Journal of Lie Theory Volume 4 (1994) 237–255 ◯^{C} 1994 Heldermann Verlag Version of May 4, 1995

Triangular Lie bialgebras and matched pairs for Lie algebras of real vector fields on S^{1}

Frank Leitenberger

Communicated by K. Schmüdgen

Abstract: Real Lie bialgebras and matched pairs for a Lie algebra of formal vector fields and for Lie algebras of smooth vector fields with fixed zeros on the circle are constructed.

1. Introduction

The construction of a quantum group generalization of the group Diff_{+}(S^{1}) (the group of orientation preserving C^{∞}-diffeomorphisms of S^{1}) is an open problem. On the classical level we have the corresponding problem of the construction of Poisson Lie groups or of matched pairs for the diffeomorphism group (cf. [D], [Ta]).

There is some progress in this direction on the Lie algebra level. In [B-M],

[M] and [T] various matched pair structures for certain Lie algebras of complex vector fields on S^{1} are obtained. The choice of different function spaces for the Lie algebra of vector fields sheds a different light on the problem. In [M] the author considers polynomial vector fields and recursive sequences for the dual Lie algebra. In [B-M] certain spaces of analytic functions are considered. However, these structures are not compatible with the subalgebra of real vector fields. Therefore, they do not admit an integration to the matched pair group level because there is no complexification of Diff_{+}(S^{1}) (cf. [P-S]).

In this article we classify additional Lie bialgebras and matched pair structures and obtain the structures of [B-M], [M] as special cases. Every structure is parametrized by a subalgebra h and by a moment b of h (or [h, h]) of the Lie algebra g for which the bilinear form h × h ∋ (x, y) ↦ b([x, y]) is invertible. The inverse plays the role of the classical r-matrix. In the case h = g the matched pair structures are self-dual. For the Lie algebra of vector fields we first choose a space of semi-infinite formal power series. Such completions of the Witt algebra occur in conformal field theory (cf. [W1]). We give an explicit description of the structures and single out the real structures. Next we choose Lie algebras g_{c} of

ISSN 0940–2268 / $2.50 ◯^{C} Heldermann Verlag smooth functions with fixed zeros. These function spaces allow the construction of real matched pairs. These Lie algebras correspond to Lie groups G_{c} with fixed points of a certain order (see the Remark after Definition 4.2). The structures correspond to subalgebras h_{c}^{(n)} and certain moments. We hope that some matched pairs allow an integration to matched pairs for the Lie groups G_{c}.

The paper is organized as follows: In Section 2 we generalize the foundations of the finite-dimensional situation to the infinite-dimensional case. In Section 3 we classify Lie bialgebra and matched pair structures for the case of semi-infinite formal vector fields on the circle. In Section 4 we obtain matched pair structures for smooth vector fields with fixed zeros on the circle.

I would like to thank the referee for his helpful comments concerning this paper.

2. Preliminaries

In this section we generalize some well-known results about Lie bialgebras and matched pairs (cf. [B-M], [D]) to the infinite-dimensional case. 2.1. Construction of matched pairs of Lie algebras. The goal of this subsection is to introduce matched pair structures for dual pairs of infinite-dimensional vector spaces without any topology. The constructions of Propositions 2.3 and 2.4 provide the foundation of Section 4.

Definition 2.1. Let g be a (possibly infinite-dimensional) Lie algebra. We say that g′ is a dual of g if and only if

(i) there is a bilinear form ⟨−, −⟩ : g × g′ → C such that g, g′, ⟨−, −⟩ forms a reflexive pair of vector spaces (i.e., g and g′ separate the points of g′ and g respectively).

(ii) the coadjoint action K of g on g′ is well-defined (i.e., ∀x ∈ g, y ∈ g′, ∃ z =: ad_{x}^{T} y = −K_{x}y ∈ g′ given by ⟨ad_{x}u, y⟩ = ⟨u, z⟩ ∀u ∈ g′).

Definition 2.2. A matched pair (g_{1}, g_{2}, α, β) is a pair of Lie algebras g_{1}, g_{2} with the Lie algebra representations α : g_{1} × g_{2} → g_{2} and β : g_{2} × g_{1} → g_{1} such that, for all x, y ∈ g_{1} and u, v ∈ g_{2},

α_{x}([u, v]) = [α_{x}(u), v] + [u, α_{x}(v)] + α_{β_{v}(x)}(u) − α_{β_{u}(x)}(v),

and

β_{u}([x, y]) = [β_{u}(x), y] + [x, β_{u}(y)] + β_{α_{y}(u)}(x) − β_{α_{x}(u)}(y).

