Years of Quantum Groups : from Definition to Classification

In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians.

In mathematics and theoretical physics, quantum groups are certain non-commutative algebras that first appeared in the theory of quantum integrable models, and which were then formalized by Drinfeld and Jimbo.
Nowadays quantum groups are one of the most popular objects in modern mathematical physics.They also play a significant role in other areas of mathematics.It turned out that quantum groups provide invariants of knots, and that they can be used in quantization of Poisson brackets on manifolds.Quantum groups gave birth to quantum geometry, quantum calculus, quantum special functions and many other "quantum" areas.
At first, quantum groups appeared to be a useful tool for the following program: Let R satisfy the quantum Yang--Baxter equation (QYBE) and let R be decomposable in a series in formal parameter ÿ.Then r, which is the coefficient at the first order term, satisfies the so-called classical Yang--Baxter equation (CYBE).In many cases CYBE can be solved and the problem is to extend the solutions of CYBE to solutions of QYBE.
However, the most important applications of quantum groups relate to the theory of integrable models in mathematical physics.The presence of quantum group symmetries (or so-called hidden symmetries) was the crucial point in explicit solutions of many sophisticated non-linear equations, such as Korteweg-de Vries or Sine-Gordon.Quantum groups changed and enriched representation theory and algebraic topology.
Quantum groups were defined by V. Drinfeld as Hopf algebra deformations of the universal enveloping algebras (and also their dual Hopf algebras).More exactly, we say that a h is a quantum group (or maybe a quasi-classical quantum group) if the following conditions are satisfied: i.The Hopf algebra A A h is isomorphic as a Hopf algebra to the universal enveloping algebra of some Lie algebra L.
ii.As a topological [ ] for some vector space V over C.
The first examples of quantum groups were quantum universal enveloping algebras U q ( ) g , quantum affine Kac-Moody algebras U q ( $) g , and Yangians Y( ) g .
Further, it is well-known that the Lie algebra L such that If A is a quantum group with A A h = U L ( ), then L possesses a new structure, which is called a Lie bialgebra structure d, and it is given by: where a is an inverse image of x in A. In particular, the classical limit of U q ( ) g is g, the classical limit of U q ( $) g is the affine Kac-Moody algebra, which is a central extension of g[ , ] u u -1 , and the classical limit of Y( ) g is g[ ] u (the corresponding Lie bialgebra structures will be described later).

The Lie bialgebra ( , )
L d is called the classical limit of A, and A is a quantization of ( , ) L d .So, any quantum group in our sense has its classical limit, which is a Lie bialgebra, and the following natural problem arises: Given a Lie bialgebra, is there any quantum group whose classical limit is the given Lie bialgebra?Or in other words: Can any Lie bialgebra be quantized?
In the mid 1990's P. Etingof and D. Kazhdan gave a positive answer to this problem.Now, let g be a simple complex finite-dimensional Lie where { } I l is an orthonormal basis of g with respect to the Killing form.Then it is well-known that the function is a rational skew-symmetric solution of the classical Yang--Baxter equation that is More generally, we call r( , ) u v a rational r-matrix if it is skew-symmetric, satisfies (2) and r( , ) ( , ) defines a Lie bialgebra structure on g[ ] u , since W is a g-invariant element of g g Ä and this implies that The Yangian Y( ) g is precisely the quantization of this Lie bialgebra.On the other hand, it is easy to see that all rational r-matrices introduced in [13] define Lie bialgebra structures on g[ ] u by the formula Of course, the next question is: Which quantum groups quantize the other "rational" Lie bialgebra structures on g[ ] u ?
Before we give an answer to this question we need the following two definitions: At this point we note that any rational r-matrix is a twist of Yang's r-matrix.Let H be a Hopf algebra and F H H Î Ä be an invertible element.Let F satisfy Such F is called a quantum twist.The formula defines a new co-multiplication on H.

Conjecture and scheme
Although it is clear that any quantum twist on a quantum group induces uniquely a classical twist on its classical limit, the converse statement remained unknown for a long time.It was formulated in [9] in 2004.

Conjecture 1.
Any classical twist can be extended to a quantum twist.
The conjecture was proved by G. Halbout in [3].In particular, we can now give an answer to the question posed in the previous section: A quantum group which quantizes any rational Lie bialgebra structure on L u = g[ ] is isomorphic to the Yangian Y( ) g as an algebra and it has a twisted co-algebra structure defined by the corresponding rational solution of CYBE.However, there might exist Lie bialgebra structures on g[ ] u of a different nature, and for classification purposes one has to find all of them.Therefore, in order to classify quantum groups which have a given Lie algebra L as the classical limit one has to solve the following four problems: 1. Describe all basic Lie bialgebra structures on L (in other words all Lie bialgebra structures up to classical twisting).2. Find quantum groups corresponding to the basic structures.3. Describe all the corresponding classical twists.4. Quantize all these classical twists.

