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On the quantum structure of the universal enveloping algebra of the lie algebra ST(2).

Abstract

The structure of Hopf co-Poisson algebra on the universal enveloping algebra U(ST(2)) of Lie algebra ST(2) is determined with the help of a solution of the Yang-Baxter equation. Using this solution, a bracket on the dual space of Lie algebra ST(2) is also determined. This cobracket on ST(2) induces a deformation of the universal enveloping algebra U(ST(2)) which has a Hopf algebra structure, as we shall verify. This Hopf algebra is called the quantum group associated to a universal enveloping algebra.

Key words: Lie bialgebras, Hopf algebras, Poisson brackets, Lie Poisson group, Hopf co-Poisson algebra, Universal enveloping algebra, r-matrix, Quantum group, Yang-Baxter equation.

Resumen

Con ayuda de una solucion de la ecuacion clasica de Yang-Baxter determinamos la estructura de algebra de Hopf-co-Poisson del algebra envolvente universal U(ST(2)) del algebra de Lie ST(2). Usando esta solucion determinamos un corchete en el espacio dual del algebra de Lie. Este co-corchete sobre ST(2) induce una deformacion del algebra envolvente universal U((ST(2)) que tiene estructura de algebra de Hopf, como probaremos. Esta algebra de Hopf es llamada el grupo cuantico asociado a un algebra envolvente universal.

Palabras claves: Bialgebra de Lie, Algebra de Hopf, Grupo de Lie Poisson, co-corchete, Algebra de Hopf co-Poisson, Algebra Envolvente Universal, r-matriz, ecuacion de Yang Baxter. Grupo Cuantico.

1. Introduction

Let G be a Lie Group and Q its Lie algebra; we can obtain quantum groups as deformations of the algebra of [C.sup.[infinity]] functions F(G) on G, or as quantizations of a Lie bialgebra G. A quantization of a Lie bialgebra G is a deformation of the universal enveloping algebra U(G) equipped with the co-Poisson Hopf algebra structure, such that the classical limit of this quantization is the Lie bialgebra structure of G. To construct a deformation of the universal enveloping algebra we need to describe the co-Poisson Hopf algebra structure on U(G) or, equivalently, we must build the bialgebra structure on the Lie algebra G.

The purpose of this work is to describe a mathematical procedure to produce a quantum group structure associated to a universal enveloping algebra, the universal enveloping algebra U(ST(2)) of Lie algebra ST(2). To achieve this purpose we use a solution of the classical Yang-Baxter equation (CYBE) on Lie algebra ST(2). We build a cobracket in ST(2) connected with this solution. This cobracket determines a bialgebra structure in ST(2) and the co-Poisson-Hopf algebra structure in the universal enveloping algebra U(ST(2)) as we shall verify.

We will deform the comultiplication in U(ST(2)) by means of a parameter h in order to build the algebra [U.sub.h](ST(2)). We shall find appropriate expressions for the coproduct [[DELTA].sub.h],, the antipode application [S.sub.h] and the bracket [[].sub.h] on [U.sub.h](ST(2)). We shall prove that [U.sub.h](ST(2)), with these applications, has a Hopf algebra structure so that when h [flecha diestra] 0 the coalgebra structure of [U.sub.h](ST(2)) coincides with the bialgebra structure of ST(2). That is, we shall prove that this algebra is a quantum group of the universal enveloping algebra type.

2. The group ST(2) and its algebra U(ST(2))

2.1. The Lie algebra ST(2)

Let ST(2) be the Lie group of upper triangular matrices 2x2 with determinant equal to 1 such that the operation of the group is the multiplication between matrices.

The Lie algebra ST(2) associated to ST(2) is the Lie algebra of upper triangular matrices 2x2 with null trace

on R, where the matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

form a basis with the Lie bracket given by

[[X.sub.1],[X.sub.2]] = -[[X.sub.2],[X.sub.1]] = 2[X.sub.2], [[X.sub.i],[X.sub.i]] = 0, i = 1, 2 (2.2)

2.2. The Lie bialgebra structure on ST(2)

A Lie bialgebra is a Lie algebra with a Lie co-algebra structure o fulfilling the 1-cocycle condition (2.5) with respect to the tensorial adjoint representation, (see [6] p. 43).