We say that (g_{1}, g_{2}, α, β) is a coadjoint matched pair if there is a bilinear form ⟨−, −⟩, such that (g_{1}; g_{2}; ⟨−, −⟩) is a dual pair of vector spaces and α, β are the corresponding coadjoint actions.

Further we say that the linear map ω : g → g′ is a 2-cocycle, if and only if the bilinear form ω : g × g → C specified by (x, y) ↦ ⟨ω(x), y⟩ is antisymmetric and a 2-cocycle, i.e., ω([p, q], r) + ω([q, r], p) + ω([r, p], q) = 0 ∀p, q, r ∈ g.

Matched pairs can be constructed by invertible 2-cocycles (cf., [W2, page 49]).

Proposition 2.3. Let g be a Lie algebra, g′ a dual to g, and ω a 2-cocycle for g. Further suppose that ω has an inverse map r : g′ → g. Set

\begin{aligned}[x, y]_{g′} & := & ω([r(x), r(y)]) & x, y ∈ g′, \\ α_{e} & := & −ad_{e}^{T} & e ∈ g, \\ β_{x} & := & r α_{r(x)}ω & x ∈ g′. \end{aligned}

Then (g, g′, α, β) forms a coadjoint matched pair of Lie algebras.

We remark that r : g′ → g is a Lie algebra isomorphism and g is self dual with respect to the bilinear form ⟨ω(−), − ⟩. Therefore we can identify g and g′ and reformulate Proposition 2.3 and obtain a self-dual representation.

Proposition 2.3.a. Under the the assumptions of Proposition 2.3, if α′ := r ∘ α ∘ (id × ω) then (g, g, α′, α′) with α′ := rαω and (g, g′, α, β) are isomorphic matched pairs of Lie algebras.

The next proposition generalizes Proposition 2.3. We show that every invertible r-matrix of a subalgebra allows the construction of a matched pair.

Let g be a Lie algebra with dual g′, and let h be a subalgebra with dual h′ ⊂ g′. We introduce the following notation: h^{⊥} := {x ∈ g′|⟨h, x⟩ = 0} while h′^{⊥} := {e ∈ g|⟨e, h′⟩ = 0}.

Proposition 2.4. Let g be a Lie algebra with dual g′, let h be a subalgebra with dual h′ ⊂ g′, and let g′ = h′ +^{.} h^{⊥} and g = h +^{.} h′^{⊥} be direct sum decompositions of g′ and g respectively, as vector spaces. Further let ω : h → h′ be a 2-cocycle with inverse r : h′ → h. Set

\begin{aligned}[x, y]_{g′} & := & \{ \begin{matrix}ω([r(x), r(y)]) & x, y ∈ h′ \\ −ad_{r(x)}^{T}y & x ∈ h′, y ∈ h^{⊥} \\ 0 & x, y ∈ h^{⊥} , \end{matrix} \\ α_{e} & := & −ad_{e}^{T} e ∈ g , \\ β_{x}(e) & := & \{ \begin{matrix}rP_{h′}α_{r(x)}ω(e) & x ∈ h′, e ∈ h \\ P_{h′}_{⊥}[r(x), e] & x ∈ h′, e ∈ h′^{⊥} \\ r(α_{e}x) & x ∈ h^{⊥}, e ∈ g. \end{matrix} \end{aligned}

Then (g, g′, α, β) is a matched pair of Lie algebras. Here P_{h′}_{⊥} denotes the projection to h′^{⊥}, and P_{h′} denotes the projection to h′.

Example 2.5. Let g = W = lin{e_{i}|i ∈ ZZ } with [e_{i}, e_{j}] = (j − i)e_{i+j} be the 2-sided Witt algebra with dual g′ = W′ = lin{f_{i}|i ∈ ZZ } where ⟨e_{i}, f_{j}⟩ = δ_{i,j}. Further let h_{n} = lin{e_{0}, e_{n}} (n ∈ ZZ , n ≠ 0) be Lie subalgebras with h_{n}′ = lin{f_{0}, f_{n}}. Up to a nonzero scalar factor there is a unique 2-cocycle ω_{n} = f_{n} ∧ f_{0} with inverse r_{n} = e_{0} ∧ e_{n}. We obtain

\begin{aligned}[f_{i}, f_{0}] & = & (2n − i)f_{i−n} & if i ≠ 0 \\ [f_{i}, f_{n}] & = & if_{i} & if i ≠ 0, n \\ [f_{i}, f_{j}] & = & 0 & if i, j ≠ 0, n , \end{aligned}

