Folded and contracted solutions of coupled classical dynamical Yang-Baxter and reflection equations

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space $\mathfrak{a}$ satisfying three coupled classical dynamical Yang-Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on $\mathfrak{a}$, generalising asymptotic boundary Knizhnik-Zamolodchikov-Bernard (KZB) equations. The recipe involves folding and contracting $\mathfrak{a}$-invariant and $\theta$-twisted symmetric classical dynamical $r$-matrices along an involutive automorphism $\theta$. In case of the universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ we determine the Etingof-Schiffmann classical dynamical $r$-matrices which are $\mathfrak{a}$-invariant and $\theta$-twisted symmetric. The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.


Introduction
Let U, A ℓ , A r be complex associative algebras, and a ⊆ U a finite dimensional subspace. In this paper we give a folding and contraction procedure which allows to construct functions r ± : a * → U ⊗ U and κ : a * → A ℓ ⊗ U ⊗ A r with (r + , r − ) satisfying three coupled classical dynamical Yang-Baxter equations CYB[t](r + , r − ) = 0 (1 ≤ t ≤ 3) and κ satisfying a classical dynamical reflection equation CR(r + , r − ; κ) = 0 relative to (r + , r − ). The explicit formulas for CYB [t] and CR are given by (3.2) and (3.3), respectively.
The core construction is as follows. Consider a pair (θ, r) with θ ∈ Aut(U ) an involution containing a in its (−1)-eigenspace, and r : a * → U ⊗ U an a-invariant θ-twisted symmetric classical dynamical r-matrix (θ-twisted symmetric means that r := (θ ⊗ id)r takes values in S 2 U ). Denote by m : U ⊗ U → U the multiplication map of U . Then folding and contracting r along θ produces a triple (1.1) (r + , r − , κ) := r + r 2 , r − r 2 , m( r) 2 with CYB[t](r + , r − ) = 0 (1 ≤ t ≤ 3) and CR(r + , r − ; κ) = 0. We also discuss extensions in which the folded solution κ of the classical dynamical reflection equation takes values in U θ ⊗ U ⊗ U θ , with U θ ⊆ U the subalgebra of elements in U fixed by θ.
Before describing the contents of the paper in more detail, we will first explain how these integrability equations and the folding and contraction procedure naturally arise in representation theory and in conformal field theory.
Triples (r + , r − , κ) satisfying CYB[t](r + , r − ) = 0 (1 ≤ t ≤ 3) and CR(r + , r − ; κ) = 0 serve as the local building blocks for consistent systems of first-order linear differential equations acting on functions a * → V , where V is a U ⊗N -module. Examples of such systems of equations are asymptotic boundary Knizhnik-Zamolodchikov-Bernard (KZB) equations, in which case we have (U, a) = (U (g), h) with U (g) the universal enveloping algebra of a complex semisimple Lie algebra g and h a Cartan subalgebra of g. These equations are satisfied by restrictions to the split Cartan subgroup of the N -point spherical functions, which form a special class of vector-valued spherical functions on split real connected semisimple Lie groups associated to g, see [28] and Subsection 2.2.
The terminology "asymptotic boundary KZB equations" is motivated by the fact that the equations are asymptotic remnants of the consistency equations for correlation functions in Wess-Zumino-Witten (WZW) conformal field theories on an elliptic curve with conformally invariant boundary conditions. This limit transition will be shortly discussed in Subsection 2.3. The boundary KZB equations themselves and the representation theoretic interpretation of N -point correlation functions in WZW conformal field theory on the elliptic curve with conformally invariant boundary conditions will be discussed in a forthcoming paper of N. Reshetikhin and the author.
As mentioned before, solutions of asymptotic KZB equations are restrictions to the split Cartan subgroup of a special class of spherical functions on split real connected semisimple Lie groups. The split real form of g determines a Chevalley involution σ ∈ Aut(g) satisfying σ| h = −id h , which extends to an involutive automorphism of U (g). It was already observed in [28] that the core triple (r + , r − , κ) underlying the asymptotic boundary KZB equations is obtained from Felder's [14] h-invariant and σ-twisted symmetric trigonometric classical dynamical r-matrix r : h * → g ⊗ g (see (2.17)) by folding and contracting along σ. Recall here that Felder's classical dynamical r-matrix itself forms the fundamental building block of the asymptotic KZB equations, which are asymptotic remnants of the consistency equations for correlation functions in WZW conformal field theory on an elliptic curve without boundaries, see [14,15,7,11]. Their solutions are given in terms of restrictions to Cartan subgroups of generalised trace functions on semisimple Lie groups, see [11].
So the context of spherical functions and asymptotic boundary KZB equations leads to a natural example of the general folding and contraction procedure that we develop in this paper. The other explicit examples we will obtain in this paper also concern U = U (g) with involution σ. In this case a will be a subspace of h and the classical dynamical r-matrices r : a * → g ⊗ g underlying the folding and contraction are from an appropriate subclass of Schiffmann's [24] family of classical dynamical r-matrices (see Definition 6.12).
For non-split real semisimple Lie groups a similar folding and contraction takes place in the theory of N -point spherical functions, but the integrability conditions for the building blocks of the associated asymptotic boundary KZB equations are more complicated, see [22] for details.
The contents of the paper is as follows. Section 2 provides a self-contained exposition explaining the role of Felder's trigonometric classical dynamical r-matrix, its folded and contracted versions, and the asymptotic (boundary) KZB equations in representation theory of semisimple Lie groups. We furthermore place the asymptotic (boundary) KZB equations in a degeneration scheme containing KZB type equations and Gaudin type Hamiltonians of types A N and C N .
In Section 3 the coupled classical dynamical Yang-Baxter equations and associated classical dynamical reflection equation are introduced. We explain how solutions of these equations give rise to commuting first order differential operators, thereby extending results from [28, §6.7].
In Section 4 we relate the usual classical dynamical Yang-Baxter equation for r + − r − to the coupled classical dynamical Yang-Baxter equations for the pair (r + , r − ). This involves a fourth classical non-dynamical Yang-Baxter type equation. We also show how extra terms can be added to the core solutions of the associated classical dynamical reflection equation. In case of the solution of the classical dynamical reflection equation appearing in the asymptotic boundary KZB equations, these extensions relate to adding non-trivial spin reflection terms at the boundaries, see [28, §6].
