scholarly journals Deformation of Integrable Systems Associated with Symmetric Space of the Lie Algebra

1991 ◽  
Vol 86 (5) ◽  
pp. 981-990
Author(s):  
A. Roy Chowdhury ◽  
D. C. Sen
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Integrable Systems

scholarly journals Integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n)

2015 ◽  
Vol 38 ◽  
pp. 1560071 ◽  
Author(s):  
Stephen C. Anco ◽  
Esmaeel Asadi ◽  
Asieh Dogonchi
Keyword(s):  
Symmetric Space ◽  
Integrable Systems ◽  
Hermitian Symmetric Space ◽  
Complex Matrix ◽  
Curve Flows ◽  
Frame Method ◽  
Parallel Frame ◽  
Natural Way ◽  
Sg Equation ◽  
Unitary Invariance

A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG) equation, involving a scalar variable coupled to a complex vector variable. The Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way from the beginning in formulating a complex matrix representation of the tangent space 𝔰𝔭(n)/𝔲(n) and its bracket relations within the symmetric Lie algebra (𝔲(n), 𝔰𝔭(n)).

scholarly journals Zero Curvature Formalism for Supersymmetric Integrable Hierarchies in Superspace

1997 ◽  
Vol 12 (34) ◽  
pp. 2623-2630 ◽  
Author(s):  
H. Aratyn ◽  
C. Rasinariu ◽  
A. Das
Keyword(s):  
Symmetric Space ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Space Structure ◽  
Hermitian Symmetric Space ◽  
Curvature Condition ◽  
Lax Equation ◽  
Abelian Gauge ◽  
Graded Algebras ◽  
Zero Curvature

We generalize the Drinfeld–Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl (n+1,n), providing the Hermitian symmetric space structure.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

scholarly journals SYMPLECTIC AND ORTHOGONAL LIE ALGEBRA TECHNOLOGY FOR BOSONIC AND FERMIONIC OSCILLATOR MODELS OF INTEGRABLE SYSTEMS

2001 ◽  
Vol 16 (07) ◽  
pp. 1199-1225 ◽  
Author(s):  
A. J. MACFARLANE ◽  
HENDRYK PFEIFFER ◽  
F. WAGNER
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bethe Ansatz ◽  
Integrable Systems ◽  
Fock Spaces ◽  
Efficient Treatment ◽  
Orthogonal Lie Algebra ◽  
Integral Systems ◽  
Quantum Integrable Models ◽  
Invariant Tensors

To provide tools, especially L-operators, for use in studies of rational Yang–Baxter algebras and quantum integrable models when the Lie algebras so (N)(bn, dn) or sp (2n)(cn) are the invariance algebras of their R matrices, this paper develops a presentation of these Lie algebras convenient for the context, and derives many properties of the matrices of their defining representations and of the ad-invariant tensors that enter their multiplication laws. Metaplectic-type representations of sp (2n) and so (N) on bosonic and on fermionic Fock spaces respectively are constructed. Concise general expressions (see (5.2) and (5.5) below) for their L-operators are obtained, and used to derive simple formulas for the T operators of the rational RTT algebra of the associated integral systems, thereby enabling their efficient treatment by means of the algebraic Bethe ansatz.

SPECTRAL RADIUS ANALYSIS OF MATRICES AND THE ASSOCIATED WITH INTEGRABLE SYSTEMS

2009 ◽  
Vol 23 (24) ◽  
pp. 4855-4879 ◽  
Author(s):  
HONWAH TAM ◽  
YUFENG ZHANG
Keyword(s):  
Lie Algebra ◽  
Spectral Radius ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Spectral Matrix ◽  
Akns Hierarchy ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Set Up

An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.