α_{e_{i}}f_{j} = (2i − j)f_{j−i} ,

and

β_{f_{i}}e_{j} = \{ \begin{matrix}(n − j)e_{j+n} & if & i = 0; j ≠ −n \\ je_{j} & & i = n; j ≠ 0, n \\ −ne_{0} & & i = n; j = 0 \\ (i − 2n)e_{0} & & j = i − n; i ≠ 0, n \\ −ie_{n} & & i = j; i ≠ 0, n \\ 0 & & i = 0; j = −n \\ & or & i = j = n \\ & or & i ≠ 0, n; j ≠ i, i − n. \end{matrix}

These structures were first obtained by Michaelis [M], then by Taft [T]. 2.2. Lie bialgebras, Classical r-matrices and CYBE. In this subsection we consider a class of infinite-dimensional graded Lie bialgebras (cf., Definition 2.9, below) for which we have at our disposal the main results of the finite-dimensional situation. The results of this subsection are applied in Section 3.

By a locally convex Lie algebra we mean a locally convex vector space with a continuous Lie bracket. By \overline{⊗}_{π} we denote the completed projective tensor product of locally convex vector spaces (cf. [K2, Chapter 41]).

Let τ, ad_{x}^{(2)} (for x ∈ g), and ξ, respectively denote the unique continuous extensions to g\overline{⊗}_{π}g and, respectively, to g\overline{⊗}_{π}g\overline{⊗}_{π}g (cf. [K2, Chapter 41]) of the continuous linear maps τ, ad_{x}^{(2)} : g ⊗ g → g ⊗ g and ξ : g ⊗ g ⊗ g → g ⊗ g ⊗ g given, respectively, by τ(x ⊗ y) = y ⊗ x, ad_{x}^{(2)}(y ⊗ z) = [x, y] ⊗ z + y ⊗ [x, z], ξ(x ⊗ y ⊗ z) = y ⊗ z ⊗ x.

Definition 2.6. A locally convex Lie bialgebra (g; [,]; △) is a locally convex Lie algebra (g; [,]) with a cobracket △, i.e. a continuous linear map △ : g → g\overline{⊗}_{π}g, which satisfies the anti-commutativity condition

τ ∘ △x = − △ f,

the Jacobi identity

(1 + ξ + ξ^{2}) ∘ (△\overline{⊗}_{π}id)△ = 0,

and the compatibility condition

△([x, y]) = ad_{x}^{(2)} △ (y) + ad_{y}^{(2)} △ (x).

Remark 2.7. The operator △\overline{⊗}_{π}id is the well-defined continuous extension of △ ⊗ id from g ⊗ (g\overline{⊗}_{π}g) to g\overline{⊗}_{π}(g\overline{⊗}_{π}g) = g\overline{⊗}_{π}g\overline{⊗}_{π}g.

Let {V_{i} | i ∈ ZZ } be a set of finite-dimensional vector spaces. Denote by ∑^{+} V_{i} the vector space of semi-infinite direct formal sums {∏_{i≤a} e_{i}|e_{i} ∈ V_{i}, a ∈ ZZ }. We endow ∑^{+} V_{i} with the topology which arises from the isomorphism ∑^{+} V_{n} ≅ ∏_{n<0} V_{n} ⊕ ∑_{n≥0} V_{n}. We define the locally convex space ∑^{−} V_{i} analogously. If V = ∑^{+} V_{i}, and if, ∀i ∈ ZZ , V_{i}^{∗} is the dual space of the finitedimensional vector space V_{i}, then we obtain V^{∗} = ∑^{−} V_{i}^{∗} as the continuous dual of V ; and (V, V^{∗}) forms a reflexive pair of locally convex vector spaces (cf. [K1, Chapter 30]).

Lemma 2.8. (i) V \overline{⊗}_{π}V ≅ L(V^{∗}, V), (ii) ∑^{+}V_{i}\overline{⊗}_{π}∑^{+}V_{i} = ∑_{i<i0(j),j<j0(i)} V_{i} ⊗ V_{j},

(iii) ∑^{+} V_{i}\overline{⊗}_{π} ∑^{+} V_{i}\overline{⊗}_{π} ∑^{+} V_{i} = ∑_{i<i0(j,k);j<j0(i,k);k<k0(i,j)} V_{i} ⊗ V_{j} ⊗ V_{k}.

Proof. Cf. [K2, Chapter 41].

Definition 2.9. A locally convex Lie algebra g is an upper semibounded graded Lie algebra (G^{+}-Lie algebra) if g = ∑^{+} g_{i} where the g_{i} are finite-dimensional subspaces of g and [g_{i}, g_{j}] = g_{i+j}.

In analogy to the finite-dimensional case we make the following definition.

