Trigonometric dynamical r-matrices over Poisson Lie base^{∗}

A. Mudrov

Dedicated to the memory of Joseph Donin

Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel, Max-Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany. e-mail: mudrov@mpim-bonn.mpg.de

Abstract

Let 𝔤 be a finite dimensional complex Lie algebra and 𝔩 ⊂ 𝔤 a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group Exp𝔩 with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra 𝔇𝔩. Let N_{𝔩}(0) ⊂ 𝔩 be an open domain parameterizing a neighborhood of the identity in Exp𝔩 by the exponential map. We present dynamical r-matrices with values in 𝔤 ∧ 𝔤 over the Poisson Lie base manifold N_{𝔩}(0).

\underline{Key words}: Dynamical Yang-Baxter equation, Poisson Lie groups, Poisson Lie manifolds.

1 Introduction

The classical dynamical Yang-Baxter equation (CDYBE) is a differential equation which generalizes the algebraic or ordinary classical Yang-Baxter equation (CYBE) associated with an arbitrary Lie algebra 𝔤. Contrary to the CYBE, a solution of CDYBE is a function of the so called dynamical parameter. Up to recently, the space of parameters was taken to be the dual 𝔩^{∗} of a Lie subalgebra 𝔩 in 𝔤, [BDF, F, ES, EV]. In [DM1, DM2], the DYBE was formulated for any quadruple (𝔤, 𝔩, 𝔩^{∗}, L), where 𝔩 ⊂ 𝔤 is a pair of Lie algebras, (𝔩, 𝔩^{∗}) is a Lie bialgebra, and L is a special Poisson Lie (PL) manifold, the space of parameters.

∗This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center ”Group Theoretic Methods in the study of Algebraic Varieties” of the Israel Science foundation, and by the RFBR grant no. 03-01-00593.

A particular case of the PL version of the CDYBE for a quasitriangular Lie bialgebra 𝔩 = 𝔤 and L being the group space Exp𝔤 appeared in [BFP, FM] in connection with a factorization problem in the chiral WZW model. In the present paper we give examples of dynamical r-matrices for the quadruple (𝔤, 𝔩, 𝔩^{∗}, L), where 𝔤 is a complex Lie algebra and (𝔩, 𝔩^{∗}) a factorizable quasitriangular Lie bialgebra. As a base manifold L we take a domain N_{𝔩}(0) in 𝔩 parameterizing a neighborhood of the identity in the group Exp𝔩 equipped with the Semenov-Tjan-Shansky (STS) Poisson bracket, [S].

We apply the Etingof-Varchenko approach of base reduction, [EV], to the PL CDYBE. Then we employ, within the PL setting, the idea of Etingof-Schiffmann which is used for constructing generalized Alekseev-Meinrenken dynamical r-matrices in [ES]. In this way we obtain PL dynamical r-matrices that are in between the r-matrices of [BFP] and trigonometric r-matrices of [EV, S].

2 Dynamical Yang-Baxter equation on PL base manifold

Let (𝔩, 𝔩^{∗}) be a finite dimensional Lie bialgebra and let δ denote the cobracket 𝔩 → 𝔩∧𝔩, [Dr1]. Let 𝔇𝔩 = 𝔩 ▷◁ 𝔩_{op}^{∗} be the double Lie algebra with the canonical invariant symmetric tensor θ := ∑_{i}(η^{i} ⊗ ξ_{i} + ξ_{i} ⊗ η^{i}), where {ξ_{i}} ⊂ 𝔩 and {η^{i}} ⊂ 𝔩^{∗} are the dual bases. Recall from [DM1] that a 𝔇𝔩-manifold is called an 𝔩-base manifold if the Casimir element θ generates the zero bidifferential operator via the action of 𝔇𝔩. It follows that a base manifold is equipped with a 𝔇𝔩-PL structure induced by the r-matrix ∑_{i} η^{i} ⊗ ξ_{i} ∈ (𝔇𝔩)^{⊗2} via the action of 𝔇𝔩.

In the present paper, by a function on L we understand an analytical or meromorphic function. Let 𝔤 be a Lie algebra containing 𝔩 as a subalgebra. Let us call a function r: L → ∧^{2}𝔤 quasi-invariant if

ξ ▷ r(λ) + [ξ ⊗ 1 + 1 ⊗ ξ, r(λ)] = −δ(ξ), ξ ∈ 𝔩,  (1)

where δ is the cobracket on 𝔩 and ▷ denotes the 𝔩-action on functions on L by vector fields. If the Lie bialgebra 𝔩 is coboundary, i.e. δ(ξ) := [ξ ⊗ 1 + 1 ⊗ ξ, r_{𝔩}] for some r_{𝔩} ∈ ∧^{2}𝔩, then a function r: L → ∧^{2}𝔤 is quasi-invariant if and only if the function r′ := r + r_{𝔩} is invariant, i.e.