The Lie bialgebra structures may be induced by the adjoint aplication of Lie algebras

ad : ST(2) [flecha diestra] ST(2) x ST(2)

defined by

[ad.sub.x](Y) = [X,Y], for X,Y [elemento de] ST(2).

The adjoint representation of ST(2) is totally determined by its representation on the {[X.sub.1],[X.sub.2]} basis of Lie algebra, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Any representation of a Lie algebra can be extended to a unique representation on the tensorial product of Lie algebras. Then we can extend the adjoint representation just defined to the adjoint tensorial representation in the following way

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With this adjoint tensorial representation and with one special r-tensor, r [elemento de] <ST(2) [producto cruzado] ST(2), we are able to define a co-Lie algebra structure.

An element r [elemento de] ST[2) [producto cruzado] ST(2) defines a Lie bialgebra structure if and only if r is skew-symmetric and [[r, r]] = 0. The equation [[r, r]] = 0 is called the clasical Yang-Baxter equation (CYBE). Moreover if r is a skew-symmetric element, r is called an r-matrix.

We know that the Lie algebra ST(2) has only one r-matrix (see [9]), and that this r-matrix is the tensor

r = [X.sub.1] [cross product] [X.sub.2] - [X.sub.2] [cross product] [X.sub.1], (2.3)

[X.sub.1], [X.sub.2] being the basis elements of ST(2). Thus r is skew-symmetric and satisfies the Yang Baxter equation,

[[r, r]] = [[r.sub.12] + [s.sub.13]] + [[r.sub.12] + [s.sub.23]] + [[r.sub.13] + [s.sub.23]] = 0.

where

[r.sub.12] = [X.sub.1] [cross product] [X.sub.2] I - [X.sub.2] [cross product] [X.sub.1] [cross product] I

[r.sub.23] = I [cross product] [X.sub.1] [cross product] [X.sub.2] - I [cross product] [X.sub.2] [producto cruzado] [X.sub.1]

[r.sub.13] = [X.sub.1] [cross product] I [cross product] [X.sub.2] - [X.sub.2] [cross product] I [cross product] [X.sub.1]

Proposition 2.1. The r-matrix

r = [X.sub.1] [cross product] [X.sub.2] - [X.sub.2] [cross product] [X.sub.1]

induces a cobracket on the Lie algebra ST(2) by the application

[delta] : ST(2) [right arrow] ST(2) [cross product] ST(2)

defined by

[delta](X) = (a[d.sub.X] [cross product] I + I [cross product] a[d.sub.X])(r) = X x r, for X [elemento de] ST(2).

Proof. We can show that J satisfies the properties of a cobracket. For this purpose it is enough to prove that 6 satisfies these properties on {[X.sub.1],[X.sub.2]}, the basis of the Lie algebra. In those elements, [delta] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.4)

Then [delta] satisfies the following cobracket properties over {[X.sub.1],[X.sub.2]}:

(1) If [delta](X) = [summation][X.sub.i] [cross product] [X.sub.j] then

[X.sub.i] [cross product] [X.sub.j] = -[X.sub.j] [cross product] [X.sub.i] for all i,j.

(2) [delta] satisfies the associative property,

(id [cross product] [delta]) o [delta] - ([delta] [cross product] id) o [delta] = 0.

(3) [delta] is one 1-cocycle, that is, [delta] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

with i,j = 1,2 and [X.sub.i], [X.sub.j] [member of] ST(2).

The proof of these properties is straightforward. However, since [delta] is defined by an r-matrix we can deduce, from Proposition 2.1.2 of [2], that [delta] is a cobracket on (ST(2)). Therefore (ST(2), [delta]) is a Lie bialgebra.

