ABOUT THE DYNAMICAL YANG-BAXTER EQUATION(S)

AN INVITATION TO DYNAMICAL QUANTUM GROUPS

DAMIEN CALAQUE

Abstract. These are the notes of a short talk given on some aspects of the dynamical Yang-Baxter equation during the meeting of the GDR “Tresses” in Clermont-Ferrand (September 3-6, 2006). It is largely inspired from the lecture notes of ICM talks by Felder [2] and Etingof [1].

I thank the organizors for giving me the occasion to give this talk, and for the excellent atmosphere during the conference.

1. The (quantum) dynamical Yang-Baxter equation

Let 𝔥 be a finite dimensional abelian Lie algebra, V a semi-simple 𝔥-module and ℏ ∈ ℂ^{×}.

For any (meromorphic) function R(λ, z) : 𝔥^{∗} × ℂ → End_{𝔥}(V ⊗ V), the quantum dynamical Yang-Baxter equation (QDYBE) with step ℏ reads:

R^{1,2}(λ − ℏh^{(3)}, z_{1} − z_{2})R^{1,3}(λ, z_{1} − z_{3})R^{2,3}(λ − ℏh^{(1)}, z_{2} − z_{3})
= R^{2,3}(λ, z_{2} − z_{3})R^{1,3}(λ − ℏh^{(2)}, z_{1} − z_{3})R^{1,2}(λ, z_{1} − z_{2}) .

Here we adopt the dynamical notation: R^{1,2}(λ − ℏh^{(3)}, z) is defined by

R^{1,2}(λ − ℏh^{(3)}, z)(v_{1} ⊗ v_{2} ⊗ v_{3}) := R(λ − ℏμ, z)(v_{1} ⊗ v_{2}) ⊗ v_{3} ,

where v_{1}, v_{2} ∈ V, v_{3} ∈ V [μ], and V [μ] denotes the weight subspace of weight μ.

1.1. An example motivated by quantum integrable models. In [2] Felder proved that any solution of the QDYBE produces solutions of the famous startriangle relation that is usefull to construct solvable models of statistical mechanics. The following example produces the so-called A_{n−1}^{(1)} face model: let V be the vector representation of 𝔤𝔩_{n}, 𝔥 the subalgera of diagonal matrices, and denote by E_{ij} the elementary matrix defined by (E_{ij})_{kl} = δ_{ik}δ_{jl}. Then we write λ = (λ_{1}, …, λ_{n}) ∈ 𝔥^{∗}, with λ_{i} = E_{ii}^{∗}, and define

R(λ, z) = ∑_{i=1}^{n} E_{ii}⊗E_{ii}+ ∑_{1≤i≠j≤n} (\frac{θ(λ_{i} − λ_{j} + ℏ)θ(z)}{θ(λ_{i} − λ_{j})θ(z − ℏ)}E_{ii}⊗E_{jj} \frac{θ(z − λ_{j} + λ_{i})θ(ℏ)}{θ(z − ℏ)θ(λ_{j} − λ_{i})}E_{ij}⊗E_{ji}) ,

where θ(z) := θ(z|τ) is the standard theta-function, with normalization ∂_{z}θ(0) = 1. It is a solution of the QDYBE. Remark. One can replace θ(z) by sin(z) or z. In these two last cases taking the limit z → ∞ one obtains new solutions of the QDYBE, that are z-independant.

1.2. An example coming from representation theory. We follow [1]. Let 𝔤 be a semi-simple Lie algebra over ℂ with Cartan subalgebra 𝔥. For any λ ∈ 𝔥^{∗} let us denote by M_{λ} the corresponding Verma module with heighest weight λ, v_{λ} its highest weight vector, and v_{λ}^{∗} the lowest weight vector of its dual module.

To any intertwining operator Φ : M_{λ} → M_{μ} ⊗ V, where V is a finite dimensional 𝔤-module and λ, μ ∈ 𝔥^{∗}, we associate its expectation value < Φ >:= v_{μ}^{∗}(Φv_{λ}) ∈ V [λ−μ]. Assume that M_{μ} be irreducible (this holds for generic μ). Then it is known that the expectation value defines an isomorphism Hom_{𝔤}(M_{λ}, M_{μ} ⊗ V) → V [λ − μ] for any λ ∈ 𝔥^{∗} and any finite dimensional 𝔤-module V. Therefore one can define the intertwining operator Φ_{λ}^{v} such that < Φ_{λ}^{v} >= v (v ∈ V of weight |v| = λ − μ).

