Some details of proofs of theorems related to the quantum dynamical Yang-Baxter equation

This paper gives some further details of proofs of some theorems related to the quantum dynamical Yang-Baxter equation. This mainly expands proofs given in"Lectures on the dynamical Yang-Baxter equation"by P. Etingof and O. Schiffmann, math.QA/9908064. This concerns the intertwining operator, the fusion matrix, the exchange matrix and the difference operators. The last part expands proofs given in"Traces of intertwiners for quantum groups and difference equations, I"by P. Etingof and A. Varchenko, math.QA/9907181. This concerns the dual Macdonald-Ruijsenaars equations. This paper does not claim originality, priority or completeness. It is meant as a service to whoever may take profit of it.


Introduction
The quantum dynamical Yang-Baxter equation (QDYBE) was first considered in 1984 by Gervais and Neveu [9], with motivation from physics (for monodromy matrices in Liouville theory). A general form of QDYBE with spectral parameter was presented by Felder [7], [8] at two major congresses in 1994. The corresponding classical dynamical Yang-Baxter equation (CDYBE) was presented there as well. Next Etingof and Varchenko started a program to give geometric interpretations of solutions of CDYBE (see [3]) and of QDYBE (see [4]) in the case without spectral parameter. In the context of this program they pointed out a method to obtain solutions of QDYBE by the so-called exchange construction (see [5]). This uses, for any simple Lie algebra g, representation theory of U (g) or of its quantized version U q (g) in order to define successively the intertwining operator, the fusion matrix and the exchange matrix. The matrix elements of the intertwining operator and of the exchange matrix generalize respectively the Clebsch-Gordan coefficients and the Racah coefficients to the case where the first tensor factor is a Verma module rather than a finite dimensional irreducible module. The exchange matrix is shown to satisfy QDYBE. Etingof and Varchenko also started in [6] a related program to connect the above objects with weighted trace functions and with solutions of the (q-)Knizhnik-Zamolodchikov-Bernard equation (KZB or qKZB).
A nice introduction to the topics indicated above was recently given by Etingof and O. Schiffmann [2]. While I was reading this paper in connection with a seminar in Amsterdam during the fall of 1999, I added some details of proofs for my own convenience, and I put these notes in TeX in order that the other participants in the seminar could take profit of it. I put these informal notes on my homepage. Since the version v2 of [2] is now referring to these notes, I decided to post the paper on QA.
I want to emphasize that these notes are purely meant as a service to whoever may take profit of it. I do not claim any originality or priority with these proofs. Neither I tried to cover the full contents of [2]. Most of my paper only treats the q = 1 case. Only the second part of the section on the exchange matrix also covers the quantum case. In general, the extension to the quantum case will ususally be straightforward.
As for the contents, Sections 2, 3 and 4 respectively deal with the intertwining operator, the fusion matrix and the exchange matrix. In [2] these topics are all covered in Section 2. My Sections 5 on difference operators and 6 on weighted trace functions address some topics in Section 9 of [2] (Transfer matrices and generalized Macdonald-Ruijsenaars equations). The details of proofs in Section 6 concern q = 1 analogues of proofs given in Section 3 of [6] in connection with the dual Macdonald-Ruijsenaars equations.
I want to call attention to one conceptual aspect. This concerns formulas (4.8), (4.9). The first formula expresses an exchange matrix R U,V ⊗W (λ) after shifted conjugation by the fusion matrix J V W (λ) as a product of R UV (λ) (with appropriately shifted λ) and R UW (λ). The second formula is analogous. These formulas are not explicitly given in [2], but they do occur in [6] without getting particular emphasis. They can be used in order to prove that R(λ) satisfies QDYBE. This is analogous to the role of the quasi-triangularity property of the (non-dynamical) universal R-matrix for proving the QYBE in that case. In fact, it is possible to see (4.8) and (4.9) in the context of a certain quasitriangular quasi-Hopf algebra, see Babelon, Bernard & Biley [1, Section 3] for the quantum sl(2) case.

The intertwining operator
First I make two preliminary remarks in preparation of the proof of [E-S], Proposition 2.2.
Let g be a Lie algebra with Lie subalgebra k, and let V be a k-module. Then: Then Frobenius reciprocity states that there is an isomorphism of linear spaces For the other remark let g be a Lie algebra and let Z, W, V be g-modules. Then there is an isomorphism of linear spaces given by F (z ⊗ w * ) = f (z), w * (z ∈ Z, w * ∈ W * ).

Proof of [E-S], Proposition 2.2. We have a composition of five isomorphisms
Proof that the coefficients of Φ v λ are rational in λ (statement in paragraph after the proof of [E-S], Proposition 2.2; the proof below is essentially due to Eric Opdam) Let α 1 , . . . , α N be the positive roots (the elements of ∆ + ). Let V be a finite-dimensional g-module, and let v ∈ V \{0} be h-homogeneous. Consider the Verma module M λ−wt (v) for generic values of λ ∈ h * , where it is irreducible. By Proposition 2.2 there is a unique The unique existence of Φ v λ satisfying the above conditions is equivalent to the unique existence of w ∈ M µ ⊗ V such that wt(w) = λ, e α i · w = 0 for i = 1, . . . , N and such that w has the form of the right-hand side of (2.1) with v 0,...,0 = v. We will show that the unique existence of w with these properties implies that the v k 1 ,...,k N are rational in λ.
Note that So the inhomogeneous system of linear equations in the coordinates of the vectors v l 1 ,...,l N (l 1 , . . . , l N nonnegative integers, not all 0) given by has for generic λ a unique solution. Since the coefficients are polynomials in λ it follows that the solution must be rational in λ.