The Lie bialgebra (ST(2), [delta]) is called a quasitriangular bialgebra because it is generated by a solution of the CYBE and triangular bialgebra because it arises from a skew-symetric solution of the CYBE. This Lie bialgebra is also called a coboundary Lie bialgebra because the cobracket [delta] is a 1-cocycle.

2.3. The universal enveloping algebra U(ST(2))

Let ST(2) be the Lie algebra of ST(2) and let T(ST(2)) be the tensorial algebra of ST(2),

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

where

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII].

The universal enveloping algebra U(ST(2)) of the Lie algebra ST(2) is the associative algebra,

U(ST(2))[congruente con] T(ST(2))/X,

where I is the ideal of T(ST(2)) engendered by

[X.sub.1] [cross product] [X.sub.2] - [X.sub.2] [cross product] [X.sub.1] - 2[X.sub.2], [X.sub.1],[X.sub.2] [elemento de] ST(2).

with the product given by the recurrent operation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note 1. It is known that a universal enveloping algebra U(G) of the Lie algebra G is a Hopf algebra (see [2], [11], [15], [16]) with the linear applications [DELTA], [epsilon] and S defined on the basis of G by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then U(ST(2)) is a Hopf algebra with the operations just defined.

Note 2. It is known that a cocycle on G induces a co-Poisson structure in the universal enveloping algebra U(G), (see [3], [4], [6], and proposition 6.2.3 in [2]). Then the cobracket induced by [delta] in (2.4) satisfies the following cobracket properties on U(ST(2))

(1) Compatibility between [delta] and [DELTA],

[delta]([X.sub.i][X.sub.j]) = [delta]([X.sub.i]) [DELTA]([X.sub.j]) + [DELTA] ([X.sub.i]) [delta]([X.sub.j]).

(2) Co-Jacobi identity, that is [delta] satisfies the co-chain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [suma] means the sum over permutations of the factors in the triple tensor product.

(3) Co-Leibniz identity,

([DELTA] [cross product] id)[delta] = (id [cross product] [delta])[DELTA] + [[sigma].sub.23] ([delta] [cross product] id)[DELTA]

where [[sigma].sub.23] means the permutations of the last two elements in ST(2) [cross product] ST(2) [cross product] ST(2).

Thus, from Note 1 and Note 2, we can infer that the universal enveloping algebra U(ST(2)) of the Lie algebra ST(2) is a Hopf co-Poisson algebra with the coproduct [DELTA], the counit [epsilon], the antipode S and the cobracket [delta]. These applications are defined on the generators [X.sub.1], [X.sub.2] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.6)

which are extended to the elements of the algebra U(ST(2)) by the following commutative diagram

2.4. The Quantum algebra [U.sub.h](ST(2))

The quantum enveloping algebra [U.sub.h](ST(2)) of the Lie bialgebra (ST(2)) is a quantization of U(ST(2)) when U(ST(2)) is considered as a co-Poisson-Hopf algebra. It means that (ST(2)) has a quasitriangular structure(see [4]). Since the bialgebra (ST(2)) has a quasitriangular structure we need work with exponential functions, it means that we should work over the ring R[[h]] of the formal series in h.

In order to build a quantization of U(ST(2)) we consider the Lie algebra U(ST(2))[[h]] = [U.sub.h](ST(2)) of formal power series in R[[h]] with coefficients in U(ST(2)), generated by [X.sub.1],[X.sub.2],I with the defining relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We must prove that U(<ST(2))[[h]] with this new product, has a Hopf *-algebra structure, so that when h [flecha diestra] 0 the coalgebra structure of [U.sub.h](ST(2)) coincides with the bialgebra structure of ST(2).