ξ ▷ r′(λ) + [ξ ⊗ 1 + 1 ⊗ ξ, r′(λ)] = 0, ξ ∈ 𝔩.  (2)

Definition 2.1. A quasi-invariant function r: L → ∧^{2}𝔤 is called a dynamical r-matrix over the base L if

∑_{i} Alt(ξ_{i} ⊗ ∇_{η^{i}}r(λ)) + CYB(r(λ)) = Z,  (3)

where ∇_{η} is the vector field on L generated by η ∈ 𝔩_{op}^{∗}, Z ⊂ (∧^{3}𝔤)^{𝔤} is an invariant element, and

CYB(A) := [A_{12}, A_{13}] + [A_{12}, A_{23}] + [A_{13}, A_{23}], A ∈ 𝔤 ⊗ 𝔤,
Alt(B) := B_{123} − B_{213} + B_{312}, B ∈ 𝔤 ⊗ 𝔤 ⊗ 𝔤.

Note that Definition 2.1 differs from the definition given in [DM1, DM2] by the change of sign r → −r. Sometimes we will use the term PL dynamical r-matrix in order to distinguish it from the particular case of abelian 𝔩^{∗}.

Remark 2.2. Suppose (𝔩, 𝔩^{∗}) is the trivial bialgebra, with zero δ (abelian 𝔩^{∗}). Then replacement Z by ϵ^{2}Z results in the transformation r(λ) → ϵr(ϵλ) of the dynamical r-matrix, the solution to (3).

Remark 2.3. A particular case of equation (3) for 𝔩 = 𝔤 and L = Exp𝔩 appeared in an equivalent form in the paper [BFP] devoted to the chiral WZW model. Motivated by equivariant deformation quantization on G-spaces, equation (3) was deduced in [DM1]. There, 𝔤 was assumed to be a Lie bialgebra containing 𝔩 as a sub-bialgebra. This assumption is relaxed in Definition 2.1, where 𝔤 is just a Lie algebra. The present definition of dynamical r-matrix was formulated in [DM2]. It was shown that such dynamical r-matrices define coboundary Lie bialgebroids of a special type.

We have the following Poisson characterization of dynamical r-matrices. Let G = Exp𝔤 be the connected simply connected Lie group corresponding to 𝔤. For ζ ∈ 𝔤 let ζ^{l} and ζ^{r} denote the left and right invariant vector fields

ζ^{l}f(g) = \frac{d}{dt}f(ge^{tζ})|_{0}, ζ^{r}f(g) = \frac{d}{dt}f(e^{tζ}g)|_{0}, f ∈ C^{∞}(G).

Consider the following two sided action of the Lie algebra 𝔩 on the Cartesian product L×G×L. The left action affects the first copy of L and the group space G (by the left invariant vector fields). The right action of 𝔩 is defined on the right copy of L via the principal anti-automorphism (antipode) and on G by the left invariant vector fields.

Let r: L → 𝔤 ∧ 𝔤 be an equivariant function. We define a bivector field π on L × G × L as follows. When restricted to the left copy of L, π coincides with the Poisson structure on L. When restricted to the right copy of L, π coincides with the opposite Poisson structure on L. Further, we set

π(f, a) := ∑_{i}(η^{i} ▷ f)(ξ_{i}^{l} + ξ_{i}^{r})(a),  (4)
π(a, b) := (−r^{l,l}(λ′) + r^{r,r}(λ′′))(a, b)  (5)

for f ∈ C^{∞}(L × L) and a, b ∈ C^{∞}(G); here (λ′, λ′′) ∈ L × L. The superscripts l and r denote the left- and right-invariant vector fields.

We consider the direct sum 𝔩 ⊕ 𝔩_{op} as a Lie bialgebra with the dual Lie algebra 𝔩^{∗} ⊕ 𝔩^{∗}.

Proposition 2.4. The bivector field π is an 𝔩 ⊕ 𝔩_{op}-Poisson Lie structure on L × G × L if and only if r is a dynamical r-matrix.