Section 5 establishes the folding and contraction procedure described in the second paragraph of the introduction. It is also shown that these folded pairs (r + , r − ) are solutions of a fourth coupled classical non-dynamical Yang-Baxter equation.
In Section 6 we consider the cases where U = U (g) is the universal enveloping algebra of a complex simple Lie algebra g and a is a subspace of a fixed Cartan subalgebra h of g. Etingof and Schiffmann [11,24] classified the associated quasi-unitary a-invariant solutions of the classical dynamical Yang-Baxter equation with values in g ⊗ g and coupling constant 1. Schiffmann's [24] explicit representatives r Sch : a * → g ⊗ g of the gauge equivalence classes of this set of solutions are labeled by a-admissible generalised Belavin-Drinfeld triples. Extremal cases are Felder's [14] solution, and the half-Casimirs. The latter ones are the standard non-dynamical solutions of the classical dynamical Yang-Baxter equation.
We show that if θ ∈ Aut(g) is an h-stabilising involution for which there exists a θ-twisted symmetric solution r Sch : a * → U (g) ⊗2 , then θ is a Chevalley involution relative to h. In this case we describe all θ-twisted symmetric solutions r Sch . By the folding and contraction procedure from Section 5, this provides new families of solutions of the coupled classical dynamical Yang-Baxter equations and of the associated classical dynamical reflection equation depending on a choice Γ of the set ∆ of simple roots and on a suitable class of subspaces a ⊆ h. The example related to N -point spherical functions on a split real simple Lie group corresponds to (Γ, a) = (∆, h).
Finally, in Section 7, we consider the twisted symmetric solution r Sch for a = 0 and use the folding and contraction procedure, together with the results from Section 3, to construct commuting asymptotic Gaudin Hamiltonians of type C N .
Conventions: For a group L and a subset S ⊂ L we write N L (S) and Z L (S) for the normaliser and centraliser of S in L, respectively. The set of conjugacy classes in L is denoted by Conj L . All modules and representations are over C unless stated otherwise. Tensor products over C are denoted by ⊗. The universal enveloping algebra of a complex Lie algebra g is denoted by U (g).
Acknowledgements: It is a pleasure to thank Nicolai Reshetikhin and Pavel Etingof for interesting discussions and comments. The author was supported by the Netherlands Organization for Scientific Research (NWO).

Representation theoretic context
2.1. Generalised trace functions and asymptotic KZB equations. In this subsection we revisit the Etingof-Kirillov-Schiffman theory on generalised trace functions for semisimple Lie groups (see, e.g, [6,7,9,20,11,10]). The only new result in this subsection is a formula for the Schrödinger operator of the spin Calogero-Moser system in terms of the radial component action of Felder's [14] trigonometric classical dynamical r-matrix (Proposition 2.10). The subsection also includes a short new derivation of the asymptotic Knizhnik-Zamolodchikov-Bernard (KZB) equations for generalised trace functions clearly separating the arguments involving the "bulk" (intertwiners) from the arguments involving the "twisted cyclic boundary conditions" (weighted traces), see Theorem 2.15.
Let g be a complex semisimple Lie algebra, h ⊂ g a Cartan subalgebra, and the corresponding root space decomposition, with root system R ⊂ h * . Let σ ∈ Aut(g) be a Chevalley involution relative to the Cartan subalgebra h (so σ| h = −id h ). Let K : g × g → C be the Killing form of g, and denote by (·, ·) the restriction of K to a nondegenerate symmetric bilinear form on h. We also write (·, ·) for the bilinear form on h * obtained by dualising (·, ·). Let {x j } n j=1 be a basis of h such that (x i , x j ) = δ i,j (it exists since (·, ·) is the complex bilinear extension of a scalar product on a suitable real form of h). For ν ∈ h * denote by t ν ∈ h the element such that ν(·) = (t ν , ·). Choose root vectors e α ∈ g α such that [e α , e −α ] = t α and σ(e α ) = −e −α (see, e.g., [19, §25.2]). Let R + = {β 1 , . . . , β m } be a choice of positive roots. Write n := α∈R + g α for the corresponding nilpotent subalgebra of g. Fix a complex connected Lie group G with Lie algebra g, and denote by T ⊂ G the connected analytic subgroup with Lie algebra h. Let Λ be the group of linear functionals µ ∈ h * that integrate to an algebraic character ξ µ : T → C × . Then Λ is a lattice satisfying Q ⊆ Λ ⊆ P , with Q and P the root and weight lattice of R, respectively. The algebra of regular functions on T is P Λ = µ∈Λ Cξ µ . The multiplication rules are ξ µ ξ ν = ξ µ+ν and ξ 0 = 1. We write P for the subalgebra spanned by ξ µ (µ ∈ Q), and Q for its quotient field.
Let U Q (g) be the Q-algebra Q⊗ U (g), and denote by D the algebra of differential operators on T with coefficients in the ring U Q (g). We write ∂ h ∈ D for the directional derivative in direction h ∈ h.
Lemma 2.1. There exists a unique complex linear map Proof. The existence and uniqueness of the map (2.1) follows easily by induction to the degree of X ∈ U (g) with respect to the standard filtration on U (g).
Example 2.2. (i) X → L X assigns to X ∈ U (h) the associated constant coefficient differential operator on T .
(ii) A direct computation using (1)-(3) gives for α ∈ R, In the following proposition we will relate the differential operator L X with the action X · f of X ∈ U (g) as left-invariant differential operator on a special class of vector-valued smooth vector-valued functions f on G.
Let (V, τ ) be a finite dimensional G-representation. Endow V with the left U (g)-module structure obtained by differentiating the G-action on V . The corresponding representation map U (g) → End(V ) is again denoted by τ . The space Ξ V (G) of V -valued class functions on G is defined to be the space of smooth functions f : G → V satisfying the equivariance property Proof. It suffices to show that for α ∈ R, X ∈ U (g), f ∈ Ξ V (G) and t ∈ T . Formula (2.4) follows from the following explicit computation, Here we used f (g ′ g) = τ (g −1 )f (gg ′ ) (g, g ′ ∈ G) for the first equality, cf. (2.2).