Lie bialgebra structures on the polynomial Lie algebras and their quantization
According to the results of an unpublished paper by Montaner and Zelmanov [11], there are four basic Lie bialgebra structures on the polynomial Lie algebra P Let us describe them (and, hence, we do the first step in the classification of Lie bialgebra structures on g[ ] u ).When it is possible we make further steps.

Case 1.
Here the one-cocycle d 1 0 = In this case it is not difficult to show that there is a one-to-one correspondence between Lie bialgebra structures of the first type and finite-dimensional quasi-Frobenius Lie subalgebras of g[ ] u .The corresponding quantum group is U u ( [ ]) g .Classical twists can be quantized following Drinfeld's quantization of skew-symmetric constant r-matrices.

Case 2.
In this case the Lie bialgebra structure is described by where G 2 ( , ) u v is Yang's rational r-matrix.
The corresponding Lie bialgebra structures are in a one-to-one correspondence with the rational solutions of CYBE described in [13].The corresponding quantum group is a Yangian Y( ) g .Quantum twists were found for g = sl n for n = 2 3 , in [9].

Case 3.
Here the basic Lie bialgebra structure is given by where r DJ is the classical Drinfeld-Jimbo modified r-matrix.
This Lie bialgebra is the classical limit of the quantum group U g u q ( [ ]), which is a certain parabolic subalgebra of a non--twisted quantum affine algebra U q ( $) g (see [15] or [10]).
There is a natural one-to-one correspondence between Lie bialgebra structures of the third type and the so-called quasi--trigonometric solutions of CYBE.A complete classification of the classical twists for sl n was presented in [9] and for the arbitrary g in [12].Details on quantization of the classical twists for sl n can be found in [9].

Case 4.
Finally, the 4-th basic Lie bialgebra structure on g[ ] u is defined as follows: It was proved in [14] that there is a natural one-to-one correspondence between Lie bialgebra structures of this kind and the so-called quasi-rational solutions of CYBE.The quasi-rational solutions of CYBE for g = sl n were classified in [14].Some ideas indicate that quasi-rational r-matrices do not exist for , , .The corresponding quantum group remains unknown, but, it is rather clear that it is related to the dual Hopf algebra of Y( ) g .

Some open questions
1. Following steps 1-4 in the classification scheme, it is natural to classify quantum groups related to affine Kac--Moody algebras.A conjecture is that in this case we have only two basic types of Lie bialgebra structures and the corresponding quantum groups are quantum affine algebras and so-called doubles of Y( ) g .

2.
More generally: Let L be a Lie algebra.Is it possible to describe the moduli space of Double(L), all Lie bialgebra structures on L modulo action of classical twists?

Conjecture 2.
Let Fun( ) M be the Poisson algebra of smooth functions on M. Then it can be quantized in an equivariant way, i.e., there exists an associative algebra A, which is a deformation of the Poisson algebra Fun( ) M , and which is a Hopf module algebra over H.This conjecture is based on some results proved in [6,7,8].A special case of this conjecture has been proved in [2].More exactly, in this paper the conjecture was proved for Lie bialgebras of a special type L D K = ( ), where K is another Lie bialgebra and L D K = ( ) is the corresponding classical double.
Recent progress in related questions was achieved in [4].
Here, a result similar to the conjecture above was proved under the following assumptions: L is the so-called coboundary Lie bialgebra and L acts freely on M (in other words M is far from being a homogeneous space).
5 Appendix: Solutions for sl (2) and deformed Hamiltonians The aim of this section is to present concrete examples of quantum twists and the corresponding Hamiltonians following the results obtained in [5].
We consider the case sl(2).Let s Recall that in sl( 2) we have two quasi-trigonometric solutions, modulo gauge equivalence.The non-trivial solution is X z z X z z z z s s .This solution is gauge equivalent to the following: The above quasi-trigonometric solution was quantized in [5].Let p 1 2 ( ) z be the two-dimensional vector representation of U sl q ( ) 2 .In this representation, the generator e -a acts as a matrix unit e 21 , e d a -as ze 21 and h a as e e ( where P 12 denotes the permutation of factors in C C Ä , is a quantization of the following rational solution of CYBE: algebra and let g[ ] u be the corresponding polynomial Lie algebra.If you ask a physicist which quantum group is a quantization of g[ ] u , you will almost certainly hear that the quantization of g[ ] u is the Yangian Y( ) g .Let us explain this in greater detail.Set W