This assertion was stated in [DM1], Proposition 3.28, in a more restrictive setting, cf. Remark 2.3. A proof can be adopted from the proof of Theorem 9.6 from [DM1]. Note that we changed the sign of r here, as compared to [DM1, DM2]. Proposition 2.4 is analogous to Proposition 1.8 from [EEM]. Note that the definition of dynamical r-matrix from [DM1] (cf. Remark 2.3) goes over to the definition of [EEM] for the case L = Exp𝔩 if one subtracts from r its value at the identity of Exp𝔩.

3 Poisson Lie CDYBE on group manifold

Suppose that the Lie bialgebra 𝔩 is quasitriangular, i.e. coboundary with an r-matrix r_{𝔩} ∈ ∧^{2}𝔩 and an element Ω_{𝔩} ∈ (S^{2}𝔩)^{𝔩} such that CYB(r_{𝔩} + \frac{1}{2}Ω_{𝔩}) = 0. Consider the connected simply connected Lie group Exp𝔩 corresponding to the Lie bialgebra 𝔩 and put L = Exp𝔩. According to [S], there is a PL structure on L considered as an Exp𝔩 -manifold with respect to conjugation. In fact, L is an 𝔩-base manifold, [DM1]. The action of double Lie algebra 𝔇𝔩 on L is defined as follows. For ξ ∈ 𝔩 denote by ξ^{l} and ξ^{r} the vector fields acting by

ξ^{l}f(g) = \frac{d}{dt}f(ge^{tξ})|_{0}, ξ^{r}f(g) = \frac{d}{dt}f(e^{tξ}g)|_{0}, f ∈ C^{∞}(L).

Then 𝔇𝔩 acts on L by the vector fields

ξ^{l} − ξ^{r} for ξ ∈ 𝔩 and r_{𝔩}^{r}(η) − r_{𝔩}^{l}(η) + \frac{1}{2}(Ω_{𝔩}^{l}(η) + Ω_{𝔩}^{r}(η)) for η ∈ 𝔩_{op}^{∗}.  (6)

Here and further on the elements from 𝔩 ⊗ 𝔩 are considered as linear operators from 𝔩^{∗} to 𝔩 via pairing with the first tensor component.

From now on we assume the Lie bialgebra (𝔩, 𝔩^{∗}) to be factorizable. This means that the Casimir element Ω_{𝔩} defines an isomorphism 𝔩^{∗} → 𝔩 of 𝔩-modules. Let {ξ_{i}} ⊂ 𝔩 be an orthonormal base with respect to the form Ω_{𝔩}^{−1}. Then the following proposition holds.

Proposition 3.1. A quasi-invariant function r: L → ∧^{2}𝔤 is a dynamical r-matrix if and only if the invariant function r′ := r + r_{𝔩} satisfies the equation

∑_{i} Alt(ξ_{i} ⊗ ∇′_{ξ_{i}}r′(λ)) + CYB(r′(λ)) = Z − Z_{l},  (7)

where Z_{𝔩} := CYB(r_{𝔩}) and the vector field ∇′_{ξ} is defined to be ∇′_{ξ} := \frac{1}{2}(ξ^{l} + ξ^{r}) for ξ ∈ 𝔩.

Proof. The elements r′ and r_{𝔩} satisfy the equality

CYB(r′ − r_{𝔩}) = CYB(r′) + CYB(r_{𝔩}) − Alt([r_{𝔩}^{12} + r_{𝔩}^{13}, r′_{23}])
= CYB(r′) + CYB(r_{𝔩}) + Alt(ξ_{i} ⊗ r_{𝔩}(η^{i}) ▷ r′).  (8)

Here we used invariance of r′ with respect to the adjoint action of 𝔩. Now recall that the vector field r_{𝔩}(η^{i})▷ on L equals r_{𝔩}(η^{i})^{l} − r_{𝔩}(η^{i})^{r}. It remains to substitute (8) into (3) and take into account the specific form of the 𝔩_{op}^{∗}-vector fields on L given by (6).

Remark 3.2. Equation (7) is more general than (3) since it makes sense for any self-dual Lie algebra 𝔩, not necessarily factorizable quasitriangular. It was derived in [BFP, FM] for the case 𝔩 = 𝔤. A solution to this equation for 𝔩 = 𝔤 (in fact, unique, up to a gauge equivalence) was found in [BFP].