Remark 2.4. The interpretation of L X as the radial component of the action of X ∈ U (g) on Ξ V (G) as left-invariant differential operator extends to V -valued class functions on (the regular part of) a split real form G R of G, cf. [4]. It implies that the map (2.1) is canonical and independent of the choices.
The space Ξ V (G) of V -valued class functions is preserved by the action of the center Z(g) of U (g). Common Z(g)-eigenfunctions in Ξ V (G) can be constructed as follows.
Let (π, M ) be a finite dimensional G-representation. We will denote by X M the corresponding infinitesimal action of X ∈ U (g) on M , and write M [µ] for the weight space of M at weight µ ∈ Λ. Let V be a finite dimensional complex vector space. The partial trace over M [µ] of a complex linear map Ψ ∈ Hom(M, M ⊗ V ) is defined by The V -valued trace functions f Φ ∈ Ξ V (G) (Φ ∈ Hom G (M, M ⊗ V )) of Etingof, Kirillov Jr. and Schiffmann [6,9,20,11] are defined by (2.5) f Φ (g) := Tr M (Φπ(g)) (g ∈ G).
Note that its restriction to T is given by If V = C is the trivial representation then f Id M = χ M is the character of (π, M ), and  L The weight decomposition of U (g), viewed as G-representation with respect to the adjoint action, is Denote by D µ ⊂ D the subspace of differential operators with coefficients in From the definition of L X it immediately follows that L X ∈ D µ if X ∈ U [µ]. In particular, L X ∈ D 0 for X ∈ Z(g). The Weyl group W := N G (T )/T acts naturally on T , P Λ , Q, U [0] and D 0 . The fact that the intersection of a regular conjugacy class in Proposition 2.5 and its extension to V -valued class functions on the regular part of G R (see Remark 2.4) then implies that Theorem 2.6. The map X → L X (see (2.1)) restricts to an algebra embedding Z(g) ֒→ D W 0 . A special role for applications in harmonic analysis, quantum integrable systems and asymptotic integrable quantum field theories is played by the radial component L Ω ∈ D W 0 of the quadratic Casimir element x 2 j + α∈R e −α e α ∈ Z(g).
The differential operator L Ω can be computed by first expressing Ω in "normally ordered form", L Ω = ∆ + with Laplacian ∆ := n j=1 ∂ 2 xj . Remark 2.8. As a special case of Proposition 2.5 one obtains the explicit second order differential equation [16] algebraic proof of Weyl's [29,30] character formula for χ M | T , the second order differential equation (2.10) appeared after direct computations. Springer [27] realised from Harish-Chandra's [17,18] analytic theory on distributional characters of real reductive groups, which appeared around the same time as Freudenthal's paper [16], that (2.10) is a consequence of the action of Ω as a biinvariant differential operator on class functions. This insight allowed Springer [27] to derive an analogue of Weyl's character formula for connected semisimple groups over algebraically closed fields of sufficiently large characteristic.
Remark 2.9. The radial component method of Harish-Chandra and Casselman-Miličić [4] provides an explicit procedure to compute the radial component of the action of left-invariant differential operators on vector-valued spherical functions. The map X → L X can be recovered by applying the radial component method to a particular space of vector-valued spherical functions associated to the symmetric pair (G × G, diag(G)), with diag(G) the group G diagonally embedded into G × G. Concretely, the relevant space of vector-valued spherical functions consists of the smooth functions f : By the right-sided trivial G-equivariance, this space of spherical functions is isomorphic to the space Ξ V (G) of generalised trace functions through the map f → f , f (g) := f (g, 1) (1 the neutral element of G).
The first order term of L Ω can be removed by gauging with the Weyl denominator Concretely, The removal of the first order term of L Ω can also be achieved in the following manner. Let m : U Q (g) ⊗ Q U Q (g) → U Q (g) be the Q-bilinear extension of the multiplication map of U (g). Extend (2.1) to a Q-linear map U Q (g) → D by f ⊗ X → f L X for f ∈ Q and X ∈ U (g).
Proof. In view of (2.9) it suffices to show that (2.14) Ω = 2(y − m(r)) with (2.15) y := 1 2 i.e., it suffices to show the identity . This is verified by a direct computation.
The action of L m(r) on C ∞ (T ) ⊗ V [0] W will be denoted by L V m(r) . It is the Schrödinger operator of the quantum spin Calogero-Moser system. For classical Lie algebras g and special representations V this quantum system describes one-dimensional quantum spin particles, with V [0] the combined internal spin spaces. Higher order quantum integrals are obtained by gauged radial components L V Z of higher Casimir elements Z ∈ Z(g). This turns it into a quantum integrable system (see, e.g., [10,Chpt. 7]).
Remark 2.11. The map h * → g ⊗ g, defined by is Felder's [14] trigonometric solution of the classical dynamical Yang-Baxter equation. We abuse notation and simply write r(λ) for r(exp(t λ )) in the remainder of the paper. Felder's r-matrix r is the classical limit of the dynamical universal R-matrix R of the quantised universal enveloping algebra U q (g) (see the proof of [10, Prop. 7.10]). The quantum dynamical Yang-Baxter equation satisfied by R reduces to the classical dynamical Yang-Baxter equation for r. The classical dynamical r-matrix r satisfies the quasi-unitarity condition where r 21 is obtained from r by interchanging the two tensor components. The element ω ∈ (S 2 g) G is called the mixed Casimir element, and satisfies m(ω) = Ω. For the derivation of the asymptotic KZB equations we also need a twisted analogue of (2.18). The twisting is with respect to the following formal analogue of the action t → Ad(t −1 ) (t ∈ T ). Definition 2.13. We write η ∈ Aut(U Q (g)) for the Q-linear automorphism satisfying for all µ ∈ Q.
The twisted analogue of (2.18) is Lemma 2.14. We have Proof. This follows by a direct computation.