We shall find appropriate expressions for the co-product [[DELTA].sub.h], the counit [[epsilon].sub.h] and the antipode application [S.sub.h] defined on U(ST(2))[[h]]. For this purpose we take Ah as the deformation of co-proctuct [DELTA] defined on U(ST(2)). Let be [[DELTA].sub.h] the linear aplication

[[DELTA].sub.h] : [U.sub.h](ST(2)) [right arrow] [U.sub.h](ST(2)) [cross product] [U.sub.h](ST(2))

defined by

[[DELTA].sub.h] = [DELTA] + h/2[delta] + O([h.sup.2]) (2.7)

such that [lim.sub.h[right arrow]0] [[DELTA].sub.h] [right arrow] [DELTA]. Here [delta] is the cobracket (2.4) and h is the parameter of the deformation. The map [[DELTA].sub.h] in [X.sub.1],[X.sub.2] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.8)

Since (I + h[X.sub.2]) and (I - h[X.sub.2]) are functions in U(ST(2))[[h]] we can write more generally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.9)

and since we must have [lim.sub.h[right arrow]0] [[DELTA].sub.h] = [DELTA], f and g must satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For these two functions we have:

Lemma 2.1. For any choice of f and g, the application

[[DELTA].sub.h] : [U.sub.h](ST(2)) [right arrow] [U.sub.h](ST(2)) [cross product] [U.sub.h](ST(2))

defined by (2.9) is an homomorphism of Lie algebras.

Proof. In fact [[DELTA].sub.h] satisfies

[[DELTA].sub.h] ([[X.sub.1],[X.sub.2]]) = [[[DELTA].sub.h] ([X.sub.1]), [[DELTA].sub.h] ([X.sub.2])] (2.10)

whereas

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.2. The application [[DELTA].sub.h] defined by (2.9) is coassociative if and only if f and g satisfy

[[DELTA].sub.h] (f) = f [cross product] f [[DELTA].sub.h] (g) = g [cross product] g

Proof. We must proof that [[DELTA].sub.h] satisfies

([[DELTA].sub.h] [cross product] id) [[DELTA].sub.h] (X) = (id [cross product] [[DELTA].sub.h]) [[DELTA].sub.h] (X) (2.11)

if and only f and g satisfy

[[DELTA].sub.h] (f) = f [cross product] f, [[DELTA].sub.h](g) = g [cross product] g (2.12)

In fact, the right-hand side of (2.11) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the left-hand side of (2.11) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We see that this two expressions are equal if, and only if,

[[DELTA].sub.h](f(h)) = f(h) [cross product] f(h) and [[DELTA].sub.h] (g(h)) = g(h) [cross product] g(h)

If we take the application

[[epsilon].sub.h] : [U.sub.h](ST(2)) [right arrow] R,

defined by

[[epsilon].sub.h] ([X.sub.1]) = [[epsilon].sub.h] ([X.sub.2]) = 0, (2.13)

then [[epsilon].sub.h] is a counit for the application [[DELTA].sub.h] in (2.9), that is, [[DELTA].sub.h] and [[epsilon].sub.h], satisfy

[[DELTA].sub.h] (id [cross product] [epsilon]) = [[DELTA].sub.h] ([epsilon] [cross product] id)

trivially. We can infer from Lemmas (2.1) and (2.2) that the map [[DELTA].sub.h] in (2.9), with [[epsilon].sub.h] in (2.13), is a co-product on the space [U.sub.h](ST(2)). Besides, if we consider the map o in (2.4) extended on [U.sub.h](ST(2)) we have the following assertion:

Proposition 2.2. The space [U.sub.h](ST(2)) generate by [X.sub.1],[X.sub.2] is a Lie bialgebra, with the coproduct [[DELTA].sub.h] the counit [[epsilon].sub.h] and the cobracket [delta] define by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14)

where f and g are functions that satisfy the following properties:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Since that [U.sub.h](ST(2)) is a co-algebra with [[DELTA].sub.h], [[epsilon].sub.h] to prove that it is a Lie bialgebra it is enough to show that [delta] is the cobracket on [U.sub.h](ST(2)). That is, we must prove that [delta] satisfies cobracket properties (see Note 2). In fact, since [delta]([X.sub.2]) = 0 we have that [delta] and [[DELTA].sub.h] are compatible, that is, they satisfy trivially

[delta]([X.sub.i][X.sub.j]) = [delta]([X.sub.1]) [[DELTA].sub.h]([X.sub.j]) + [[DELTA].sub.h]([X.sub.i]) [delta]([X.sub.j]). (2.15)

Since the co-Jacobi identity,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

does not depend of [[DELTA].sub.h], when we extend [delta] over [U.sub.h](ST(2)), [delta] satisfies this property.