4 Trigonometric dynamical r-matrix over abelian base

Let 𝔤 be a complex semisimple Lie algebra and 𝔥 a Cartan subalgebra in 𝔤. Denote by Δ(𝔤) the root system of 𝔤 and by the Δ_{+}(𝔤) the subset of positive roots. Let {e_{α}}, α ∈ Δ(𝔤) be a base of root vectors from the root subspaces 𝔤_{α} normalized to (e_{α}, e_{−α}) = 1 for all α ∈ Δ_{+}(𝔤) with respect to a non-degenerate invariant inner product on 𝔤. Let Ω_{𝔤} denote the split-Casimir of this form,

Ω_{𝔤} := ∑_{i} x_{i} ⊗ x_{i} + ∑_{α∈Δ(𝔤)}e_{α} ⊗ e_{−α},

where {x_{i}} is an orthonormal base of 𝔥. Recall from [EV], Theorem 3.1, that the meromorphic function 𝔥 ∋ λ ↦ ρ(𝔤, ϵ, λ) ∈ ∧^{2}𝔤 defined by

ρ(𝔤, ϵ, λ) := ∑_{α∈Δ(𝔤)} \frac{ϵ}{2}coth(\frac{ϵ}{2}(α, λ))e_{α} ⊗ e_{−α}  (9)

is a solution to the DYBE with Z = Z_{Ω_{𝔤}}^{ϵ} := \frac{ϵ^{2}}{4} [Ω_{𝔤}^{12}, Ω_{𝔤}^{23}]. Here and further on we identify 𝔥 and 𝔥^{∗} via the restriction of the Killing form.

We may assume 𝔤 to be reductive and extend this solution to a meromorphic function on 𝔥 using projection along the center of 𝔤.

5 Base reduction in trigonometric case

It is shown in [EV] that the dynamical Yang-Baxter r-matrix admits a reduction of base when 𝔩 is a reductive Lie algebra with the trivial Lie bialgebra structure (abelian 𝔩^{∗}). We will show that an analogous statement also holds if 𝔩 is a factorizable Lie bialgebra.

Let 𝔩 be a reductive subalgebra of a complex Lie algebra 𝔤. Suppose that 𝔩 is equipped with a factorizable quasitriangular Lie bialgebra structure. Denote by r_{𝔩} ⊂ ∧^{2}𝔩 its classical r-matrix and by Ω_{𝔩} ∈ (S^{2}𝔩)^{𝔩} the corresponding Casimir element. Let N_{𝔩}(0) ⊂ 𝔩 be an open domain parameterizing a neighborhood of the identity in Exp𝔩 by the exponential map (we require the operator 1 − adξ to be non-degenerate for ξ ∈ N_{𝔩}(0)). We put the 𝔩-base manifold L to be N_{𝔩}(0) ⊂ 𝔩 equipped with the STS Poisson bracket. Let 𝔥_{𝔩} denote the Cartan subalgebra of 𝔩. The open domain N_{𝔥_{𝔩}}(0) := N_{𝔩}(0) ∩ 𝔥_{𝔩} ⊂ 𝔥_{𝔩} is a base manifold for 𝔥_{𝔩}. Theorem 5.1. A quasi-invariant function r: N_{𝔩}(0) → ∧^{2}𝔤 is a dynamical r-matrix over N_{𝔩}(0) if and only if the function r̃(λ) := r|_{𝔥_{𝔩}}(λ)+ρ(λ)+r_{𝔩}, where ρ(λ) = ρ(𝔤, 1, λ), λ ∈ N_{𝔥_{𝔩}}(0), is a dynamical r-matrix over the abelian base 𝔥_{𝔩}. Proof. Equation (3) on the quasi-invariant function r is equivalent to equation (7) on the invariant function r′ := r+r_{𝔩}. We assume in (7) that the orthonormal base {ξ_{k}} is compatible with the root decomposition. Equation (7) gives rise to the following equation for the restriction of r′ to 𝔥_{𝔩}:

∑_{i=1}^{rk 𝔩} Alt(x_{i} ⊗ ∇′_{x_{i}}r′(λ)) + ∑_{α∈Δ(𝔩)}Alt(e_{α} ⊗ ∇′_{e_{−α}}r′(λ)) + CYB(r′(λ)) = Z_{𝔤} − CYB(r_{𝔩}) (10)

where λ ∈ 𝔥_{𝔩}. For all ξ ∈ 𝔩 the vector fields ∇′_{ζ}, on N_{𝔩}(0) read

∇′_{ξ}f(λ) = ∂f(λ)(\frac{1}{2}adλ)coth(\frac{1}{2}adλ)ξ,

where ∂f denotes the differential of a function f. On the other hand, ξ▷f(λ) = ∂f(λ)(adλ)ξ. This implies, by 𝔩-invariance of r′,

e_{α} ⊗ ∇′_{e_{−α}}r′(λ) = \frac{1}{2}coth(\frac{1}{2}(α, λ))e_{α} ⊗ [e_{−α} ⊗ 1 + 1 ⊗ e_{−α}, r′(λ)], α ∈ Δ(𝔩), λ ∈ 𝔥_{𝔩}.

