scholarly journals Traces for Star Products on the Dual of a Lie Algebra

2003 ◽  
Vol 15 (05) ◽  
pp. 425-445 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Simone Gutt ◽  
Martin Bordemann ◽  
Stefan Waldmann
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Star Product ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Compact Lie Group ◽  
Regular Orbit ◽  
Elementary Argument

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

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 ^ .

scholarly journals Ruang Fase Tereduksi Grup Lie Aff (1)

2021 ◽  
Vol 3 (2) ◽  
pp. 180-186
Author(s):  
Edi Kurniadi
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Group Action ◽  
Lie Group ◽  
Dual Space ◽  
Coadjoint Orbits ◽  
Coadjoint Representation

ABSTRAKDalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine  berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari  melalui orbit coadjoint buka di titik tertentu pada ruang dual  dari aljabar Lie . Aksi dari grup Lie    pada ruang dual  menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi  tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine     tepat mempunyai dua buah orbit coadjoint buka.  Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine  berdimensi  dan untuk kasus grup Lie lainnya.ABSTRACTIn this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.

scholarly journals Comparison and continuity of Wick-type star products on certain coadjoint orbits

2019 ◽  
Vol 31 (5) ◽  
pp. 1203-1223
Author(s):  
Chiara Esposito ◽  
Philipp Schmitt ◽  
Stefan Waldmann
Keyword(s):  
Star Product ◽  
Coadjoint Orbit ◽  
Type Star ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Continuity Properties ◽  
Fréchet Algebra

AbstractIn this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

scholarly journals Representasi Unitar Tak Tereduksi Grup Lie Dari Aljabar Lie Filiform Real Berdimensi 5

2020 ◽  
Vol 17 (1) ◽  
pp. 100-108
Author(s):  
E Kurniadi
Keyword(s):  
Harmonic Analysis ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Lie Group ◽  
Dual Space ◽  
Irreducible Unitary Representation ◽  
Coadjoint Orbits ◽  
Dimensional Representation ◽  
Induced Representation ◽  
Filiform Lie Algebra

In this paper, we study a harmonic analysis of a Lie group  of a real filiform Lie algebra of dimension 5. Particularly, we study its  irreducible unitary representation (IUR) and contruct this IUR corresponds to its coadjoint orbits through coadjoint actions of its group to its dual space.  Using induced representation of  a 1-dimensional representation of its subgroup we obtain its IUR of its Lie group

scholarly journals Calabi quasimorphisms for monotone coadjoint orbits

2017 ◽  
Vol 09 (04) ◽  
pp. 689-706
Author(s):  
Alexander Caviedes Castro
Keyword(s):  
Lie Group ◽  
Coadjoint Orbit ◽  
Universal Covering ◽  
Coadjoint Orbits ◽  
Compact Lie Group ◽  
Hamiltonian Diffeomorphisms ◽  
Schubert Classes

We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group with Kostant–Kirillov–Souriau form. We show that this result follows from positivity results of Gromov–Witten invariants and the fact that the quantum product of Schubert classes is never zero.

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

Sign in / Sign up

Export Citation Format

Share Document