# Quantum groups and functional relations for higher rank

###### Abstract.

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.

###### Contents

## 1. Introduction

The method of functional relations was proposed by Baxter to solve statistical models which can or cannot be treated with the help of the Bethe ansatz, see, for example, [1, 2]. It appears that its main ingredients, transfer matrices and -operators, are essential not only for the integration of the corresponding quantum statistical models in the sense of calculating the partition function in the thermodynamic limit. One of the remarkable recent applications is their usage in the construction of the fermionic basis [3, 4, 5, 6] for the observables of the XXZ spin chain.

It seems that the most productive, although not comprehensive, modern approach to the theory of quantum integrable systems is the approach based on the concept of quantum group invented by Drinfeld and Jimbo [7, 8]. In accordance with this approach, the transfer matrices and -operators are constructed by choosing first the representations for the factors of the tensor product of two copies of the underlying quantum group. Then these representations are applied to the universal -matrix, and finally the trace over one of the representation spaces is taken. Here the functional relations are consequences of the properties of the used representations of the quantum group. For the first time, it was conceived by Bazhanov, Lukyanov and Zamolodchikov [9, 10, 11].

Following the physical tradition, we call the representation space corresponding to the first factor of the tensor product the auxiliary space, and the representation space corresponding to the second factor the quantum space. The representation on the auxiliary space defines the integrability object, while the representation on the quantum space defines the concrete physical integrable system. It appears convenient for our purposes to fix the representation for the first factor only, see, for example, [12, 13, 14, 15, 16]. We use for the objects obtained in such a way the term “universal”. The relations obtained for them can be used for any physical model related to the quantum group under consideration.

In the papers [9, 10, 11] the case of the quantum group was considered and as the quantum space the state space of a conformally invariant two dimensional field theory was taken. We reconsidered the case of this quantum group in the papers [15, 16] where we obtained the functional relations in the universal form and then chose as the quantum space the state space of the XXZ-spin chain.

A two dimensional field theory with extended conformal symmetry related to the quantum group was analysed in the paper [17]. Here as the quantum space the corresponding state space of the quantum continuous field theory under consideration was taken again. In the present paper we define for the case of the quantum group universal integrability objects and derive the corresponding functional relations. Here to define the integrability objects we use the general gradation and general twisting. However, the main difference from [17] is that we give a new and detailed proof of the functional relations. In the paper [17] proofs are often skipped, or given in schematic form. This was one of the reasons for writing our paper. Another reason was the desire to have the specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain.

Below we use the notation

so the definition of the -deformed number can be written as

We denote by the loop Lie algebra of a finite dimensional simple Lie algebra and by its standard central extension, see, for example, the book by Kac [18]. The symbol means the set of natural numbers and the symbol the set of non-negative integers.

To construct integrability objects one uses spectral parameters. They are introduced by defining a -gradation of the quantum group. In the case under consideration a -gradation of is determined by three integers , and . We use the notation and denote by some fixed th root of , so that .

## 2. Integrability objects

In this paper we consider integrable systems related to the quantum group . Depending on the sense of the “deformation parameter” , there are at least three definitions of a quantum group. According to the first definition, , where is an indeterminate, according to the second one, is indeterminate, and according to the third one, , where is a complex number. In the first case a quantum group is a -algebra, in the second case a -algebra, and in the third case it is just a complex algebra. We define the quantum group as a -algebra, see, for example, the books [19, 20].

To construct representations of the quantum group we use the Jimbo’s homomorphism from to the quantum group . Therefore, we first remind the definition of and then discuss .

### 2.1. Quantum group

Denote by the standard Cartan subalgebra of the Lie algebra and by , , the elements forming the standard basis of .^{1}^{1}1We use the usual notation for the matrix units. The root system of relative to is generated by the simple roots , , given by the relations

(2.1) |

where

The Lie algebra is a subalgebra of , and the standard Cartan subalgebra of is a subalgebra of . Here the standard Cartan generators , , of are

(2.2) |

and we have

where

(2.3) |

is the Cartan matrix of .

Let be a complex number and . We define the quantum group as a unital associative -algebra generated by the elements , , , and , , with the relations^{2}^{2}2It is necessary to assume that . In fact, we assume that is not any root of unity.