We now turn to asymptotic analogues of KZB equations. Up to this moment we looked at f Φ | T as the restriction to the Cartan subgroup T of an (elementary) spherical function on G. Instead we will now think of Φ ∈ Hom G (M, M ⊗ V ) as an asymptotic vertex operator, and view the operation as imposing twisted cyclic boundary conditions. Of special interest is the case that Φ is the composition of N intertwiners. We then have Then Etingof and Schiffmann [11,Thm. 3.1] determined the following system of first order linear differential equations for f Φ | T .
for i = 1, . . . , N , with the sub-labels indicating on which tensor components of V the elements x j , r and y are acting.
Proof. We give a proof that separates the "bulk" computations, only involving the product Φ of N intertwiners Φ j , from the computations involving the "boundary condition" (2.20).
with S the anti-automorphism of g such that S| g = −id g . Extending scalars to Q, the decomposition (2.14) of Ω and Ω Mj = c Mj (Ω)id Mj give Consider the composition (2.21) with Φ i replaced by (2.23), and push the first leg of r in rΦ i (respectively the second leg of r in Φ i * r) to the far left (respectively far right), we get the identity As a consequence we have We call (2.22) the asymptotic Knizhnik-Zamolodchikov-Bernard (KZB) equations (see Subsection 2.3 for a justification of the terminology). The consistency of the asymptotic KZB equations follows either directly from the classical dynamical Yang-Baxter equation satisfied by r (see Section 3), or from an argument similar to the one leading to Theorem 2.6.

2.2.
Spherical functions and boundary asymptotic KZB equations. In this subsection we will discuss some of the recent results in [28, §6]. We focus here on the emergence of folding and contraction and on the parallels with the theory of generalised trace functions, as discussed in the previous subsection.
Consider the decomposition Denote by K ⊂ G the analytic subgroup with Lie algebra k.
To derive a spherical/boundary analogue of the asymptotic KZB equations, we need to split off an appropriate symmetric factor from r to serve the altered boundary conditions.
The bulk computations in the proof of Theorem 2.15 depend on a particular choice of splitting of the action of the quadratic Casimir, see (2.23). The following lemma analyses the flexibility in the choice of splitting. Then Proof. Start with the right hand side of (2.27) and remove the action on V using the intertwining property of Φ. The right hand side of (2.27) then becomes The proof now immediately follows.
Remark 2.17. (i) There is a natural version of the lemma with the ground field extended to Q, which will be used in the proof of Theorem 2.21.
The special case s ′ = 0 was used in the proof of Theorem 2.15 with s = r.
In the present boundary case we apply Lemma 2.16 to s = r with a suitably chosen nonzero correction term s ′ . The left hand side of (2.27) then still admits an expression in terms of the action of the quadratic Casimir up to first order correction terms, cf. (2.14).
To have boundary conditions that are compatible with the Chevalley involution σ we choose In this case we denote s + and s − by r + and r − . They are explicitly given by (2.28) Note that (r 21 ) σ2 ∈ Q ⊗ S 2 g, so (r 21 ) σ2 = r σ1 and r ± = 1 2 r σ1 ± r . Note that the restricted root system for the symmetric pair (G, K) is 2R, since the underlying involution is the Chevalley involution σ. To compensate for the factor 2 we define for s ∈ Q⊗g ⊗2 , viewed as rational function on T with values in g ⊗2 , the element s ∈ Q ⊗ g ⊗2 by When removing boundary terms in the present context, the role of Lemma 2.14 is replaced by the following Then and hence Proof. This follows by a direct computation.
Remark 2.19. It follows from (2.30) that with k Q := Q ⊗ k and p Q := Q ⊗ p. So it lies in the two anti-diagonal blocks when splitting the tensor product space The Q-linear map (2.1) is now replaced by a Q-linear map U Q (g) → D ′ , X → D X with D ′ the algebra of differential operators on T with coefficients in U Q (k ⊕ k), and D X (X ∈ U (g)) the radial component of the action of X as left-invariant differential operator on spherical functions on the split real semisimple Lie group G R (see [4] and [22,Thm. 3.4]). Then the analogue of Proposition 2.10 in the present context is with y α := e α − e −α ∈ k, cf. [28]. See [28] for a further analysis of the associated quantum spin Calogero-Moser system.
We are now prepared to consider the analogue of Theorem 2.15 for special classes of spherical functions on G. We keep the notations and the setup of the previous subsection. In particular we cf. [28, §6]. It is the natural analogue of the V -valued trace function f Φ (see (2.5)) for the symmetric pair (G, K), satisfying trivial transformation behaviour with respect to the right-regular K-action while the left-regular K-action transforms according to V . In [21] Oblomkov related special cases of 1-point spherical functions on complex Grassmannians to BC n -type Heckman-Opdam polynomials.
The restriction f φ0,mN to T amounts to applying the boundary operation We can now formulate the asymptotic boundary KZB equations (it is a special case of [28, Prop. 6.13]).
Proof. In [28, §6] two proofs are given. The first proof is based on the radial component map for (G, K), mentioned in Remark 2.20. The second proof follows the proof of Theorem 2.15 in the bulk and then uses the K-fixed vectors φ 0 and m N to reflect the boundary terms and merge them back together (see the proof of [28, Prop. 6.13]). We give here a third proof which is based on Lemma 2.16 and Lemma 2.18. Take s = r in Lemma 2.16. For s ′ ∈ Q ⊗ g ⊗2 set s + := 1 2 (s ′ 21 + r) and s − := 1 2 (s ′ − r) to obtain the operator KZB-type equation (2.27). As in the first part of the proof of Theorem 2.15 we then get Now take s ′ := ( r 21 ) σ2 = r σ1 ∈ Q ⊗ S 2 g, so that s ± = r ± , and impose the boundary condition (2.32). Since r + ∈ Q ⊗ k ⊗ g and φ 0 is K-fixed the first of the two boundary terms vanishes, For the second boundary term, note first that The latter space is {0} due to the infinitesimal Cartan decomposition The infinitesimal Cartan decomposition plays an important role in computing the radial components of G-invariant differential operators, see [4,28,22].
In [28] sufficiently many solutions of the asymptotic boundary KZB equations (2.33) were constructed to conclude consistency. The consistency translates in four explicit equations for its local factors (r + , r − , m(r σ1 )/2), viewed as functions on h * (cf. Remark 2.11). Three equations only involve (r + , r − ) and are coupled versions of the classical dynamical Yang-Baxter equations. The fourth equation is a classical dynamical reflection type equation for m(r σ1 )/2 relative to (r + , r − ) (see [28, §6] for details).
