS AND PROPER MOMENTUM MAPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250021 ◽  
Author(s):  
SALMA NASRIN
Keyword(s):  
Lie Group ◽  
Discrete Series ◽  
Geometric Quantization ◽  
Series Representation ◽  
Coadjoint Orbit ◽  
Irreducible Unitary Representation ◽  
Symmetric Pair ◽  
Holomorphic Discrete Series ◽  
Branching Laws ◽  
Scalar Type

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions. (1) The restriction π|H is discretely decomposable in the sense of Kobayashi. (2) The momentum map [Formula: see text] is proper. In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

scholarly journals SU(3)L ⋊ (ℤ3 × ℤ3) GAUGE SYMMETRY AND TRI-BIMAXIMAL MIXING

2010 ◽  
Vol 25 (35) ◽  
pp. 2981-2989
Author(s):  
CHUICHIRO HATTORI ◽  
MAMORU MATSUNAGA ◽  
TAKEO MATSUOKA ◽  
KENICHI NAKANISHI
Keyword(s):  
Finite Group ◽  
Gauge Theory ◽  
Gauge Symmetry ◽  
Gauge Group ◽  
Semidirect Product ◽  
Lie Group ◽  
Irreducible Representations ◽  
Mixing Matrix ◽  
A Finite Group ◽  
Connected Lie Group

We study an effective gauge theory whose gauge group is a semidirect product G = G c ⋊ Γ with G c and Γ being a connected Lie group and a finite group, respectively. The semidirect product is defined through a projective homomorphism (i.e. homomorphism up to the center of G c ) from Γ into G c . To be specific, we take SU (3) L as G c and ℤ3 × ℤ3 as Γ. We notice that the irreducible representations of the gauge group G necessarily contain three G c -multiplets in spite of the Abelian nature of Γ = ℤ3 × ℤ3. This triplication phenomenon is due to the semidirect product structure of G. We suggest that the appearance of three families is attributable to this triplication. We give a toy model on the lepton mixing and show that under a particular vacuum alignment the tri-bimaximal mixing matrix is reproduced.

The Classification of Pin4-Bundles over a 4-Complex

1999 ◽  
Vol 42 (2) ◽  
pp. 248-256
Author(s):  
Christian Weber
Keyword(s):  
Semidirect Product ◽  
Lie Group ◽  
Structure Theorem ◽  
Classification Theorem

AbstractIn this paper we show that the Lie-group Pin4 is isomorphic to the semidirect product (SU2 × SU2) Z/2 where Z/2 operates by flipping the factors. Using this structure theorem we prove a classification theorem for Pin4-bundles over a finite 4-complex X.

scholarly journals INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF

2011 ◽  
Vol 83 (3) ◽  
pp. 439-449 ◽  
Author(s):  
ANDREW DOUGLAS ◽  
JOE REPKA
Keyword(s):  
Euclidean Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Dimensional Euclidean Space ◽  
Euclidean Group ◽  
New Class ◽  
Euclidean Algebra

AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

Geometric Realizations of Representations of Finite Length

1997 ◽  
Vol 09 (07) ◽  
pp. 821-851 ◽  
Author(s):  
Charles H. Conley
Keyword(s):  
Semidirect Product ◽  
Finite Length ◽  
Lie Group ◽  
Vector Bundles ◽  
Indecomposable Representations

Let G=H×ℝn be a semidirect product Lie group. We reduce the problem of deciding which indecomposable representations of G may be realized in subquotients of spaces of sections of vector bundles over infinitesimal neighborhoods of orbits of H in the dual of ℝn to a problem involving only representations of the H-stabilizers of the orbits.

scholarly journals Another proof for the continuity of the Lipsman mapping

2020 ◽  
Vol 72 (7) ◽  
pp. 945-951
Author(s):  
A. Messaoud ◽  
A. Rahali
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Inner Product ◽  
Coadjoint Orbits ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Dimensional ◽  
Unitary Dual

UDC 515.1 We consider the semidirect product G = K ⋉ V where K is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an inner product 〈 , 〉 . By G ^ we denote the unitary dual of G and by 𝔤 ‡ / G the space of admissible coadjoint orbits, where 𝔤 is the Lie algebra of G . It was pointed out by Lipsman that the correspondence between G ^ and 𝔤 ‡ / G is bijective. Under some assumption on G , we give another proof for the continuity of the orbit mapping (Lipsman mapping) Θ : 𝔤 ‡ / G - → G ^ .

Sign in / Sign up

Export Citation Format

Share Document