(2.4) | |||

(2.5) | |||

(2.6) |

satisfied for any and , and the Serre relations

(2.7) |

satisfied for any distinct and . Note that is just a convenient notation. There are no elements of corresponding to the elements of . In fact, this notation means a set of elements of parametrized by . It is convenient to use the notations

and

(2.8) |

for any and . Here equation (2.6) takes the form

Similar notations are used below for the case of the quantum groups and .

With respect to the properly defined coproduct, counit and antipode the quantum group is a Hopf algebra.

Looking at (2.5) one can say that the generators and are related to the roots and respectively. Define the elements related to the roots and as

(2.9) |

The Serre relations (2.7) give

(2.10) | ||||

(2.11) |

One can also verify that

and, besides,

(2.12) | ||||

(2.13) |

Using the above relations, one can find explicit expressions for the action of the generators of on Verma -modules, see appendix A.

### 2.2. Quantum group

#### 2.2.1. Definition

We start with the quantum group . Recall that the Cartan subalgebra of is

where is the standard Cartan subalgebra of and is the central element [18]. Define the Cartan elements

so that one has

(2.14) |

and

The simple roots , , , , are given by the equation

where

is the Cartan matrix of the Lie algebra .

As before, let be a complex number and . The quantum group is a unital associative -algebra generated by the elements , , , and , , with the relations

(2.15) | |||

(2.16) | |||

(2.17) |

satisfied for all and , and the Serre relations

(2.18) |

satisfied for all distinct and .

The quantum group is a Hopf algebra with the comultiplication , the antipode , and the counit defined by the relations

(2.19) | |||

(2.20) | |||

(2.21) |

The quantum group can be defined as the quotient algebra of by the two-sided ideal generated by the elements of the form , . In terms of generators and relations the quantum group is a -algebra generated by the elements , , , and , , with relations (2.15)–(2.18) and an additional relation

(2.22) |

where . It is a Hopf algebra with the Hopf structure defined by (2.19)–(2.21). One of the reasons to use the quantum group instead of is that in the case of we have no expression for the universal -matrix.

#### 2.2.2. Universal -matrix

As any Hopf algebra the quantum group has another comultiplication called the opposite comultiplication. It can be defined explicitly by the equations

(2.23) | |||

(2.24) |

When the quantum group is defined as a -algebra it is a quasitriangular Hopf algebra. It means that there exists an element , called the universal -matrix, such that

for all , and^{3}^{3}3For the explanation of the notation see, for example, the book [21] or the papers [22, 15].

(2.25) |

The expression for the universal -matrix of considered as a -algebra can be constructed using the procedure proposed by Khoroshkin and Tolstoy [23]. Note that here the universal -matrix is an element of , where is the Borel subalgebra of generated by , , and , , and is the Borel subalgebra of generated by , , and , .

In fact, one can use the expression for the universal -matrix from the paper [23] also for the case of the quantum group defined as a -algebra having in mind that in this case the quantum group is quasitriangular only in some restricted sense. Namely, all the relations involving the universal -matrix should be considered as valid only for the weight representations of , see in this respect the paper [24] and the discussion below.

Recall that a representation of on the vector space is a weight representation if

where

Taking into account relations (2.22) and (2.14), we conclude that only if

(2.26) |

Let and be weight representations of on the vector spaces and with the weight decompositions

In the tensor product the role of the universal -matrix is played by the operator

(2.27) |

Here is an element of , where and are the subalgebras of generated by , , and , , respectively. The operator acts on a vector in accordance with the equation

(2.28) |

where

is the inverse matrix of the Cartan matrix (2.3) of the Lie algebra . It follows from (2.26) that (2.28) can be written in a simpler form

(2.29) |

### 2.3. -operators

To obtain -operators, or, as they also called, -matrices, one uses for both factors of the tensor product of the two copies of the quantum group one and the same finite dimensional representation. We do not use in this paper the explicit form of the -operators for the quantum group . The corresponding calculations for this case and for some other quantum groups can be found in the papers [25, 26, 27, 28, 29, 22, 30].

### 2.4. Universal monodromy and universal transfer operators

#### 2.4.1. General remarks

To construct universal monodromy and transfer operators we endow with a -gradation, see, for example, [15, 16]. The usual way to do it is as follows.

