Pair Of Lie Algebras Research Articles

Lie Atoms and Their Deformations

A Lie atom is essentially a pair of Lie algebras and its deformation theory is that of a deformation with respect to the first algebra, endowed with a trivialization with respect to the second. Such deformations occur commonly in algebraic geometry, for instance as deformations of subvarieties of a fixed ambient variety. Here we study some basic notions related to Lie atoms, focussing especially on their deformation theory, in particular the universal deformation. We introduce Jacobi–Bernoulli cohomology, which yields the deformation ring, and show that, under suitable hypotheses, infinitesimal deformations are classified by certain Kodaira–Spencer data.

Multiplicity-free branching rules for outer automorphisms of simple Lie algebras

We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras 𝔤 to the fixed point subalgebras 𝔤 σ of outer automorphisms σ . The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras ( 𝔤 , 𝔤 σ ) includes an exceptional symmetric pair ( E 6 , F 4 ) and also a non-symmetric pair ( D 4 , G 2 ) as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of 𝔤 in terms of the subalgebras 𝔤 σ on one hand, and the length function on the other hand.

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G 1 ⊆ G are complex linear simple Lie groups. Let 𝔤 1 ⊆ 𝔤 be the corresponding pair of Lie algebras. For the Killing-orthogonal 𝔭 of 𝔤 1 in 𝔤 we have a vector space direct sum 𝔤 = 𝔤 1⊕ 𝔭, which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G 1) = (SL (4,ℂ),Sp (2,ℂ)); here 𝔤 = 𝔰 𝔩(4,ℂ), 𝔤 1 = 𝔰 𝔭(2,ℂ), and 𝔭 = {X ∈ 𝔤 | X ♯ = X}, where X↦ X ♯ is the symplectic involution. We prove that G = G 1exp 𝔭 ∪ i G 1exp 𝔭. The key point of the proof is to study in detail the set exp 𝔭; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ♯ = X, which is here a natural counterpart of the standard Pfaffian.

Two pairs of Lie algebras and the integrable couplings as well as the Hamiltonian structure of the Yang hierarchy

Two types of Lie algebras, which are the subalgebras of the Lie algebra A 2 , A 3 respectively, are presented. The resulting loop algebras are following. As their applications, two different integrable couplings of the Yang hierarchy are obtained, called them the double integrable couplings. The Hamiltonian structure of one of them is worked out by a proper linear isomorphic transformation and the quadratic-form identity.

Construction of cocycles for bicrossproduct of Lie groups

We obtain an explicit formula for finding cocycles on a matched pair of Lie groups by using cocycles on the corresponding pair of Lie algebras. This formula for cocycles allows one to construct examples of locally compact quantum groups via bicrossproduct of Lie groups.

From dynamical to numerical R-matrices: a case study for the Calogero models

Within the class of integrable Calogero models associated with (semi-)simple Lie algebras and with symmetric pairs of Lie algebras identified in a previous paper, we analyze whether and to what extent it is possible to find a gauge transformation that takes the traditional Lax pair with its dynamical R-matrix to a new Lax pair with a numerical R-matrix.

Dynamical R-matrices for Calogero models

We present a systematic study of the integrability of the Calogero models, degenerate as well as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras, where “integrability” is understood to encompass not only the existence of a Lax representation for the equations of motion but also the—more far-reaching—existence of a (dynamical) R-matrix. Using the standard group-theoretical machinery available in this context, we show that integrability of these models, in this sense, can be reduced to the existence of a certain function, denoted here by F, defined on the relevant root system and taking values in the respective Cartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, all given by explicit formulas. We also show that among the simple Lie algebras, only those belonging to the A-series admit a solution of these constraints, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU( p, q)/S(U( p)×U( q)), allows non-trivial solutions when | p− q|⩽1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our method provides new ones, namely for the degenerate models when | p− q|=1 and for the elliptic models when | p− q|=1 or p= q.

Colour valued scattering matrices

We describe a general construction principle which allows to add colour values to a coupling constant dependent scattering matrix. As a concrete realization of this mechanism we provide a new type of S-matrix which generalizes the one of affine Toda field theory, being related to a pair of Lie algebras. A characteristic feature of this S-matrix is that in general it violates parity invariance. For particular choices of the two Lie algebras involved this scattering matrix coincides with the one related to the scaling models described by the minimal affine Toda S-matrices and for other choices with the one of the Homogeneous sine-Gordon models with vanishing resonance parameters. We carry out the thermodynamic Bethe ansatz and identify the corresponding ultraviolet effective central charges.