Likewise, [delta] satisfies the co-Leibniz identity , ([[DELTA].sub.h] [cross product] id) [delta] = (id [cross product] [delta])[[DELTA].sub.h] + [[sigma].sub.23]([delta] [producto cruzado] id) [DELTA]h, (2.16) where [[sigma].sub.23] means the permutations of the last two elements. Since [sigma]([X.sub.2]) = 0 the identity is zero for [X.sub.2]. While for [X.sub.1] the left side of (2.16) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the right expression becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore [U.sub.h](ST(2)) has a Lie bialgebra structure with the bracket [,] in (2.2), the cob racket [delta] and the coproduct

Since the quantum enveloping algebra is an algebra over formal power series in h, we can describe it by exponential expressions such as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To perform this description we need the following lemma:

Lemma 2.3. The functions on [U.sub.h](ST(2))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Since [[DELTA].sub.h] is linear we can calculate [[DELTA].sub.h] on the exponential function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In a similar fashion we can prove the lemma for the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we can take in (2.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, [[DELTA].sub.h] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note 3. Since [delta]([X.sub.2]) = 0 we must take the special expression for [[DELTA].sub.h] ([X.sub.2]). That is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

In this case [delta] in (2.4) is a cobracket on the [U.sub.h](ST(2) that does not satisfy the co-Leibniz identity (2.16). However, ([U.sub.h](ST(2), [[DELTA].sub.h], [delta], [member of], [,]) is still a Lie bialgebra with [[DELTA].sub.h] in (2.17).

With [[DELTA].sub.h] defined in (2.17) we can find an antipode application on [U.sub.h](ST(2). Let

[S.sub.h] : [U.sub.h](ST(2)) [right arrow] [U.sub.h](ST(2))

be this application. Since [S.sub.h] must satisfy the properties of the antipode application, it satisfies in particular the identity

m([S.sub.h] [cross product] I) [[DELTA].sub.h] = m (I [cross product] [S.sub.h]) [[DELTA].sub.h] = 0

where m is the multiplication on [U.sub.h](ST(2)). Thus [S.sub.h] satisfies the following lemma:

Lemma 2.4. The application

[S.sub.h] : [U.sub.h](ST(2))[right arrow] [U.sub.h](ST(2))

defined on the Lie bialgebra ([U.sub.h](ST(2)), [[DELTA].sub.h], [[epsilon].sub.h], [delta]) satisfies

m([S.sub.h] [cross product] I)[[DELTA].sub.h] = m(I [cross product] [S.sub.h])[[DELTA].sub.h] = 0, (2.18) if, and only if,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.19)

Proof. The left-hand of (2.18) in [X.sub.1] takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the same way we can obtain [S.sub.h]([X.sub.2]) = -[X.sub.2]

Now we must prove that [S.sub.h] defined in (2.19) is the antipode application on [U.sub.h](ST(2)).

Lemma 2.5. The application [S.sub.h]

[S.sub.h]: [U.sub.h](ST(2))[right arrow] [U.sub.h](ST(2))

defined on {[X.sub.1], [X.sub.2]} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an antipode application on [U.sub.h](ST(2)).

Proof. In fact, [S.sub.h] satisfies the following properties

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first property is the property (2.18), which we used to find [S.sub.h]([X.sub.1]) and [S.sub.h]([X.sub.2]); thus [S.sub.h] satisfies this property for [X.sub.1], [X.sub.2]. The second property is obtained as a result of the following two expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we have used that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from the definition of the exponential function. In the same form we prove the last property of antipode,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Lemmas (2.4) and (2.5) complete the proof of the following proposition:

Proposition 2.3. The algebra [U.sub.h](ST(2)) generated by [X.sub.1], [X.sub.2], I with the operations defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

has the structure of a Hopf algebra.