Observe that ∇′_{x_{i}} = ∂_{i} := \frac{∂}{∂x_{i}} at λ ∈ 𝔥_{𝔩}. So equation (10) can be rewritten as

∑_{i=1}^{rk 𝔩} Alt(x_{i} ⊗ ∂_{i}r′(λ)) + Alt([ρ_{12} + ρ_{13}, r′_{23}]) + CYB(r′(λ)) = Z_{𝔤} − CYB(r_{𝔩})

or, upon the substitution r′ = r̃ − ρ, as

∑_{i=1}^{rk 𝔩} Alt(x_{i} ⊗ ∂_{i}r̃(λ)) − ∑_{i=1}^{rk 𝔩} Alt(x_{i} ⊗ ∂_{i}ρ(λ)) +
+ Alt([ρ_{12} + ρ_{13}, r̃_{23} − ρ_{23}])+CYB(r̃ − ρ) = Z_{𝔤} − CYB(r_{𝔩}). (11)

Since ρ is skew, we find CYB(r̃ − ρ) to be equal to

CYB(r̃) + CYB(ρ) − ([r̃_{12}, ρ_{13}] + [r̃_{12}, ρ_{23}] + [r̃_{13}, ρ_{23}] + [ρ_{12}, r̃_{13}] + [ρ_{12}, r̃_{23}] + [ρ_{13}, r̃_{23}]) = CYB(r̃) + CYB(ρ) − Alt([ρ_{12} + ρ_{13}, r̃_{23}]).

Also, it easy to see that Alt([ρ_{12}+ρ_{13}, ρ_{23}]) = 2CYB(ρ). Taking this into account, we rewrite (11) as

∑_{i=1}^{rk 𝔩} Alt(x_{i} ⊗ ∂_{i}r̃(λ)) + CYB(r̃) − ∑_{i=1}^{rk 𝔩} Alt(x_{i} ⊗ ∂_{i}ρ(λ)) − CYB(ρ(λ))= Z_{𝔤} − CYB(r_{𝔩}).

We have CYB(r_{𝔩}) = −CYB(Ω_{𝔩}) = \frac{1}{4}[Ω_{𝔩}^{12}, Ω_{𝔩}^{23}] = Z_{𝔩}^{ϵ=1}. Since ρ solves the CDYBE for ϵ = 1, this equation reduces to the CDYBE for the function r̃(λ) = r|_{𝔥_{𝔩}}(λ)+ρ(λ)+r_{𝔩}, were λ varies in N_{𝔥_{𝔩}}(0).

Conversely, suppose the function r̃(λ) satisfies the CDYBE over base N_{𝔥_{𝔩}}(0). Then the function r′ := r̃|_{N_{𝔥_{𝔩}} (0)} − ρ satisfies equation (7). Observe that equation (7) is 𝔩-invariant and, by assumption, r′ is the restriction of an invariant function defined on N_{𝔩}(0). Therefore the function r′ solves (7) when restricted to semisimple elements from N_{𝔩}(0) and hence everywhere in N_{𝔩}(0) since semisimple elements are dense in 𝔩.

Theorem 5.1 allows to construct a trigonometric dynamical r-matrix over a Levi subalgebra in a complex simple Lie algebra. Recall once again that we use identification 𝔤 ⊗ 𝔤 ≃ 𝔤^{∗} ⊗ 𝔤 ≃ End(𝔤) via the invariant pairing on 𝔤.

Corollary 5.2. Let 𝔩 be a Levi subalgebra in a complex simple Lie algebra 𝔤 and let 𝔩^{⊥} be the orthogonal complement of 𝔩 in 𝔤 with respect to the Killing form. The meromorphic function r: N_{𝔩}(0) ∋ λ ↦ f_{𝔩}(adλ) ⊕ f_{𝔩^{⊥}}(adλ) ∈ End(𝔩) ⊕ End(𝔩^{⊥}) ⊂ End(𝔤) with

f_{𝔩}(λ) = \frac{1}{2}coth(\frac{1}{2}adλ) − \frac{ϵ}{2}coth(\frac{ϵ}{2}adλ) − r_{𝔩},
f_{𝔩^{⊥}}(λ) = −\frac{ϵ}{2}coth(\frac{ϵ}{2}adλ).

is a PL dynamical r-matrix for Z_{𝔤}^{ϵ} = \frac{ϵ^{2}}{4} [Ω_{𝔤}^{12}, Ω_{𝔤}^{23}].