Symmetric boundary conditions in boundary critical phenomena

Conformally invariant boundary conditions for minimal models on a cylinder are classified by pairs of Lie algebras $(A,G)$ of ADE type. For each model, we consider the action of its (discrete) symmetry group on the boundary conditions. We find that the invariant ones correspond to the nodes in the product graph $A \otimes G$ that are fixed by some automorphism. We proceed to determine the charges of the fields in the various Hilbert spaces, but, in a general minimal model, many consistent solutions occur. In the unitary models $(A,A)$, we show that there is a unique solution with the property that the ground state in each sector of boundary conditions is invariant under the symmetry group. In contrast, a solution with this property does not exist in the unitary models of the series $(A,D)$ and $(A,E_6)$. A possible interpretation of this fact is that a certain (large) number of invariant boundary conditions have unphysical (negative) classical boundary Boltzmann weights. We give a tentative characterization of the problematic boundary conditions.

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an etale form of the vacancy condition.

Integrable boundaries, conformal boundary conditions and A-D-E fusion rules

The $sl(2)$ minimal theories are labelled by a Lie algebra pair $(A,G)$ where $G$ is of $A$-$D$-$E$ type. For these theories on a cylinder we conjecture a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph $A\otimes G$. The cylinder partition functions are given by fusion rules arising from the graph fusion algebra of $A\otimes G$. We further conjecture that, for each conformal boundary condition, an integrable boundary condition exists as a solution of the boundary Yang-Baxter equation for the associated lattice model. The theory is illustrated using the $(A_4,D_4)$ or 3-state Potts model.

Matched pairs of Lie algebroids

AbstractWe extend to Lie algebroids the notion variously known as a double Lie algebra (Lu and Weinstein), matched pair of Lie algebras (Majid), or twilled extension of Lie algebras (Kosmann-Schwarzbach and Magri). It is proved that a matched pair of Lie groupoids induces a matched pair of Lie algebroids. Conversely, we show that under certain conditions a matched pair of Lie algebroids integrates to a matched pair of Lie groupoids. The importance of matched pairs of Lie algebroids has been recently demonstrated by Lu.

Some Structural Results for Nonsymmetric Pairs of Lie Algebras

Some Structural Results for Nonsymmetric Pairs of Lie Algebras

Step algebras of quantum algebras of typeA,BandD

We study the step algebras for the quantum algebras corresponding to the Lie algebra pairs and .

Solvable lattice models labelled by Dynkin diagrams

An equivalence between generalised restricted solid-on-solid (RSOS) models, associated with sets of graphs, and multi-colour loop models is established. As an application we consider solvable loop models and in this way obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply-laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A$_L$,A$_L$) and related to the C$_2^{(1)}$ vertex model, we give an off-critical extension, which breaks the Z$_2$ symmetry of the Dynkin diagrams.

Quantum R matrix for G2and a solvable 175-vertex model

The quantum R matrix for Cartan's exceptional simple Lie algebra G2 in its seven-dimensional (minimal) representation is presented. The R matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra Uq(G2). This defines a new solvable seven-state 175-vertex model on the planar square lattice whose Boltzmann weights satisfy the Yang-Baxter equation. For the equivalent face model, the local state probability is obtained in terms of a relative of a level-1 string function related to the affine Lie algebra pair B3(1) contains/implies G2(1).

Quantum equivalence of coset space models

Many conformal field theories are defined by a pair of Lie algebras g ⊃ h. They represent the continuum limit of two-dimensional integrable lattice models. There are a few known examples, such as the three-state Potts model, which are obtained by more than one of these coset constructions. We propose a simple way of systematically building equivalent coset models, and we illustrate it in a lot of cases. We also explain how it can be translated into identities relating the partition functions.

Solvable lattice models whose states are dominant integral weights of A it?1 (1)

A new hierarchy of solvable IRF models is presented. It is generated from Belavin's Z n ×Z n symmetric model. The site variables take values in the set of level l dominant integral weights of A −1 (1) . It is conjectured that the local state probabilities are given through the irreducible decomposition of characters for the affine Lie algebra pair (A n−1 (1) ⊕A n−1 (1) ,A n−1 (1) ).

On the integrability of systems of nonlinear ordinary differential equations with superposition principles

A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincaré surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.