Lie Bialgebra Structures Research Articles

Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ℝ n . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W n of formal vector fields on ℝ n . We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ℝ n , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff(ℝ m ) × FDiffℝ n ) gives rise to classes of compatible Poisson structures on the space J ∞(ℝ m ,ℝ n ) of infinite jets of smooth maps ℝ m → ℝ n , which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.

Classification of the Lie bialgebra structures on the Witt and Virasoro algebras

We prove that all the Lie bialgebra structures on the one sided Witt algebra W 1, on the Witt algebra W and on the Virasoro algebra V are triangular coboundary Lie bialgebra structures associated to skew-symmetric solutions r of the classical Yang-Baxter equation of the form r= a∧ b. In particular, for the one-sided Witt algebra W 1=Der k[t] over an algebraically closed field k of characteristic zero, the Lie bialgebra structures discovered in Michaelis (Adv. Math. 107 (1994) 365–392) and Taft (J. Pure Appl. Algebra 87 (1993) 301–312) are all the Lie bialgebra structures on W 1 up to isomorphism. We prove the analogous result for a class of Lie subalgebras of W which includes W 1.

(1+1) Schrödinger Lie bialgebras and their Poisson-Lie groups

All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universal R-matrices, are constructed.

Dual affine quantum groups

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.

Some remarks on Lie bialgebra structures on simple complex Lie algebras

Let g be a complex simple Lie algebra with root system ▵. We prove that any classical double D (g) is graded by ▵.As a consequence of this fact we obtain that D (g)≅g⊗A, where A is a unital com-muative associative algebra of dimension 2. Therefore we have two possibilities for A nilpotent and semisimple. The first case leads to solutions of CYBE and the second case leads to solutions of mCYBE. We obtain an explicit description of Lie bialgebra structures on g in both cases.

Multiparametric quantumgl(2): Lie bialgebras, quantumR-matrices and non-relativistic limits

Multiparametric quantum deformations of $gl(2)$ are studied through a complete classification of $gl(2)$ Lie bialgebra structures. From them, the non-relativistic limit leading to harmonic oscillator Lie bialgebras is implemented by means of a contraction procedure. New quantum deformations of $gl(2)$ together with their associated quantum $R$-matrices are obtained and other known quantizations are recovered and classified. Several connections with integrable models are outlined.

Existence of triangular Lie bialgebra structures

We characterize finite-dimensional Lie algebras over an algebraically closed field of arbitrary characteristic which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

Poisson–Lie structures on infinite-dimensional jet groups and quantum groups related to them

We study the problem of classifying all Poisson–Lie structures on the group G∞ of formal diffeomorphisms of the real line R1 which leave the origin fixed, as well as the extended group of diffeomorphisms G0∞⊃G∞ whose action on R1 does not necessarily fix the origin. A complete local classification of all Poisson–Lie structures on the groups G∞ and G0∞ is given. This includes a classification of all Lie-bialgebra structures on the Lie algebra G∞ of G∞, which we prove to be all of the coboundary type, and a classification of all Lie-bialgebra strucutures on the Lie algebra G0∞ (the Witt algebra) of G0∞ which also turned out to be all of the coboundary type. A large class of Poisson structures on the space Vλ of λ-densities on the real line is found such that Vλ becomes a homogeneous Poisson space under the action of the Poisson–Lie group G∞. We construct a series of quantum semigroups whose quasiclassical limits are finite-dimensional Poisson–Lie quotient groups of G∞ and G0∞.

Lie bi-algebra structures for centrally extended two-dimensional Galilei algebra and their Lie-Poisson counterparts

All bi-algebra structures for centrally extended Galilei algebra are classified. The corresponding Lie-Poisson structures on centrally extended Galilei group are found.

Poisson structures on double Lie groups

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauss-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.

YANGIAN DEFORMATION OF THE LIE ALGEBRA dĩff(T2)

New solution of a classical Yang–Baxter equation on a loop extended, "modified" Lie algebra of a diffeomorphism group of a torus dĩff(T2)[λ]is given. Lie bialgebra structure on dĩff(T2)[λ] is defined. By quantization of a defined Lie bialgebra, a Yangian algebra Y(dĩff(T2)) is obtained. Existence of a quantum double and a quantum R-matrix for Y(dĩff(T2)) is considered.

Drinfeld-Sokolov Reduction for Difference Operators and Deformations of W -Algebras¶ II. The General Semisimple Case

The paper is the sequel to [9]. We extend the Drinfeld--Sokolov reduction procedure to q-difference operators associated with arbitrary semisimple Lie algebras. This leads to a new elliptic deformation of the Lie bialgebra structure on the associated loop algebra. The related classical r-matrix is explicitly described in terms of the Coxeter transformation. We also present a cross-section theorem for q-gauge transformations which generalizes a theorem due to R. Steinberg.