Proof. Since 𝔤 is simple, the restriction of the matrix adλ ∈ End(𝔤) to the invariant subspace 𝔩^{⊥} is invertible for λ belonging to a dense open subset in N_{𝔩}(0). Hence λ ↦ r(λ) is a correctly defined meromorphic function on N_{𝔩}(0). The function r + r_{𝔩} is 𝔩-invariant and r|_{𝔥_{𝔩}} + r_{𝔩} + ρ(𝔩, 1, λ) = ρ(𝔤, ϵ, λ) for λ ∈ 𝔥_{𝔩} .

6 Generalized Balog-Fehér-Palla dynamical r-matrices

Let 𝔤 be a finite dimensional complex Lie algebra and B: 𝔤 → 𝔤 an automorphism of order n. Then 𝔤 = ⊕_{j∈ℤ/nℤ} 𝔤_{j}, where 𝔤_{j} := ker(B − e^{2iπj/n}). The Lie algebra 𝔤_{0} acts on 𝔤_{j} for all j.

Let 𝔤 be equipped with an ad- and B-invariant form; denote by Ω_{𝔤} ∈ (S^{2}𝔤)^{𝔤} the corresponding Casimir element. According to [ES], there exists a dynamical r-matrix over the base 𝔩^{∗} for the trivial Lie bialgebra 𝔩 (abelian 𝔩^{∗}). Under the identification 𝔤⊗𝔤 ≃ 𝔤^{∗}⊗𝔤 ≃ End(𝔤), it is given by the invariant function ρ: N_{𝔩}(0) → End(𝔤), ρ(A)|_{𝔤_{j}} = f_{j}(adA), with

f_{0}(s) = \frac{1}{s} − \frac{1}{2}coth(\frac{1}{2}s),  (12)
f_{j}(s) = −\frac{1}{2}coth(\frac{1}{2}s + \frac{iπj}{n} ), j ≠ 0.  (13)

This solution corresponds to Z_{𝔤} = \frac{1}{4}[Ω_{𝔤}^{12}, Ω_{𝔤}^{23}]. For B = id, formulas (12-13) give the rational-trigonometric dynamical r-matrix of Alekseev-Meinrenken, [AM].

The trigonometric analog of this construction is as follows. Consider the analytical function r′_{trig}: N_{𝔩}(0) → End(𝔤), r′_{trig}(A)|_{𝔤_{j}} = f_{j}^{trig}(adA), with

f_{0}^{trig}(s) = \frac{1}{2}coth(\frac{1}{2}s) − \frac{ϵ}{2}coth(\frac{ϵ}{2}s),  (14)
f_{j}^{trig}(s) = −\frac{ϵ}{2}coth(\frac{ϵ}{2}s + \frac{iπj}{n} ), j ≠ 0.  (15)

Let Ω_{𝔩} denote the Casimir element of the restriction of the invariant form to 𝔩.

Theorem 6.1. The equivariant function r′_{trig} satisfies equation (7) for Z_{𝔤}^{ϵ} = \frac{ϵ^{2}}{4} [Ω_{𝔤}^{12}, Ω_{𝔤}^{23}] and Z_{𝔩} = \frac{1}{4}[Ω_{𝔩}^{12}, Ω_{𝔩}^{23}].

Proof. The proof is an appropriate modification of the proof of Theorem A.1 of [ES]. At the first step one considers 𝔤 = 𝔩 ⊕ … ⊕ 𝔩, the direct sum of n-copies of a reductive Lie algebra 𝔩 and B the cyclic permutation of these copies. One proves that the function r′_{trig}|_{𝔥_{l}} +ρ, where ρ(λ) := ρ(𝔩, 1, λ) from (9), is a dynamical r-matrix over the 𝔥_{l} for Z = Z_{𝔤}^{ϵ}. At ϵ = 1, this function coincides with the corresponding function from [ES], see the proof of Proposition A.1; for ϵ ≠ 1 cf. Remark 2.2. By Theorem 5.1, r′ solves equation (7) for these specific 𝔤 and B. The case of general 𝔤 is derived from this one similarly to [ES]. Namely, define a map W: N_{𝔩}(0) → ∧^{3}𝔤 setting

W (A) = Alt(∇′r′_{trig}(A)) + CYB(r′_{trig}(A)) − Z_{𝔤}^{ϵ} + \frac{1}{4}[Ω_{𝔩}^{12}, Ω_{𝔩}^{13}].