Given , we define an automorphism of by its action on the generators of as

where are arbitrary integers. The family of automorphisms , , generates the -gradation with the grading subspaces

Taking into account (2.19) we see that

(2.30) |

It also follows from the explicit form of the universal -matrix obtained with the help of the Tolstoy–Khoroshkin construction, see [23, 22], that for any we have

(2.31) |

Following the physical tradition, we call the spectral parameter.

If is a representation of , then for any the mapping is also a representation of . Below, for any homomorphism from to some algebra we use the notation

(2.32) |

If is a -module corresponding to the representation , we denote by the -module corresponding to the representation . Certainly, as vector spaces and coincide.

Now let be a representation of the quantum group on a vector space . The universal monodromy operator corresponding to the representation is defined by the relation

It is clear that is an element of .

Universal monodromy operators are auxiliary objects needed for construction of universal transfer operators. The universal transfer operator corresponding to the universal monodromy operator is defined as

where is a group-like element of called a twist element. Note that is an element of . An important property of the universal transfer operators is that they commute for all representations and all values of . They also commute with all generators , , see, for example, our papers [15, 16].

As we noted above, to construct representations of we are going to use Jimbo’s homomorphism. It is a homomorphism defined by the equations^{4}^{4}4Recall that the Cartan generators of are related to those of by relation (2.2).

(2.33) | |||||

(2.34) | |||||

(2.35) |

see the paper [31]. If is a representation of , then is a representation of . Define the universal monodromy operator

being an element of . It is evident that

For the corresponding transfer operator we have

Introduce the notation

Note that is a trace on . This means that it is a linear mapping from to satisfying the cyclicity condition

for any . One can write

Thus, to obtain the universal transfer operators , one can use different universal monodromy operators corresponding to different representations , or use one and the same universal monodromy operator but different traces corresponding to different representations .

#### 2.4.2. More universal monodromy operators

Additional universal monodromy operators can be defined with the help of automorphisms of . There are two special automorphisms of . The first one is defined by the relations

(2.36) | |||||

(2.37) | |||||

(2.38) |

and the second one is given by

(2.39) | |||||

(2.40) | |||||

(2.41) |

These automorphisms generate a subgroup of the automorphism group of isomorphic to the dihedral group . Using the automorphisms and , we define two families of homomorphisms from to generalizing the Jimbo’s homomorphism as

and the corresponding universal monodromy operators as

The prime means that the corresponding homomorphisms and the objects related to them are redefined below to have simpler form of the functional relations. Below we often define objects with the help of the powers of the automorphism . Different powers are marked by the values of the corresponding index. We assume that if the index is omitted it means that the object is taken for the index value , in particular, we have . Since is the identity automorphism of , we have

Therefore, there are only six different universal monodromy operators of such kind.

It follows from (2.19) that

Similarly, (2.23) and (2.24) give

Using the definition of the universal -matrix (2.25), we obtain the equation

Taking into account the uniqueness theorem for the universal -matrix [25], we conclude that

(2.42) |

Using this relation, it is not difficult to demonstrate that

(2.43) |

where stands for

One can also show that

(2.44) |

This relation, together with the equation

(2.45) |

gives

(2.46) |

where stands for

and we take into account that .

Starting with the infinite dimensional representation of the quantum group described in appendix A, we define the infinite dimensional representations

of the quantum group . Slightly abusing notation we denote the corresponding -modules by and . We define two families of universal monodromy operators:

In the same way one defines two families of universal monodromy operators associated with the finite dimensional representation of :

where

The newly defined universal monodromy operators satisfy relations similar to and .

#### 2.4.3. Universal transfer operators

Recall that a universal transfer operator is constructed by taking the trace over the auxiliary space. In the case of an infinite dimensional representation there is a problem of convergence which can be solved with the help of a nontrivial twist element. We use a twist element of the form

(2.47) |

where , and are complex numbers. Taking into account (2.22)and (2.14), we assume that

We define two families of universal transfer operators associated with the infinite dimensional representations of as

where

and two families of universal transfer operators associated with the finite dimensional representations of as

where

Note that the mappings and are traces on the algebra .

Let us discuss the dependence of the universal transfer operators on the spectral parameter . Consider, for example, the universal transfer operator . From the structure of the universal -matrix, it follows that the dependence on is determined by the dependence on of the elements of the form , where . Any such element is a linear combination of monomials each of which is a product of , , and for some . Let be such a monomial. We have

where , and are the numbers of , and