The quantum bialgebra associated with the eight-vertexR-matrix

The quantum bialgebra related to the Baxter's eight-vertexR-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus.     

Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann–Schwarzbach. This approach provides a definition of an L _∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L _∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L _∞ (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L _∞-version of a Manin (quasi) triple and get a correspondence theorem with L _∞-(quasi)bialgebras. This paper is dedicated to Jean-Louis Loday on the occasion of his 60th birthday with admiration and gratitude.     

Quantization of the Lie Bialgebra of String Topology

Let M be a smooth, simply-connected, closed oriented manifold, and LM the free loop space of M. Using a Poincaré duality model for M, we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.     

Deformations of multiparameter quantum gl(N)

Multiparameter quantum gl(N) is not a rigid structure. This Letter 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 (constantr-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.     

Some remarks on producing Hopf algebras

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to aq-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure ofsl _q (2) can be completely determined by requiring theq-oscillator algebra to be its covariant comodule, in analogy with Manin's approach to defineSL _q (2) as a symmetry algebra of the bosonic and fermionic quantum planes.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.This work was supported in part by International Sciences Foundation (grant RFF-300) and by Russian Basic Research Foundation (grant 95-02-05679).I acknowledge helpful discussions with A. Isaev, P. Kulish, V. Lyakhovsky, O. Ogievetsky, P. Pyatov, and V. Tolstoy.     

New <Emphasis Type="Italic">r</Emphasis>-Matrices for Lie Bialgebra Structures over Polynomials

For a finite dimensional simple complex Lie algebra \mathfrakg{\mathfrak{g}} , Lie bialgebra structures on \mathfrakg[[u ]]{\mathfrak{g}\left[\left[u \right]\right]} and \mathfrakg[u]{\mathfrak{g}\left[u\right]} were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to produce r-matrices which correspond to Lie bialgebra structures over polynomials.     

Free differential algebras: Their use in field theory and dual formulation

The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials (p>1). We study here the algebra of FDA transformations. To every p-form in the FDA, we associate an extended Lie derivative l generating a corresponding gauge transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's.     

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 of formal power series is given. All Lie bialgebra structures on the Lie algebra G of G are also classified.     

Nonstandard 2+1 conformal Lie bialgebras and their quantum deformations

The nonstandard and so(2, 2) Lie bialgebras are generalized to the so(3, 2) case in two natural ways by considering this algebra as the conformal algebra of the 2+1 Minkowskian spacetime. Lie bialgebra contractions are analyzed providing conformal bialgebras of the 2+1 Galilean and Carroll spacetimes. The corresponding quantum Hopf so(3, 2) algebras are presented and contractions are performed at the quantum level.     

Superalgebras in N = 1 gauge theories

N = 1 supersymmetric gauge theories with global flavor symmetries contain a gauge invariant W-superalgebra which acts on its moduli space of gauge invariants. With adjoint matter, this superalgebra reduces to a graded Lie algebra. When the gauge group is SO(n_c), with vector matter, it is a W-algebra, and the primary invariants form one of its representation. The same superalgebra exists in the dual theory, but its construction in terms of the dual fields suggests that duality may be understood in terms of a charge conjugation within the algebra. We extend the analysis to the gauge group E₆.     

Differential calculus on the inhomogeneous quantum groups IGL q (n)

We obtain the inhomogeneousq-groups IGL _q (n) via a projection from GL _q (n + 1). The bicovariant differential calculus of IGL _q (n) is constructed, and the corresponding quantum Lie algebra is given explicitly.     

Quantum harmonic oscillator algebras as non-relativistic limits of multiparametric gl(2) quantizations

Multiparametric quantum gl(2) algebras are presented according to a classification based on their corresponding Lie bialgebra structures. From them, the non-relativistic limit leading to quantum harmonic oscillator algebras is implemented in the form of generalized Lie bialgebra contractions.     

Compatible Lie Bialgebras

A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket.We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra.They can also be regarded as a "compatible version" of Lie bialgebras,that is,a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra.Many properties of compatible Lie bialgebras as the "compatible version" of the corresponding properties of Lie bialgebras are presented.In particular,there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in Lie algebras.Furthermore,a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter.As a byproduct,the compatible Lie bialgebras St into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.     

Solutions of the Yang–Baxter equations on quadratic Lie groups: The case of oscillator groups

A Lie group is called quadratic if it carries a bi-invariant semi-Riemannian metric. Oscillator Lie groups constitute a subclass of the class of quadratic Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang–Baxter equation on a generic class of oscillator Lie algebras. Moreover, we show that any solution of the generalized classical Yang–Baxter equation (resp. classical Yang–Baxter equation) on a quadratic Lie group determines a left invariant locally symmetric (resp. flat) semi-Riemannian metric on the corresponding dual Lie groups.     

Relating the approaches to quantised algebras and quantum groups

This paper constructs two representations of the quantum groupU _q g' by exploiting its quotient structure and the quantum double construction. Here the quantum group is taken as the dual to the quantised algebraU _q g, a one parameter deformation of the universal enveloping algebra of the Lie algebra g, as in Drinfel'd [6] and Jimbo [10]. From the two representations, the Hopf structure of the quantised algebraU _q g is reexpressed in a matrix format. This is the very structure given by Faddeev et al. [7], in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated Lie group to g.Supported by a SERC studentship     

Hamiltonian and Super-Hamiltonian Extensions Related to Broer-Kaup-Kupershmidt System

Based on the Lie algebra A ₁, the integrable Broer-Kaup-Kupershmidt (BKK) system is revisited. The bi-Hamiltonian structure is constructed by the trace identity. Two extensions of the Lie algebra A ₁ are considered, i.e., the non-semi-simple Lie algebra of 4×4 matrix and the super-Lie algebra of 3×3 matrix, from which two hierarchies of soliton equations related to BKK system are given. With the aid of the generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures of the resulting systems are constructed.     

Quantization of U q [so(2n + 1)] with deformed para-Fermi operators

The observation thatn pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n + 1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. With this background U _q [so(2n + 1)] and its Cartan-Weyl generators are written down entirely in terms of deformed para-Fermi operators.     

Remarks on bicovariant differential calculi and exterior Hopf algebras

We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.     

Poisson Geometry of Discrete Series Orbits,and Momentum Convexity for Noncompact Group Actions

The main result of this paper is a convexity theorem for momentum mappings of certain Hamiltonian actions of noncompact semisimple Lie groups. The image is required to fall within a certain open subset D of the (dual of the) Lie algebra, and the momentum map itself is required to be proper as a map to D. The set D corresponds roughly, via the orbit method, to the discrete series of representations of the group, Much of the paper is devoted to the study of D itself, which consists of the Lie algebra elements which have compact centralizer. When the group is Sp(2n), these elements are the ones which are called 'strongly stable' in the theory of linear Hamiltonian dynamical systems, and our results may be seen as a generalization of some of that theory to arbitrary semisimple Lie groups. As an application, we prove a new convexity theorem for the frequency sets of sums of positive definite Hamiltonians with prescribed frequencies.     

Vertex Operators Arising from the Homogeneous Realization for

We use the underlying Fock space for the homogeneous vertex operator representation of the affine Lie algebra to construct a family of vertex operators. As an application, an irreducible module for an extended affine Lie algebra of type A _{N −1} coordinatized by a quantum torus ℂ _q of 2 variables (or 3 variables) is obtained. Moreover, this module turns out to be a highest weight module which is an analog of the basic module for affine Lie algebras. Received: 16 August 1999 / Accepted: 18 January 2000