The fusion matrix
Proof of [E-S], Proposition 2.3, part 2 It follows that

Proof of [E-S], Proposition 2.3, part 3
On the one hand we have On the other hand, expression (3.1) also equals Hence, by equality of expressions (3.2) and (3.3), we have Hence we arrive at the dynamical 2-cocycle condition, which was to be proved:

The exchange matrix
as an identity of operators on V ⊗ W ⊗ U . In preparation of the proof recall that Φ w,v Now we have on the one hand and accordingly On the other hand we have and accordingly It follows from (4.2) and (4.4) that Hence the right-hand sides of (4.3) and (4.5) are equal. Thus the left-hand sides of (4.3) and (4.5) are also equal.
As pointed out in [E-S], §2.2 the construction of intertwining operators, fusion and exchange matrices admit natural quantum analogues. Most definitions, results and proofs go on essentially unchanged compared to the q = 1 case. However, in the definition of the exchange matrix the R-matrix R V W associated to U q (g)-modules V and W , and induced by the universal R-matrix R, is also needed. I will use the notation This is different from the notation R 21 The exchange matrix in the quantum case is now defined by The dynamical two-cocycle condition (3.4) will remain valid in the quantum case. I will now discuss a second proof of the QDYBE (4.1), which is briefly sketched in the remark in [E-S] after Proposition 2.4, and which also holds in the quantum case. In the following, when being in the q = 1 case, just put R V W equal to 1 (for any V, W ).
I derive first the following two important formulas (not given in [E-S]) for the exchange matrix: where both sides in (4.8) and (4.9) are acting on U ⊗ V ⊗ W . Here we have adapted the notation introduced in [E-S] just before Proposition 2.3 as follows. If U = A i then F (λ − h (U) ) will mean F (λ − h (i) ). One of the formulas (4.8) and (4.9) can be obtained by specialization of formula (2.42) in [E-V]. Note that (4.8) and (4.9) are also dynamical analogues of the formulas obtained from the following formulas for the universal R-matrix: which belong to the defining properties of a quasitriangular Hopf algebra. Another defining property of a quasitriangular Hopf algebra is that which implies for the universal fusion matrix J(λ) (see [E-S], §8) that (4.11) In the proof of (4.8) and (4.9) I will need (4.10) and (4.11).

Difference operators
Next I give a proof for the q = 1 case of the formula stated at the end of §9.1 in [E-S] for the quantum case. Let g be a simple Lie algebra. For any two finite-dimensional g-modules U and V let R V U (λ) be the exchange matrix. (which block will be zero unless λ + µ = ν + σ).
Proof of (5.1) We can rewrite (4.9) as Then Note that it was possible to apply (4.9) in the above proof above because we had assumed thet D λ,U First consider the proof of Lemma 2.14 in [E-V]. Let W be a finite-dimensional gmodule. By the properties of the intertwining operator we can uniquely define a bilinear form B λ,W : W × W * → C by the formula Define a generalized element Q(λ) in U (g) in terms of the universal fusion matrix by This induces an endomorphism Q W (λ) of W given by where C W denotes contraction of an endomorphism of W ⊗ W to an endomorphism of W . Now we have B λ,W (w, w * ) = Q W (λ) w, w * .
It follows from (6.5) that Q W (λ) is a weight preserving endomorphism of W . Next we have Combination of (6.6) with (6.5) yields that It follows from (6.5) and (6.1) that Q U⊗W (λ) = Q 21 W ⊗U (λ). Hence we can rewrite (6.7) as which is clearly independent of the choice of B. Define the isomorphism , z ∈ M µ+ν . (6.11) Proposition 3.1 together with formula (3.2) in [E-V] can now be formulated as follows: Proof of (6.12) Write R W V (µ + ν) = i p i ⊗ q t i . Let y ∈ M µ+ν and w ∈ W [ν]. Then For λ ∈ h * and U a g-module let e λ : u → e λ,wt(u) u: U → U . Let V be a finite dimensional g-module and let v ∈ V [0]. Let {y i } be a basis of M µ consisting of weight vectors. Since Φ v µ : A difference equation for Ψ V (λ, µ) in the variable µ can be derived from (6.12). First multiply both sides of (6.12) with e λ,µ+ν , observe that η W (µ) • (id W ⊗ e λ ) = (e λ ⊗ e λ ) • η W (µ), sum both sides of (6.12) with respect to ν, and next multiply both sides of (6.12) on the left with (η W (µ) ⊗ id V ⊗ id V * ) −1 . Then we obtain the following identity of linear Now take the trace with respect to ⊕ ν (W [ν] ⊗ M µ+ν ) on both sides of (6.16) and use (6.13). Then µ+ν • e λ .
Now substitute (6.14) and (6.15), and take inside the sum on the right-hand side the transpose with respect to W . Then On the right-hand side of formula (6.17) substitute (6.9). Next also substitute R V U (λ) := R V U (−λ − ρ) and Q V (λ) := Q V (−λ − ρ). Then