This is an analog of the function W from Appendix of [ES]. Then one defines K_{ij}(A, X, Y) := (id⊗X ⊗Y, W (A)) ∈ 𝔤_{i+j} for all i, j ∈ ℤ/nℤ, A ∈ 𝔩, X ∈ 𝔤_{i} and Y ∈ 𝔤_{j}, which are expressed by universal Lie series in A, X, and Y for each pair i, j. For K_{ij}(A, X, Y) one can prove the same assertions as Proposition A.2 and Lemma A.2 of [ES] and find that they vanish for all A ∈ 𝔩, X ∈ 𝔤_{i}, and Y_{j} ∈ 𝔤_{j}. Thus W (A) = 0 and this proves the theorem.

Corollary 6.2. Suppose 𝔩 = ker B has a quasitriangular structure (r_{𝔩}, Ω_{𝔩}). Then r_{trig} = r′_{trig} − r_{𝔩} is a PL dynamical r-matrix.

For B = id, formulas (14-15) give the trigonometric PL dynamical r-matrix of [BFP].

7 Concluding remarks

In conclusion, we make some comments on the dynamical r-matrices over general base.

7.1. Let us call an 𝔩-base manifold homogeneous, if it is 𝔇𝔩-homogeneous. The following proposition gives a local description of homogeneous base manifolds.

Proposition 7.1. Local homogeneous 𝔩-base manifolds are in one-to-one correspondence with Lie subalgebras 𝔨 ⊂ 𝔇𝔩 containing their orthogonal complement 𝔨^{⊥} with respect to the canonical inner product in 𝔇𝔩. Over the complex field, such Lie subalgebras are exactly those containing a lagrangian (maximal isotropic) subspace.

This proposition implies that base manifolds are quite numerous. The simplest examples are 𝔨 = 𝔩 and 𝔨 = 𝔩^{∗}, which are themselves lagrangian.

All the examples of dynamical r-matrices known (including those constructed in the present paper) correspond to 𝔨 ≃ 𝔩. It is interesting to find dynamical r-matrices over other bases.

7.2. The homogeneous base manifolds corresponding to 𝔨 ⊃ 𝔩 are alike, to a certain extent. Each point of such a manifold has a neighborhood with a normal system of coordinates {λ_{i}}, where 𝔩 acts by linear (coadjoint) transformations. The local ring of formal functions on a normal neighborhood of a point admits a decreasing 𝔩-invariant filtration by powers of the maximal ideal. The dual Lie algebra acts on the normal neighborhood by vector fields ξ^{i} ↦ ∂_{λ_{i}}+ terms of higher degrees. All this makes such base manifolds close to 𝔩^{∗} for the trivial Lie bialgebra 𝔩 (abelian 𝔩^{∗}). It is possible to develop a classification theory of dynamical r-matrices over such base manifolds along the line of [EV, ES].

7.3. In [Dr2], Drinfeld established a correspondence between local homogeneous Poisson Lie 𝔤-manifolds with lagrangian Lie algebras 𝔣 in 𝔇𝔤. The same objects describe Lie quasibialgebras, the infinitesimal deformations of U(𝔣) in the class of quasi-Hopf algebras, [Dr3]. Drinfeld set up a problem of finding relations between quasi-Hopf algebras and quantized Poisson Lie manifolds. A partial answer to this problem is achieved in the framework of dynamical Yang-Baxter equation. The covered cases correspond to 𝔣 ≃ 𝔤.

Let 𝔤 be a Lie bialgebra and 𝔩 its Lie sub-bialgebra. Let U_{ℏ}(𝔤) and U_{ℏ}(𝔩) be quantized universal enveloping algebras with U_{ℏ}(𝔩) ↪ U_{ℏ}(𝔤). One may admit 𝔤 or 𝔩 to be trivial Lie bialgebras; then U_{ℏ}(𝔤) or U_{ℏ}(𝔩) would be the ordinary universal enveloping algebras extended by ℂ[[ℏ]]. Let L be a base algebra over U_{ℏ}(𝔩), [DM1]. It plays a role of a function algebra on a quantized base manifold. Given a dynamical twist F = 1 ⊗ 1 ⊗ 1 + O(ℏ) with values in U_{ℏ}(𝔤) ⊗ U_{ℏ}(𝔤) ⊗ L and a U_{ℏ}(𝔩)-invariant character λ of L one can reduce F to F_{λ} ∈ U_{ℏ}(𝔤) ⊗ U_{ℏ}(𝔤) and interpret it as a quasi-Hopf algebra twist of U_{ℏ}(𝔤).