One of the goals of this paper is to obtain these four coupled classical dynamical Yang-Baxter and reflection equations by direct algebraic manipulations from the classical dynamical Yang-Baxter equation satisfied by r, and to reveal the basic underlying algebraic principles.
Here {λ j } n j=1 is the linear basis of h * dual to {x j } n j=1 , ∂ λj is the directional derivative in direction λ j ∈ h * , h ∨ is the dual Coxeter number, c ∈ C the level, τ lies in the upper half plane (fixing an elliptic curve C × /p Z with p := e 2πiτ ), z = (z 1 , . . . , z N ) ∈ (C × ) N are the spectral parameters, ∂/∂z i the associated partial derivatives, and A [14] classical elliptic dynamical r-matrix r ell : h * → g⊗g with spectral parameter (we write r ell explicitly down in the following paragraph). The explicit expressions for A (N ) i (·|τ ) in case of the KZB equations are given in [14]. For the boundary KZB equations the building blocks of A (N ) i (·|τ ) are folded and contracted versions of r ell , with the folding and contraction relative to the Chevalley involution of the affine Lie algebra. This will be discussed in a forthcoming paper by N. Reshetikhin and the author. The spectral parameters arise representation theoretically from the realisation of the spin space V as a N -fold tensor product V 1 (z 1 ) ⊗ · · · ⊗ V N (z N ) of evaluation representations of the affine Lie algebra. Note that under the identification h ≃ h * via the Killing form, ∂ λj corresponds to ∂ xj .
The limit from trigonometric (boundary) KZB operators to trigonometric (boundary) KZB operators is ℑ(τ ) → ∞. The role of r ell (z; λ|τ ) is then taken over by the trigonometric classical dynamical r-matrix The limit from trigonometric (boundary) KZB operators to (boundary) Gaudin Hamiltonians is as follows. Write h * R for the real subspace of h * generated by R. It is a real form of h * . In the asymptotic region of h * where the real part of λ tends to infinity in the Weyl chamber opposite to the fundamental Weyl chamber relative to R + (i.e., ℜ((α, λ)) → −∞ for α ∈ R + ), the r-matrix r trig (z; ·) reduces to the Belavin-Drinfeld [1,2] classical trigonometric r-matrix with spectral parameter, where Ω ± := 1 2 n j=1 x j ⊗ x j + α∈R ± e α ⊗ e −α are the half-Casimirs. In this asymptotic region the trigonometric (boundary) KZB operators reduce to a family of commuting Gaudin type Hamiltonians B (N ) j (z) (1 ≤ j ≤ N ) on V , which are of type C N in the boundary case. Gaudin Hamiltonians of this form appeared before in, e.g., [25,26].
Finally, asymptotic Gaudin Hamiltonians are obtained from Gaudin Hamiltonians by the limit |z i /z i+1 | → 0 and |z N | → 0, and from the asymptotic KZB operators when the real part of λ ∈ h * tends to infinity in the Weyl chamber opposite to the positive Weyl chamber. The corresponding limits of r-matrices are where λ → −∞ stands for ℜ(λ) ∈ h * R tending to infinity in the Weyl chamber opposite to the positive Weyl chamber. See Section 7 for a direct approach to asymptotic Gaudin type Hamiltonians of type C N .

Commuting first order differential operators
Let A ℓ , U, A r be three unital complex associative algebras. Let a ⊆ U be a finite dimensional linear subspace. Let M be the field of meromorphic functions on a * . We call elements in the algebraic tensor product M ⊗ U meromorphic U -valued functions on a * , and denote them as maps a * → U . Functions are assumed to be meromorphic unless stated explicitly otherwise.
Let N ∈ Z >0 . If {i 1 , . . . , i k } ⊆ {1, . . . , N } and z ∈ U ⊗k then we write z i1i2···i k ∈ U ⊗N for the element obtained from z by placing the j-th tensor component of z in the i j -th tensor component and placing ones everywhere else (N is implicit from context). For instance, if z = i a i ⊗ b i ∈ U ⊗2 then z 31 = i b i ⊗ 1 ⊗ a i . We use the same convention when placing elements from A ℓ ⊗U ⊗k ⊗A r in A ℓ ⊗U ⊗N ⊗A r , omitting sublabels for the A ℓ and A r -components unless confusion may arise (which may happen, for instance, when A ℓ = A r or A ℓ = U = A r ).
Denote by D N the algebra of differential operators on a * with coefficients in the ring of A ℓ ⊗ U ⊗N ⊗ A r -valued meromorphic functions on a * . We write ∂ λ ∈ D N for the directional derivative in direction λ ∈ a * . The gradient type first order differential terms occurring in the differential operators we define below, are defined as follows.
where {x j } d j=1 is a linear basis of a and {λ k } d k=1 is the corresponding dual basis of a * . We used here the conventions of the previous paragraph, so Note that E i is independent of the choice of basis {x j } d j=1 of a. We consider now a special class of first order differential operators in D N whose constant terms are sums of local tensor product contributions r ± : a * → U ⊗2 and κ : a * → A ℓ ⊗ U ⊗ A r : Definition 3.2. For meromorphic functions r ± : a * → U ⊗2 and κ : a * → A ℓ ⊗ U ⊗ A r define the first-order differential operator D r − is (we suppress the dependence on r ± and κ in the notation for D We suppress r ± , κ from the notations if no confusion can arise. Proof. This follows by a direct computation (compare with the proof of [22, Thm. 6.12]).
As an immediate consequence, we have Corollary 3.4. The following statements are equivalent:
The classical dynamical reflection equation relative to (r + , r − ) is a classical dynamical analog of the compatibility condition for factorised scattering of quantum particles on a half-line bouncing off the wall, known as the reflection equation [5]. We call a pair (r + , r − ) satisfying CYB[t](r + , r − ) = 0 (1 ≤ t ≤ 3) a classical dynamical r-matrix pair, and we call κ satisfying CR(r + , r − ; κ) = 0 a classical dynamical k-matrix relative to (r + , r − ).
