Showing papers on "Lie bialgebra published in 1999"

The quantum duality principle

Abstract: The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (QUEA) - provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via a quantum formal series Hopf algebra (QFSHA) - and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more precisely, there exist functors QUEA --> QFSHA and QFSHA --> QUEA, inverse of each other, such that in either case the Lie bialgebra associated to the target object is the dual of that of the source object. Such a result was claimed true by Drinfeld, but seems to be unproved in literature: we give here a complete detailed proof of it.

Some remarks on Lie bialgebra structures on simple complex Lie algebras

Abstract: 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 quantum gl(2): Lie bialgebras, quantum R-matrices and non-relativistic limits

Abstract: Multiparametric quantum deformations of gl(2) are studied through a complete classification of gl(2) Lie bialgebra structures. From these, 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.

PBW and duality theorems for quantum groups and quantum current algebras

Abstract: We give proofs of the PBW and duality theorems for the quantum Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the $A_1^{(1)}$ case.

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

Abstract: 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.

Two-photon algebra deformations

Abstract: In order to obtain a classification of all possible quantum deformations of thetwo-photon algebra h 6 , we introduce its corresponding general Lie bialgebra,which is a coboundary one. Two non-standard quantum deformations of h 6 ,together with their associated quantum universal R-matrix, are presented;each of them contains either a quantum harmonic oscillator subalgebra or aquantum gl(2) subalgebra. One-boson representations for these quantum two-photon algebras are derived and translated into Fock–Bargmann realizations.In this way, a systematic study of ‘deformed’ states of light in quantum opticscan be developed. 1 Introduction The two-photon Lie algebra h 6 is generated by the operators {N,A + ,A − ,B + ,B − ,M} endowed with the following commutation rules [1]:[N,A + ] = A + [N,A − ] = −A − [A − ,A + ] = M[N,B + ] = 2B + [N,B − ] = −2B − [B − ,B + ] = 4N +2M[A + ,B − ] = −2A − [A + ,B + ] = 0 [M, ·] = 0[A − ,B + ] = 2A + [A − ,B − ] = 0.(1)The Lie algebra h 6 contains several remarkable Lie subalgebras: the Heisenberg–Weyl algebra h

The global quantum duality principle

Abstract: Let R be an integral domain, let a non-zero h in R be such that k := R/hR is a field, and let HA be the category of torsionless (or flat) Hopf algebras over R. We call H in HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=QrUEA), at h if - roughly speaking - H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. Extending a result of Drinfeld, we establish an "inner" Galois' correspondence on HA, via two endofunctors, ( )^\vee and ( )' , of HA such that H^\vee is a QrUEA and H' is a QFA (for all H in HA). In addition: (a) the image of ( )^\vee , resp. of ( )' , is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(R/hR) = 0 , the restrictions of ( )^\vee to the QFAs and of ( )' to the QrUEAs yield equivalences inverse to each other; (c) if p = 0 , starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra g, the functor ( )^\vee , resp. ( )' , gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. Several, far-reaching applications are developed in detail in [Ga2-4].