The dual algebra U_{ℏ}^{∗}(𝔤) is a quantum analog of functions on the group. It is a two sided module over U_{ℏ}(𝔤). Apply the quasi-Hopf algebra twist to U_{ℏ}^{∗}(𝔤) from the left. The result will be a quasi-associative algebra. The deviation from associativity is controlled by an associator canonically constructed out of F_{λ}. This associator reduces to identity on the trivial U_{ℏ}(𝔩)-submodule in U_{ℏ}^{∗}(𝔤). So the subspace of U_{ℏ}(𝔩)-invariants in U_{ℏ}^{∗}(𝔤) becomes an associative algebra. This algebra preserves the right U_{ℏ}(𝔤)-action intact. In this way it is an equivariant quantization of the Poisson Lie homogeneous 𝔤-manifold with the stabilizer 𝔩.

7.4. The dynamical twists have been constructed for all pairs (𝔤, 𝔩) where 𝔤 is simple Lie algebra and 𝔩 is a Levi subalgebra in 𝔤, [DM1, EE, EEM]. The choice of Levi subalgebras among other reductive Lie subalgebras is conditioned by the following. For Levi subalgebras, there exists a powerful computational tool, the generalized Verma modules. Categorically, the construction works as follows. Parabolic induction establishes an additive isomorphism between two module categories over the monoidal category of U_{ℏ}(𝔤)-modules. In [DM1] this isomorphism is called dynamical adjoint functor (to the restriction functor from U_{ℏ}(𝔤)- modules to U_{ℏ}(𝔩)-modules). A dynamical adjoint functor induces a new multiplication on morphisms in the dynamical extension of the category of U_{ℏ}(𝔤)-modules over U_{ℏ}(𝔩)-modules. This new multiplication is equivalent to a dynamical twist.

We believe that reductive Lie subalgebras other than Levi can be treated using more sophisticated modules than the generalized Verma modules.

7.5. We conclude that for a good deal of important cases the Drinfeld problem has been solved within the theory of dynamical Yang-Baxter equation. The natural question is about limitations of this approach. This concerns both quantum and classical setting. First of all, what is the exact correspondence between dynamical r-matrices over Poisson Lie bases and Poisson Lie manifolds in Drinfeld’s description?

References

[AM] A. Alekseev, E. Meinrenken: The non-commutative Weil algebra, Invent. Math. 135 (2000) 135–172.

[BDF] J. Balog, L. Dabrowski, and L. Fehér: Classical r-matrix and exchange algebra in WZNW and Toda field theories, Phys. Lett. B 244 (1990) 227–234.

[BFP] J. Balog, L. Fehér, and L. Palla: The chiral WZNW phase space and its Poisson-Lie groupoid, Phys. Lett. B 463 (1999) 83–92.

[Dr1] V. Drinfeld: Quantum Groups, in Proc. Int. Congress of Mathematicians, Berkeley, 1986, ed. A. V. Gleason, AMS, Providence (1987) 798–820.

[Dr2] V. Drinfeld: On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys. 95 (1993) #2, 524–525.

[Dr3] V. Drinfeld: Quasi-Hopf algebras, Leningrad Math. J. 1 (1990) # 6, 1419–1457.

[DM1] J. Donin, A. Mudrov: Dynamical Yang-Baxter equation and quantum vector bundles, accepted to Commun. Math. Phys., arXiv: math.QA/0306028.

[DM2] J. Donin, A. Mudrov: Quantum groupoids associated with dynamical categories, math.QA/0311116.

[EE] B. Enriquez, P. Etingof: Quantization of classical dynamical r-matrices withnonabelian base, arXiv:math.QA/0311224.

[EEM] B. Enriquez, P. Etingof, I. Marshall: Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces, arXiv: math.QA/0403283.

[ES] P. Etingof, O. Schiffmann: On the moduli space of classical dynamical r-matrices, Math. Res. Lett. 8 (2001) 157–170.

[EV] P. Etingof, A. Varchenko: Geometry and classification of solutions to the classical dynamical Yang-Baxter equation, Commun. Math. Phys. 192 (1998) 77–120.

[F] G. Felder: Conformal field theories and integrable models associated to elliptic curves, Proc. ICM Zurich, (1994) 1247–1255.

[FM] L. Fehér and I. Marshall: On a Poisson-Lie analogue of the classical dynamical Yang-Baxter equation for self dual Lie algebras, Lett. Math. Phys. 62 (2002) 51–62.

[S] M. Semenov-Tian-Shansky: Poisson-Lie Groups, Quantum Duality Principle, and the Quantum Double, Contemp. Math. 175 (1994) 219–248.