Let V, W be finite dimensional 𝔤-modules and consider v ∈ V, w ∈ W homogeneous vectors. The expectation value < Φ_{λ}^{v,w} > of the composition of two intertwining operators

Φ_{λ}^{v,w} := (Φ_{λ−|w|}^{v} ⊗ id) ∘ Φ_{λ}^{w} : M_{λ} → M_{λ−|v|−|w|} ⊗ V ⊗ W

is a bilinear function of v and w. Therefore there exists J_{V,W} (λ) ∈ End_{𝔥}(V ⊗ V) such that < Φ_{λ}^{v,w} >= J_{V,W} (λ)(v ⊗ w).

Then one can prove that J_{V,W} (λ) is an invertible meromorphic function of λ, and satisfies the dynamical twists equation (DTE):

J_{V_{1}⊗V_{2},V_{3}}(λ)J_{V_{1},V_{2}}(λ − h^{(3)}) = J_{V_{1},V_{2}⊗V_{3}}(λ)J_{V_{2},V_{3}}(λ) .

The DTE implies that R(λ) := J_{V V} (λ)^{−1}J_{V,V}^{2,1} (λ) is a (z-independant) solution of the QDYBE with step 1.

1.3. Categorical interpretation of the DTE. Let C = Rep(U𝔤) and M = Rep(Mer(𝔥^{∗})). For any finite dimensional 𝔤-module V one has an algebra morphism Mer(𝔥^{∗}) → End(V) ⊗ Mer(𝔥^{∗}) ; f(λ) ↦ f(λ − h). Therefore one has a functor

⊗ : C × M → M .

Let us now interprete J_{V,W} (λ) as a natural “associativity isomorphism”

V ⊗ (W ⊗ M) ˜ ⟶(V ⊗ W) ⊗ M (V, W ∈ C , M ∈ M) .

Then the DTE implies that J defines a structure of a C-module category on M (in fact it is equivalent). Namely, the following diagram commutes:

[figure: − ⊗ (− ⊗ (− ⊗ •)); 1⊗J_{V_{2},V_{3}} (λ); J_{V_{1},V_{2}} (λ−h^{(3)}); − ⊗ ((− ⊗ −) ⊗ •); J_{V_{1},V_{2}⊗V_{3}} (λ); (− ⊗ (− ⊗ −)) ⊗ •; (− ⊗ −) ⊗ (− ⊗ •); J_{V_{1}⊗V_{2},V_{3}} (λ); ((− ⊗ −) ⊗ −) ⊗ •]

2. Classical limit

If R(λ, z) = id_{V ⊗V} −ℏr(λ, z)+O(ℏ^{2}) ∈ Mer(𝔥^{∗}×ℂ, End_{𝔥}(V ⊗V)) is a solution of the QDYBE with step ℏ, then r(λ, z) satisfies the classical dynamical Yang-Baxter equation (CDYBE):

[r^{1,2}(λ, z_{1}−z_{2}), r^{2,3}(λ, z_{2}−z_{3})]+[r^{1,2}(λ, z_{1}−z_{2}), r^{1,3}(λ, z_{1}−z_{3})]+[r^{1,3}(λ, z_{1}−z_{3}), r^{2,3}(λ, z_{2}−z_{3})]

+ ∑_{ν} (h_{ν}^{(1)} \frac{∂r^{2,3}}{∂λ^{ν}} (λ, z_{2} − z_{3}) − h_{ν}^{(2)} \frac{∂r^{1,3}}{∂λ^{ν}} (λ, z_{1} − z_{3}) + h_{ν}^{(3)} \frac{∂r^{1,2}}{∂λ^{ν}} (λ, z_{1} − z_{2})) = 0

2.1. Relation to integrable systems. In [2] Felder proved that given a solution of the CDYBE one can define a compatible system of differential equations as follows:

