Feigin–Frenkel Image of Witten–Kontsevich Points

2020 ◽  
Author(s):  
Martin T Luu
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Affine Algebra ◽  
Simple Complex ◽  
Dual Roles ◽  
Tau Function ◽  
Basic Representation ◽  
Complete Set ◽  
Topological Gravity

Abstract The Witten–Kontsevich KdV tau function of topological gravity has a generalization to an arbitrary Drinfeld–Sokolov hierarchy associated to a simple complex Lie algebra. Using the Feigin–Frenkel isomorphism we describe the affine opers describing such generalized Witten–Kontsevich functions in terms of Segal–Sugawara operators associated to the Langlands dual Lie algebra. In the case where the Lie algebra is simply laced there is a second role these Segal–Sugawara operators play: their action, in the basic representation of the affine algebra associated to the Lie algebra, singles out the Witten–Kontsevich tau function within the phase space. We show that these two Langlands dual roles of Segal–Sugawara operators correspond to a duality between the first and last operator for a complete set of Segal–Sugawara operators.

scholarly journals Heisenberg Doubles for Snyder-Type Models

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1055
Author(s):  
Stjepan Meljanac ◽  
Anna Pachoł
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Dual Pair ◽  
One Dimension ◽  
Lorentz Algebra ◽  
Explicit Formulae ◽  
Heisenberg Double

A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

THE QUADRATIC TIME-DEPENDENT HAMILTONIAN: EVOLUTION OPERATOR, SQUEEZING REGIONS IN PHASE SPACE AND TRAJECTORIES

1994 ◽  
Vol 08 (11n12) ◽  
pp. 1563-1576 ◽  
Author(s):  
S.S. MIZRAHI ◽  
M.H.Y. MOUSSA ◽  
B. BASEIA
Keyword(s):  
Phase Space ◽  
Unitary Transformation ◽  
Uniform Magnetic Field ◽  
Evolution Operator ◽  
Single Mode ◽  
Time Dependent ◽  
Special Cases ◽  
Complete Set ◽  
Hamiltonian Evolution ◽  
General Time

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

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 Combinatorial descriptions of the crystal structure on certain PBW bases

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Ben Salisbury ◽  
Adam Schultze ◽  
Peter Tingley
Keyword(s):  
Crystal Structure ◽  
Weyl Group ◽  
Lie Algebra ◽  
Simple Complex ◽  
International Audience ◽  
The Relationship

International audience Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.

ON SEMIREGULAR NILPOTENT ORBITS IN SIMPLE LIE ALGEBRAS OF TYPE D

2002 ◽  
Vol 01 (03) ◽  
pp. 341-356 ◽  
Author(s):  
BENOÎT ARBOUR ◽  
DRAGOMIR Ž. ĐOKOVIĆ
Keyword(s):  
Linear Combination ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Nilpotent Orbits ◽  
Connected Component ◽  
Real Form ◽  
Simple Complex ◽  
Type D ◽  
Explicit Formulae

We derive explicit formulae for the characteristics H(k) of the semiregular nilpotent orbits Dn(ak) of the simple complex Lie algebra [Formula: see text] of type Dn. These formulae express H(k) as an integral linear combination of a basis of the Cartan subalgebra [Formula: see text] of [Formula: see text]. For that purpose we use several suitable bases of [Formula: see text] consisting of coroots. We also construct several explicit standard triples (E, H, F) with H = H(k), and E, F ∈ Dn(ak). Similar triples are constructed also for each connected component of the intersection of the orbit Dn(ak) with the split real form [Formula: see text] and the real form [Formula: see text] of [Formula: see text].

scholarly journals GEOMETRICAL DESCRIPTION OF THE LOCAL INTEGRALS OF MOTION OF MAXWELL-BLOCH EQUATION

1995 ◽  
Vol 10 (17) ◽  
pp. 1209-1223 ◽  
Author(s):  
A.V. ANTONOV ◽  
B.L. FEIGIN ◽  
A.A. BELOV
Keyword(s):  
Phase Space ◽  
Homogeneous Space ◽  
Lie Algebra ◽  
Bloch Equation ◽  
Integrals Of Motion ◽  
Positive Part ◽  
Abelian Subalgebra ◽  
Hamiltonian Flows ◽  
Nilpotent Subalgebra ◽  
Lattice Setting

We represent a classical Maxwell-Bloch equation and relate it to positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra n+ of affine Lie algebra [Formula: see text] on a Maxwell–Bloch phase space treated as a homogeneous space of n+. A space of local integrals of motion is described using cohomology methods. We show that Hamiltonian flows associated with the Maxwell–Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an Abelian subalgebra of the nilpotent subalgebra n− on a Maxwell–Bloch phase space. Possibilities of quantization and lattice setting of Maxwell–Bloch equation are discussed.

THE CRITICAL REPRESENTATIONS OF AFFINE LIE ALGEBRAS

1986 ◽  
Vol 01 (10) ◽  
pp. 557-564 ◽  
Author(s):  
D. ALTSCHÜLER
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Charge ◽  
Affine Lie Algebras ◽  
Affine Algebra ◽  
Open Problems ◽  
Simple Lie Algebra ◽  
Coxeter Number ◽  
Dual Coxeter Number ◽  
Current Algebras

A critical representation of an affine algebra Ĝ is a representation with central charge k=−g, g being the dual Coxeter number of the underlying simple Lie algebra G. These representations arise naturally in the study of conformal current algebras and BRS cohomology. The author shows how to construct them explicitly in a number of cases, and some intriguing open problems are mentioned.

VI.—Complex Four-dimensional Lie Algebras

1958 ◽  
Vol 65 (1) ◽  
pp. 72-83
Author(s):  
E. W. Wallace
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Canonical Forms ◽  
Complete Set

SynopsisCanonical forms of the four-dimensional complex Lie algebras are obtained by considering the roots of certain well-defined vectors of the algebras. A complete set of characters of the algebras is also given, enabling any given four-dimensional complex Lie algebra to be identified with one of the canonical forms.

Invariant Polynomials of Weyl Groups and Applications to the Centres of Universal Enveloping Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 583-592 ◽  
Author(s):  
C. Y. Lee
Keyword(s):  
Lie Algebra ◽  
Weyl Groups ◽  
Killing Form ◽  
Dimensional Representation ◽  
Universal Enveloping Algebras ◽  
Semisimple Lie Algebra ◽  
Finite Dimensional Representation ◽  
Polynomial Invariants ◽  
Finite Dimensional ◽  
Complete Set

An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).

Quantum Torus Lie Algebra in the q-Deformed Kadomtsev–Petviashvili System

2019 ◽  
Vol 26 (04) ◽  
pp. 579-588
Author(s):  
Chuanzhong Li ◽  
Xinyue Li ◽  
Fushan Li
Keyword(s):  
Lie Algebra ◽  
Kp Hierarchy ◽  
Quantum Torus ◽  
Tau Function ◽  
Ghost Symmetry

Based on the W∞ symmetry of the q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy, which is a q-deformation of the KP hierarchy, we construct the quantum torus symmetry of the q-KP hierarchy, which further leads to the quantum torus constraint of its tau function. Moreover, we generalize the system to a multi-component q-KP hierarchy that also has the well-known ghost symmetry.

Sign in / Sign up

Export Citation Format

Share Document