We now look at the special case when r − = −r + , in which case the usual classical dynamical Yang-Baxter equation and classical dynamical reflection equation naturally appear. For a meromorphic function r : a * → U ⊗2 define YB(r) : a * → U ⊗3 by If in addition κ : a * → A ℓ ⊗ U ⊗ A r is a meromorphic function, then define the function R(r; κ) : as meromorphic functions a * → U ⊗3 , and Proof. This follows by a direct computation.
Define first order differential operators L Corollary 3.6. The following statements are equivalent.
A meromorphic solution r : a * → U of the classical dynamical Yang-Baxter equation is called a classical dynamical r-matrix. For (U, a) = (U (g), h) as in Section 2, r : h * → g ⊗ g given by (2.17) is an example of an h-invariant classical dynamical r-matrix. A meromorphic solution κ : a * → A ℓ ⊗U ⊗A r of the classical dynamical reflection equation R(r; κ) = 0 is called a classical dynamical k-matrix relative to r.  (1 ≤ i ≤ N ). Note that indeed y is a classical dynamical k-matrix relative to r since r is h-invariant and with e α : h * → C the exponential map λ → e (α,λ) .
4. Some properties of CYB[t](r + , r − ) and CR(r + , r − ; κ) Let r ± : a * → U ⊗2 be two meromorphic U ⊗2 -valued functions. Consider   Proof. Write r := r + − r − . The differential contribution to the right hand side of (4.2) is which is the differential contribution to YB(r). Hence it remains to show that  The function CYB(r + , r − ) will naturally reappear in Section 5 when constructing classical dynamical r-matrix pairs by folding classical dynamical r-matrices along an involution.
A classical dynamical k-matrix relative to a classical dynamical r-matrix pair (r + , r − ) is said to be a core classical dynamical k-matrix if the boundary algebras A ℓ and A r are trivial, A ℓ = C = A r . We now present a method to promote core classical classical dynamical k-matrices to classical dynamical k-matrices involving nontrivial boundary algebras A ℓ and A r .
(1) This follows from the fact that (2) This is a direct check, using that [κ 1 , κ core The following result relates resCR to CYB. In this case A ℓ and A r are both equal to U .
With the same notational conventions as in Lemma 4.4, we have the following consequence.

Folding along an involution
We now construct classical dynamical r-matrix pairs and associated classical dynamical kmatrices from a given solution r of the classical dynamical Yang-Baxter equation. We use for this folding and contraction along an involution. It is likely that this folding and contraction procedure extend in a natural way to classical dynamical r-matrices with spectral parameter, but we do not pursue this here.
Fix an involutive unital algebra automorphism θ ∈ Aut(U ) (if U = U (g) is the universal enveloping algebra of a Lie algebra g then one typically takes the extension to U (g) of an involutive automorphism of g). Write U = U + ⊕ U − for the decomposition of U in (+1)-and (−1)-eigenspaces of θ. Then U + is a unital subalgebra, U − is an U + -bimodule and U − · U − ⊆ U + . We assume in this section that a ⊆ U − .
Definition 5.1. We call r θ-twisted symmetric if r is symmetric, i.e., if r 21 = r. We say that r is a-invariant if r takes values in the subspace Remark 5.2. (1) Note that (θ ⊗ θ)r = r 21 if r is θ-twisted symmetric.
(2) Let L(U ) be the algebra U viewed as Lie algebra, with Lie bracket the commutator bracket. Let a be the Lie subalgebra of L(U ) generated by a. Then (U ⊗ U ) a = (U ⊗ U ) a , and (U ⊗ U ) a is the subspace of elements in U ⊗ U that are invariant for the diagonal adjoint a-action on U ⊗ U .
We now consider meromorphic functions r ± : a * → U ⊗2 that are of the form for some meromorphic r : a * → U ⊗2 . Note that r ± takes values in U ± ⊗ U . Furthermore, Proof. We give the proof of the first equation (5.4) CYB [1](r + , r − ) = 1 4 YB(r) + θ 1 (YB(r)) + θ 2 (YB(r) 213 ) − θ 3 (YB(r) 312 ) , the others are proved by a similar computation. Let us first establish that the differential contributions on both sides of (5.4) match. Since a ⊆ U − and r 21 = r, the differential contribution to the right hand side of (5.4) is Collecting terms and using that r ± = 1 2 ( r ± r) this expression reduces to E 3 (r + 12 ) − E 2 (r + 13 ), which is the differential contribution to CYB [1](r + , r − ).
To complete the proof of (5.4) it remains to show that where we use the notation (4.3). Substituting r ± = 1 2 ( r ± r) into CYB 0 [1](r + , r − ) we get In this expression we rewrite the commutators involving r in terms of commutators that only involve r, for which we sometimes need to use the θ-twisted symmetry r = r 21 of r. We get where the last equality follows by a direct computation. This completes the proof of (5.5), and hence of (5.4).
We next construct an accompanying core classical dynamical k-matrix by a θ-twisted contraction procedure. Denote by m : U ⊗ U → U the multiplication map of U .
Proposition 5.5. Suppose that a ⊆ U − . Let r : a * → U ⊗ U be θ-twisted symmetric and a-invariant, and write r ± := 1 2 ( r ± r) : a * → U ± ⊗ U . Then Proof. Formula (5.6) follows from the a-invariance of r and the fact that a ⊆ U − . Write κ core := m( r)/2. We first check that the differential contributions to the left and right hand side of (5.7) match. The differential contribution to Reordering the terms and using that r = r + + r − it can be written as Applying m ⊗ m we obtain the expression By (5.6) this reduces to E 2 (κ core , which is the differential contribution to the left hand side of (5.7). Hence it remains to show that Since κ core = 1 2 m(r + + r − ) the left hand side of (5.8) is Now for s, s ′ ∈ U ⊗2 one verifies by direct computations that Proof. We have to show that CR(r + , r − ; κ core ) = 0.
This follows immediately from Proposition 5.5 and Theorem 5.4.
Then κ : a * → U + ⊗ U ⊗ U + , with κ one of the following three meromorphic functions Proof. We have to show that CR(r + , r − ; κ) = 0 for κ either one of the three meromorphic functions listed in the corollary. This follows from Theorem 5.6 and Corollary 4.5(1).