Since when h [right arrow] 0, the coalgebra structure of [U.sub.h](ST(2)) coincides with the bialgebra ST(2).

Finally, we verify the *-algebra structure.

Proposition 2.4. The algebra [U.sub.h](ST(2)) is a Hopf *-algebra with [X.sub.1] = [X.sub.1], [X.sub.2] = [X.sup.*.sub.2].

Proof. Let * : [X.sub.i] [flecha diestra] [X.sup.*.sub.i], i = 1, 2 be the involution. Then the operations defined in the proposition (2.3) are *-algebra maps. In fact, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similary we can see that [DELTA] and [epsilon] are *-algebra maps.

Because of propositions (2.4) and (2.3) we can affirm that the Hopf algebra [U.sub.h](ST(2)) is the quantum group of the universal enveloping algebra U(ST(2)).

References

[1] G.E. Arutyunov & P.B. Medvedev, Quantization of the external algebra on a Poisson-Lie group, hep-th/9311096 (1993), 1-20.

[2] V. Chari 8? A. Pressley, A guide to Quantum Groups, Cambridge University Press, Cambridge University, 1994.

[3] J. Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996.

[4] V. Drinfeld, Quantum groups, ICM-86, 1986, 798-820.

[5] V. Drinfeld, On some unsolved problems in quantum group theory, Lecture Notes in Mathematics 1510, Springer-Verlag, 1992, 1-8.

[6] H. D. Doebner, J. D. Hennig & W. Liicke, Quantum Groups, Lecture Notes in Physics 370, Springer-Verlag, 1990.

[7] P. Etingof &? David Kazhdan, Quantization of Poisson algebraic groups and Poisson homogeneous spaces, q-alg 9510020 (1995), 1-9.

[8] Berenice Guerrero, Sobre una estructura diferencial cuantica. Reporte interno No. 56, Departamento de Matematicas y Estadistica, Universidad Nacional de Colombia, 1997.

[9] Berenice Guerrero, Cuantizacion no estandar del grupo triangular ST(S), Lecturas Matematicas 18 (1997), 23-44.

[10] D. Gurevich &? V. Rubtsov, Yang-Baxter equation and deformation of associative and Lie algebras. Lectures Notes in Mathematics 1510,'Springer-Verlag, 1992, 9-46.

[11] [12] C. Kassel, Quantum Groups, Springer-Verlag, Berlin, 1995.

[12] J. H. Lu A. Weinstein, Poisson Lie groups, dressing trasnformations and Bruhat decompositions, J. Differential Geometry 31 (1990), 501-526.

[13] S. Majid Foundations of Quantum Group Theory, Cambridge University Press, 1995.

[14] L. A. Takthajan, Quantum groups and integrable models, Advanced Studies in Pure Mathematics 19 (1990), 435457.

[15] L. A. Takthajan, Lectures on Quantum Groups, Nakai Institute Series in Mathematical Physics (1990), 193-225.

[16] N. Yu. Reshetikhin, L. Takthajan & L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra and Analyis (1989), 178-206.

[16] M. A. Semenov-Tian-Shnsky, Lectures on R-matrices, Poisson-Lie Groups and Integrable Systems, Proceedings of the CIMPA School 1991 Nice (France), World Scientific, 1994, 269-318.

Berenice Guerrero (1)

1991 Mathematics Subject Clasification. Primary 17B37. Secundary 16W30, 17B66.

(1) Departamento de Matematicas y Estadistica, Universidad Nacional de Colombia, Bogota. Email: aguerrer@ciencias.ciencias.unal.edu.co, beregue@matematicas.unal.edu.co.

The author wishes to thank the support of CINDEC and ICFES. She also wants to acknowledge the important remarks of the referee which helped to improve substantially the text of this article.
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