CONFORMALLY REDUCED WZNW THEORY, NEW EXTENDED CHIRAL ALGEBRAS AND THEIR ASSOCIATED TODA TYPE INTEGRABLE SYSTEMS

1992 ◽  
Vol 07 (28) ◽  
pp. 7015-7043 ◽  
Author(s):  
BO-YU HOU ◽  
LIU CHAO
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Integrable Systems ◽  
Equations Of Motion ◽  
Chiral Algebras ◽  
Block Diagonal

We propose and analyze a large class of conformal reductions Cons [g(H, d)] of WZNW theory based on the integral gradations of the underlying Lie algebra g. The W bases of the associated W algebras W[g(H, d)] are constructed under the generalized Drinfeld-Sokolov gauge which we call O’Raifeartaigh gauge of the constrained Kac-Moody currents, and the equations of motion of the extended Toda type integrable systems corresponding to these W algebras are also derived. As an example, we construct explicitly the W algebra associated with the (pqp) block diagonal decomposition of sl2p+q, namely W[(pqp)2], and discuss some of the properties thereof.

scholarly journals CONVOLUTION OF ORBITAL MEASURES IN SYMMETRIC SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 470-485 ◽  
Author(s):  
BOUDJEMÂA ANCHOUCHE ◽  
SANJIV KUMAR GUPTA
Keyword(s):  
Positive Integer ◽  
Symmetric Space ◽  
Lie Algebra ◽  
Haar Measure ◽  
Symmetric Spaces ◽  
Double Coset ◽  
Cartan Decomposition ◽  
Absolutely Continuous ◽  
Orbital Measure ◽  
Noncompact Symmetric Space

AbstractLet G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

scholarly journals ADAMS-IWASAWA $\mathcal{N} = 8$ BLACK HOLES

2012 ◽  
Vol 13 ◽  
pp. 44-53 ◽  
Author(s):  
SERGIO LUIGI CACCIATORI ◽  
BIANCA LETIZIA CERCHIAI ◽  
ALESSIO MARRANI
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Symmetric Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Cartan Subalgebra ◽  
Discrete Subgroup ◽  
Iwasawa Decomposition ◽  
Extremal Black Hole ◽  
Supergravity Theory

We study some of the properties of the geometry of the exceptional Lie group E7(7), which describes the U-duality of the [Formula: see text], d = 4 supergravity. In particular, based on a symplectic construction of the Lie algebra 𝔢7(7) due to Adams, we compute the Iwasawa decomposition of the symmetric space [Formula: see text], which gives the vector multiplets' scalar manifold of the corresponding supergravity theory. The explicit expression of the Lie algebra is then used to analyze the origin of [Formula: see text] as scalar configuration of the "large" ⅛-BPS extremal black hole attractors. In this framework it turns out that the U(1) symmetry spanning such attractors is broken down to a discrete subgroup ℤ4, spoiling their dyonic nature near the origin of the scalar manifold. This is a consequence of the fact that the maximal manifest off-shell symmetry of the Iwasawa parametrization is determined by a completely non-compact Cartan subalgebra of the maximal subgroup SL(8, ℝ) of E7(7), which breaks down the maximal possible covariance SL(8, ℝ) to a smaller SL(7, ℝ) subgroup. These results are compared with the ones obtained in other known bases, such as the Sezgin-van Nieuwenhuizen and the Cremmer-Julia /de Wit-Nicolai frames.

scholarly journals HIDDEN ALGEBRA OF THREE-BODY INTEGRABLE SYSTEMS

1998 ◽  
Vol 13 (18) ◽  
pp. 1473-1483 ◽  
Author(s):  
ALEXANDER TURBINER
Keyword(s):  
Lie Algebra ◽  
Algebraic Structure ◽  
Integrable Systems ◽  
Differential Operators ◽  
Reduction Method ◽  
Hamiltonian Reduction ◽  
Degree Polynomial ◽  
Infinite Dimensional ◽  
Three Body ◽  
Infinite Dimensional Lie Algebra

It is shown that all three-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial in generators of an infinite-dimensional Lie algebra of differential operators.

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