Quantum Heisenberg - Weyl algebras

All Lie bialgebra structures on the Heisenberg--Weyl algebra $[A_+,A_-]=M$ are classified and explicitly quantized. The complete list of quantum Heisenberg--Weyl algebras so obtained includes new multiparameter deformations, most of them being of the non-coboundary type.

QuantumE(2) groups and Lie bialgebra structures

Lie bialgebra structures on e(2) are classified. For two Lie bialgebra structures which are not coboundaries (i.e. those which are not determined by a classical r-matrix) we solve the cocycle condition, find the Lie - Poisson brackets and obtain quantum group relations. There is one to one correspondence between Lie bialgebra structures on e(2) and possible quantum deformations of U(e(2)) and E(2).

Classification of all Poisson-Lie structures on an infinite-dimensional jet group

A local classification of all Poisson-Lie structures on an infinite-dimensional group $G_{\infty}$ of formal power series is given. All Lie bialgebra structures on the Lie algebra ${\Cal G}_{\infty}$ of $G_{\infty}$ are also classified.

Quantization of a loop extended SU(2) affine Kac–Moody algebra

A quantization of a Lie bialgebra structure on a loop extended SU■(2) algebra is considered. An asymptotic expansion for R-matrix is given.

Deformation of the Poisson-Lie structure in the SU(2) WZW model

A Lie bialgebra structure in the WZW model is considered. By the deformation of the Poisson structure in the SU(2) WZW model, a commutative noncocommutative Hopf algebra with nonlinear Poisson structure is obtained.

Deformations of multiparameter quantum gl(N)

Multiparameter quantum gl(N) is not a rigid structure. This paper defines an essential deformation as one that cannot be interpreted in terms of a similarity transformation, nor as a perturbation of the parameters. All the equivalence classes of first order essential deformations are found, as well as a class of exact deformations. This work provides quantization of all the classical Lie bialgebra structures (constant r-matrices) found by Belavin and Drinfeld for sl(n). A special case, that requires the Hecke parameter to be a cubic root of unity, stands out.

A poisson?lie structure on the diffeomorphism group of a circle

Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S1) of the vector fields on the circleS1 and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr ∈X(S1) ∧X(S1) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S1), and a sort of Poisson-Lie structure on the group\(\widetilde{Diff}(S^1 )\) of diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to ‘quantum diffeomorphism groups’.

A Class of Infinite-Dimensional Lie Bialgebras Containing the Virasoro Algebra

In this note, we show explicitly how to obtain the structure of a Lie bialgebra on the Virasoro algebra (with or without a central extension), on the Witt algebra, and on many other Lie algebras. Previously, V. G. Drinfel′d (in a fundamental paper (1983, Soviet Math. Dokl. 27, No. 1, 68-71)), introduced the notion of what later (1987, in "Proceedings, International Congress of Mathematicians, 1986, Berkeley, CA," pp. 798-820, Amer. Math. Soc., Providence, RI) came to be known as a triangular, coboundary Lie bialgebra G associated to a solution r ∈ G ⊗ G of the Classical Yang-Baxter Equation; and together with A. A. Belavin he showed (1982, Functional Anal. Appl. 16, No. 3, 159-180) that all invertible solutions r of that equation could be obtained as inverses of 2-cocycles. E. Witten (1988, Comm. Math. Phys. 114, 1-53) considered the case of such invertible r for diff( S 1), the Lie algebra of smooth vector fields on the circle. E. Beggs and S. Majid (1990, Ann. Inst. H. Poincaré Phys. Theor. 53, No. 1, 15 34) considered the invertible case further (in a topological framework) and found solutions for the complexification of diff( S 1) but not for the real Lie algebra diff( S 1). In this note, we exhibit a Lie bialgebra structure corresponding to a non-invertible solution r. The method by which we obtain such a structure works for both the real and the complex case of diff( S 1), the Virasoro algebra (with or without a central extension), the Witt algebra, and many other Lie algebras. We present an explicit expression for the diagonal in all these cases. In fact, we show, more generally, how to obtain the structure of a triangular, coboundary Lie bialgebra on any Lie algebra containing linearly independent elements a and b satisfying [ a, b] = k · b for some non-zero k ∈ K. Depending on the particular Lie algebra under consideration, the flexibility provided by allowing k to vary enables us to obtain different Lie bialgebra structures (some locally finite, others not) on the same underlying Lie algebra.