∂_{z_{i}}F (z_{1}, …, z_{n}) = ∑_{j|j≠i} r^{i,j}(λ, z_{i} − z_{j}) · F − ∑_{ν} h_{ν}^{(i)} · \frac{∂F}{∂λ^{ν}} (i = 1, …, n) ,

where F (z_{1}, …, z_{n}) : ℂ^{n} → V^{⊗n}. 2.2. The universal classical dynamical Yang-Baxter equation and infinitesimal braids on the torus. Definition. The Lie algebra of infinitesimal (pure) braids on the torus is the graded Lie algebra generated by x_{i}’s and y_{i}’s (1 ≤ i ≤ n) in degree 1 and t_{ij}’s (1 ≤ i ≠ j ≤ n) in degree 2, with relations

(1)  [x_{i}, x_{j}] = [y_{i}, y_{j}] = [x_{i}, y_{j}] = 0 and [x_{i}, y_{j}] = t_{ij} = [x_{j}, y_{i}] (i ≠ j) ;

(2) [x_{i}, y_{i}] = − ∑_{j:j≠i} t_{ij} (∀i) ; [x_{i}, t_{jk}] = [y_{i}, t_{jk}] = 0 (#{i, j, k} = 3) .

One can easily check that the usual infinitesimal pure braid relations on the plane (t_{ij} = t_{ji}, [t_{ij}, t_{ik} + t_{jk}] = 0, [t_{ij}, t_{kl}] = 0) are consequences of (1-2).

Let i ≠ j ∈ {1, …, n}. There is a Lie algebra morphism 𝔱_{1,2} → 𝔱_{1,n}; α ↦ α^{i,j} defined by x_{k}^{i,j} = δ_{1k}x_{i} + δ_{2k}x_{j} and y_{k}^{i,j} = δ_{1k}y_{i} + δ_{2k}y_{j} (k = 1 , 2).

For a (meromorphic) function r(z) : ℂ → 𝔱̂_{1,2}, the universal CDYBE reads

[r(z_{1} −z_{2})^{1,2}, r(z_{2} −z_{3})^{2,3}]+[r(z_{1} −z_{2})^{1,2}, r(z_{1} −z_{3})^{1,3}]+[r(z_{1} −z_{3})^{1,3}, r(z_{2} −z_{3})^{2,3}]

= [y_{1}, r(z_{2} − z_{3})^{2,3}] + [y_{2}, r(z_{1} − z_{3})^{1,3}] + [y_{3}, r(z_{1} − z_{1})^{1,2}] .

Here means “the degree completion of” and the universal CDYBE takes place in 𝔱̂_{1,3}. It implies that the following system of differential equations is compatible:

∂_{z_{i}}F (z_{1}, …, z_{n}) = ∑_{j|j≠i} r(z_{i} − z_{j})^{i,j} · F − y_{i} · F (i = 1, …, n) ,

where F (z_{1}, …, z_{n}) : ℂ^{n} → 𝔱̂_{1,n}.

Example ([3]). Let θ(z) = θ(z|τ) be, as before, the standard theta-function. Then

r(z) = (\frac{θ(x_{1} + z)}{θ(x_{1})θ(z)} − \frac{1}{x_{1}} )(t_{12})

is a solution of the universal CDYBE. The system of differential equations that we obtain actually defines a holomorphic flat connection on a principal exp(𝔱̂_{1,n})-bundle over the configuration space of n points on the elliptic curve E_{τ} = ℂ/(ℤ+τℤ). Then one has a group homomorphism P B_{1,n} → exp(𝔱̂_{1,n}), where P B_{1,n} denotes the pure braid group of the torus. If we denote by 𝔭𝔟_{1,n} be the Malcev Lie algebra of P B_{1,n} (i.e. the Lie algebra of its prounipotent completion), then it induces a Lie algebra morphism 𝔭𝔟_{1,n} → 𝔱̂_{1,n}. Proposition ([3]). It is an isomorphism.

References

[1] P. Etingof, On the dynamical Yang-Baxter equation, Proceedings of the ICM, Beijing (2002),

   arXiv:math.QA/0207008.

[2] G. Felder, Conformal field theory and integrable systems associated to elliptic curves, Proceedings of the ICM, Zürich (1994), arXiv:hep-th/9407154.

[3] D. Calaque, B. Enriquez, and P. Etingof, Universal KZB equations, in preparation.