Remark 5.9. For the example discussed in Remark 5.7, the extensions of the core classical dynamical k-matrix provided by Corollary 5.8 correspond representation theoretically to adding spin boundary behaviour to the N -point spherical functions, see [28, §6]. For this particular example additional extensions of the associated core classical dynamical k-matrix can be made, see Proposition 6.16.
Remark 5.10. Assume that r : a * → U ⊗2 is a θ-twisted symmetric and a-invariant classical dynamical r-matrix. Write r ± := 1 2 ( r ± r) : a * → U ± ⊗ U and suppose that κ : a * → A ℓ ⊗ U ⊗ A r is a classical dynamical k-matrix with relative to (r + , r − ) . The resulting commuting first order differential operators {D Definition 3.2) can then be expressed as If r is quasi-unitary with coupling constant t, i.e., (5.12) r + r 21 = t̟, This applies in particular to the asymptotic boundary KZB operators, when r is Felder's classical dynamical r-matrix (2.17), which is quasi-unitary with coupling constant −1 (see (2.18)). Note that if r is skew-symmetric, then 6. Examples 6.1. Schiffmann's classical dynamical r-matrices. We freely use the notations from Section 2. We assume though g to be simple, and we will allow different normalisations for the root vectors e α .
Recall that h ⊂ g is a fixed Cartan subalgebra. The pair (g, h) is the complexification of a pair (g R , h R ) with g R a split real simple Lie algebra and Cartan subalgebra h R = span R {t β | β ∈ R}. The restriction (·, ·)| h R ×h R is positive definite. We write g R,α := g R ∩ g α for α ∈ R.
We call a complex subspace a ⊆ h of real type if it has a real form contained in h R . We write t for its orthogonal complement in h with respect to (·, ·). Note that if a is of real type, then t is also of real type and h = a ⊕ t. For subspaces a ⊆ h of real type we embed a * ֒→ h * by extending λ ∈ a * to a linear functional on h by λ(x + y) := λ(x) for x ∈ a and y ∈ t.
Let ∆ be a basis of R. Write R ± for the corresponding set of positive and negative roots, and b ± for the associated positive and negative Borel sub-algebras of g. For a subset Γ ⊆ ∆ of simple roots write R Γ := ZΓ ∩ R, which is a root subsystem with basis Γ. We write R ± Γ for the corresponding positive and negative roots in R Γ . Let g Γ ⊆ g be the (semisimple) Lie subalgebra generated by g α (α ∈ Γ).
We now recall the classical dynamical r-matrices r : a * → g ⊗ g for subspaces a ⊆ h of real type constructed in [24,11] (to make contact with the setup in the earlier sections, one views r as function r : a * → U (g) ⊗ U (g) using the canonical embedding of g into U (g)). We follow here [12, App. A], which is slightly more general than the setup in [24].
For α ∈ R Γ1 define ϕ α : a * → g by This formula should be interpreted as follows. If α ∈ R \ R Γ1,τ then τ k (α) ∈ R \ R Γ1 for some k ∈ Z >0 , and hence τ j (e α ) = 0 for j ≥ k. In this case the sum (6.1) terminates and defines an analytic g-valued function on a * . If α ∈ R Γ1,τ then write k < ℓ for the strictly positive integers with k + ℓ as small as possible such that τ k (α) = τ ℓ (α). Then ϕ α (λ) should be read as which is a meromorphic g Γ1 -valued function on a * .
Lemma 6.3. Let a ⊆ h be a subspace of real type and (Γ 1 , Γ 2 , τ ) an a-admissible generalised Belavin-Drinfeld triple. Write S ⊆ t ⊗ t for the set of elements r t ∈ t ⊗ t such that
Note that r Sch does not depend on the particular choice of root vectors e α ∈ g α (α ∈ R) as long as [e α , e −α ] = t α for all α ∈ R. Furthermore, note that r Sch is quasi-unitary with coupling constant one, i.e., (6.6) r Sch + r Sch 21 = ̟, iff r t ∈ S ∩ ∧ 2 t.
Note that the a-invariance of r Sch is due to the fact that τ (α)| a = α| a for all α ∈ R Γ1 , see Definition 6.1 (1). Up to an appropriate notion of gauge equivalence, the r Sch with r t ∈ S ∩ ∧ 2 t are all the a-invariant quasi-unitary classical dynamical r-matrices with values in g ⊗2 having coupling constant 1, see [12, App. A].

6.2.
Coupled classical dynamical r-and k-matrices associated to r Sch . Let Aut(g, h) be the group of automorphisms of g stabilising the Cartan subalgebra h. Denote by Aut + (g, h) the normal subgroup consisting of automorphisms θ ∈ Aut(g, h) satisfying θ| h = id h . We denote the extension of an automorphism θ ∈ Aut(g) to an automorphism of U (g) by θ again. By the isomorphism theorem [19, Thm. 14.2] we have with P ∨ ⊂ h the co-weight lattice and Ad y ∈ Aut + (g, h) for y ∈ h characterised be Ad y | gα = e α(y) id gα for all α ∈ R. The isomorphism theorem also gives the group isomorphism (6.7) Aut(g, h)/Aut + (g, h) with t θ ∈ Gl(h * ) the transpose of θ| h ∈ Gl(h). Recall furthermore that with W the Weyl group of R and Ξ the group of Dynkin diagram automorphisms (extended to automorphisms of R in the natural way), see, e.g., [19, §12.2]. Write Inv(g, h) ⊆ Aut(g, h) for the subset of involutive automorphisms of g stabilising h. Note that Inv(g, h) is invariant under conjugation by Aut(g, h). For θ ∈ Inv(g, h) denote by g ± θ ⊆ g the (±1)-eigenspaces of θ. Then Writing h ± θ := g ± θ ∩h, we also have h = h + θ ⊕h − θ and φ(h ± θ ) = h ± φθφ −1 . A special class of involutions are the Chevalley involutions: Definition 6.6. An involution θ ∈ Inv(g, h) is called a Chevalley involution of g relative to h if h − θ = h. The set of Chevalley involutions relative to h will be denoted by Ch(g, h). The following lemma is well known (it is an easy consequence of the isomorphism theorem [19, Thm. 14.2]). Lemma 6.7. Ch(g, h) is the Aut + (g, h)-coset in Aut(g, h) corresponding to −1 ∈ Aut(R) under the isomorphism (6.7). Furthermore, Ch(g, h) is a single Aut + (g, h)-orbit for the conjugation action of Aut + (g, h) on Inv(g, h).
