Hopf Algebra Deformation Research Articles

(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.

A Jordanian quantum two-photon/Schrödinger algebra

A non-standard quantum deformation of the two-photon algebra $h_6$ is constructed, and its quantum universal R-matrix is given. Representations of this new quantum algebra are studied on the Fock space and translated into Fock-Bargmann realizations that provide a direct formalism for the definition of deformed states of light. Finally, the isomorphism between $h_6$ and the (1+1) Schr\"odinger algebra is used to introduce a new (non-standard) Hopf algebra deformation of this latter symmetry algebra.

Lie Stack Deformations

Enveloping algebras of Lie stacks give irreducible Hopf algebra deformations ofU(g) which are neither commutative nor cocommutative. In this paper we present and study a large class of examples of Lie stacks. In particular, we show that the PBW-bases of these Hopf algebras do not have to be finite in general. Further, we construct a non-cocommutative Hopf structure onU(g) (usually with antipode of infinite order) whenever g has a codimension one Lie ideal h such that the quotient has the h-weight of an eigenvector of ⋀2h.

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.

INFINITE QUANTUM GROUP SYMMETRY OF FIELDS IN MASSIVE 2D QUANTUM FIELD THEORY

Starting from a given S-matrix of an integrable quantum field theory in 1 + 1 dimensions, and knowledge of its on-shell quantum group symmetries, we describe how to extend the symmetry to the space of fields. This is accomplished by introducing an adjoint action of the symmetry generators on fields, and specifying the form factors of descendents. The braiding relations of quantum field multiplets is shown to be given by the universal ℛ-matrix. We develop in some detail the case of infinite-dimensional Yangian symmetry. We show that the quantum double of the Yangian is a Hopf algebra deformation of a level zero Kac–Moody algebra that preserves its finite-dimensional Lie subalgebra. The fields form infinite-dimensional Verma module representations; in particular, the energy–momentum tensor and isotopic current are in the same multiplet.

Universal deformations of simple Lie algebras

This Letter examines the question of the structure of the Hopf algebra deformations of the universal enveloping algebras of the simple Lie algebras. Deformations of a complex algebra A are viewed as algebras defined over formal power series rings that specialize to A when the parameters go to 0. Only the case of U(sl(2,C)) is treated but the methods are general. Under the Ansatz that the two Borel subalgebras are deformed as Hopf algebras but possibly differently, we construct a universal two-parameter deformation.

Tangent cohomology, Hopf algebra actions and deformations

The tangent cohomology of commutative coalgebras is applied to questions concerning extensions of coalgebra maps, the integrability of (infinitesimal) formal curves, actions of Hopf algebras on coalgebras and (formal) deformations of algebraic systems such as representations of an algebra or (formal) deformations of algebra structures on a vector space. The connection between the tangent cohomology and these deformations is realized via the classifying coalgebras of the algebraic systems. The more geometric aspects of coalgebras (formal schemes) are reflected in the results.