We consider the following special class of classical dynamical r-matrix pairs and related classical dynamical k-matrices. Definition 6.8. Let θ ∈ Inv(g, h), and fix a subspace a ⊆ h − θ . We denote by M(θ, a) the set of triples (r + , r − , κ) of meromorphic functions r ± : a * → g ± θ ⊗ g and κ : a * → U (g + θ ) ⊗ U (g) ⊗ U (g + θ ) satisfying the following two conditions: (1) (r + , r − ) is a classical dynamical r-matrix pair satisfying the h-compatibility condition (2) κ is a classical dynamical k-matrix relative to (r + , r − ).
We give here two natural classes of gauge transformations for triples (r + , r − , κ) in M(θ, a) (a more thorough analysis of gauge transformations and the corresponding classification problems, as in [11,24] for the ordinary classical dynamical Yang-Baxter equations, will not be pursued further in this paper). Lemma 6.9. Fix θ ∈ Inv(g, h) and a subspace a ⊆ h − θ . For (r + , r − , κ) ∈ M(θ, a) we have with t (φ| a ) : φ(a) * → a * the transpose of the linear isomorphism φ| a : a ∼ −→ φ(a). Proof.
(1) By a direct computation one verifies that for 1 ≤ t ≤ 3, from which the result immediately follows.
We now construct explicit solutions (r + , r − , κ) ∈ M(θ, a) by folding and contracting Schiffmann's [24] classical dynamical r-matrices r Sch . As we have seen in Section 5, the key additional requirement for this to work is that r Sch is θ-twisted symmetric.
Let Γ ⊆ ∆. Write V Γ for the orthocomplement of the subspace α∈Γ Ker(α) ⊆ h with respect to (·, ·). Note that V Γ ⊆ h is a subspace of real type. Theorem 6.10. Let θ ∈ Inv(g, h) and fix a subspace a ⊆ h of real type. Suppose that (Γ 1 , Γ 2 , τ ) is an a-admissible generalised Belavin-Drinfeld triple. Let r t ∈ S. Write r Sch : a * → (g⊗g) a for the associated classical dynamical r-matrix, see (6.5). The following two statements are equivalent: (1) r Sch is θ-twisted symmetric.
Remark 6.11. Theorem 6.10 shows that the folding and contraction procedure of Section 5 in case U = U (g) is naturally related to the split real form of g. This should be compared with [22], where it was shown that the asymptotic KZB equations for N -point spherical functions for real semisimple Lie groups G are governed by triples (r + , r − , κ) satisfying glued versions of the coupled classical dynamical Yang-Baxter and reflection equations, which only decouple when G is split. In the non-split case the triple (r + , r − , κ) is obtainable from a single r : a * → g ⊗ g by folding and contraction along the Cartan involution θ, but r does not satisfy the classical dynamical Yang-Baxter equation.
It is convenient to make the dependence of Schiffmann's classical dynamical r-matrix r Sch on the initial data explicit for the special class of generalised Belavin-Drinfeld triples arising in Theorem 6.10. Recall from Lemma 6.3 that S 2 t ⊆ S for a-admissible Belavin-Drinfeld triples of the form (Γ, Γ, id). Definition 6.12. Let (Γ, a, r t ) be a triple consisting of (1) a subset Γ ⊆ ∆, (2) a subspace a ⊆ h of real type containing V Γ , (3) an element r t ∈ S 2 t. We write (6.9) r (Γ,a,rt) (λ) = 1 2 for the a-invariant classical dynamical r-matrix r Sch : a * → (g⊗g) a associated to the a-admissible generalised Belavin-Drinfeld triple (Γ, Γ, id) and the element r t ∈ S 2 t.
Note that r (Γ,a,rt) + (r (Γ,a,rt) ) 21 = ̟ + 2r t , in particular r (Γ,a,rt) is quasi-unitary with coupling constant 1 iff r t = 0. Note furthermore that all terms in the sum over α ∈ R Γ in (6.9) depend nontrivially on λ ∈ a * since α| a = 0 for all α ∈ R Γ by the assumption that V Γ ⊆ a. Finally, by Theorem 6.10, the r (Γ,a,rt) form the subclass of classical dynamical r-matrices r Sch that are θ-twisted symmetric for all θ ∈ Ch(g, h). Note that for Γ = ∆ we are forced to take a = h, hence r t = 0. Then r (∆,h,0) : h * → (g ⊗ g) h is essentially Felder's classical dynamical r-matrix r, see Remark 6.5. Recall that r (Γ,a,rt) does not depend on the choice of e α ∈ g α (α ∈ R) such that [e α , e −α ] = t α for all α ∈ R.
We next explicitly describe the dependence of the classical dynamical r-matrix pairs and the associated classical dynamical k-matrices, obtained from r (Γ,a,rt) by folding and contracting, on the normalisation of the root vectors and on the Chevalley involution. For this it is convenient to choose the e α 's to be well behaved with respect to a fixed Chevalley involution: Convention 6.13. In the remainder of this section we fix a Chevalley involution σ ∈ Ch(g, h) and fix root vectors e α ∈ g α (α ∈ R) such that [e α , e −α ] = t α and σ(e α ) = −e −α for all α ∈ R (this is the same convention as used in Section 2). We denote the other Chevalley involutions in Aut(g, h) by σ y := σ • Ad y , y ∈ h three coupled classical dynamical Yang-Baxter equations and a fourth classical non-dynamical Yang-Baxter type equation.
In the degeneration scheme (2.37), Skrypnyk's folding theory takes place in the "Gaudin" box and the present folding theory in the "asymptotic KZB" box. It is an open problem whether these folding theories are degenerations of (a single) folding theory in the "KZB" box, where both the dynamical and spectral parameters are present.