Definition 2.10. Let g be a G^{+}-Lie algebra. We say that r = ∑_{i,j,n} a_{i}^{n} ⊗ b_{j}^{n} ∈ g\overline{⊗}_{π}g, (a_{i}^{n} ⊗ b_{j}^{n} ∈ g_{i} ⊗ g_{j}) is a classical r-matrix if it satisfies the Classical Yang Baxter Equation (CYBE)

∑_{i,j,k,l,n} [a_{i}^{n}, a_{k}^{n}] ⊗ b_{j}^{n} ⊗ b_{k}^{n} + a_{i}^{n} ⊗ [b_{j}^{n}, a_{k}^{n}] ⊗ b_{l}^{n} + a_{i}^{n} ⊗ a_{k}^{n} ⊗ [b_{k}^{n}, b_{l}^{n}] = 0.  (1)

Remark 2.11. Because of Lemma 2.4(iii) we have in the CYBE only finite summations in the graded components g_{i} ⊗ g_{j} ⊗ g_{k}.

Classical r-matrices can be constructed by certain 2-cocycles.

Proposition 2.12. Let g be a G^{+}-Lie algebra, let h be a Lie subalgebra, and let ω : h → h^{∗} be an invertible 2-cocycle. Further let i : h → g be the imbedding operator. Then the extension r = iω^{−1}i^{T} : g^{∗} → g of ω^{−1} : h^{∗} → h is a classical r-matrix for g.

Lie bialgebras can be given by classical r-matrices.

Proposition 2.13. Let g be a G^{+}-Lie algebra, and let r ∈ g\overline{⊗}_{π}g be a classical r-matrix. Then △ : g → g\overline{⊗}_{π}g defined by △(x) := ad_{x}^{(2)}r is a well-defined continuous operator and (g; [,]; △) is a locally convex Lie bialgebra. (We call g a triangular G^{+}-Lie bialgebra (cf. [D, page 804]).)

Finally we consider the connection between Lie bialgebras and coadjoint matched pairs. Both concepts are equivalent in the finite-dimensional case whereas in the infinite-dimensional situation matched pairs are more general.

Lemma 2.14. (i) Let (g; [,]; △) be a G^{+}-Lie bialgebra. Let [,]_{g^{∗}} := △^{T}, let α be the coadjoint action of g on g^{∗}, and let β be the coadjoint action of g^{∗} on g. Then (g, g^{∗}, α, β) is a coadjoint matched pair of Lie algebras.

(ii) Let (g, g^{∗}, α, β) be a finite-dimensional coadjoint matched pair. Then

(g; [,]_{g}; [,]_{g^{∗}}^{T}) is a Lie bialgebra. 2.3. Real structures. Let g be a complex Lie algebra with dual g′. Let I : g → g be an antilinear involution given by x ↦ \overline{x} and let g_{IR} := \{x ∈ g|x = \overline{x}\} be the real subalgebra. We say that I defines a real structure on g. We remark that I^{T} : g′ → g′ defines a real structure on g′. We say that a linear map a : g → g′ is real if a ∘ I^{T} = I ∘ a.

Lemma 2.15. (i) Let g be a G^{+}-Lie algebra (over C) and let r be an rmatrix. Then (g_{IR}; [,]|g_{IR}; △|g_{IR}) is a G^{+}-Lie algebra over IR if and only if r is real.

(ii) Under the assumptions of Proposition 2.2 let I be an real structure of g. Then (g_{IR}, g′_{IR}, α|g_{IR}, β|g_{′}_{IR}) is a real matched pair if and only if r : h → h′ is real.

3. The case of formal power series

3.1. Construction of r-matrices. In this section we consider completions of the Witt algebra W and of its finite dual W′ defined in Example 2.5.

Let

W^{+} := ∑^{+} Ce_{i}

W^{−} := ∑^{−} Cf_{i}

be the spaces of semi-infinite formal linear combinations of the elements e_{i} and f_{i}, respectively. W^{+} is a Lie algebra with respect to the canonical extension of the Lie bracket of W, namely,

[∑_{i≤a} a_{i}e_{i}, ∑_{i≤b} b_{i}e_{i}] = ∑_{i≤a,j≤b}(j − i)a_{i}b_{j}e_{i+j}.

W^{−} is the continuous dual of W^{+} (i.e. W^{+∗} = W^{−}) and if we set

⟨ ∑_{i=−∞}^{a} a_{i}e_{i}, ∑_{i=b}^{∞} b_{i}f_{i}⟩ := ∑_{i=−∞}^{∞} a_{i}b_{i} = ∑_{i=b}^{a} a_{i}b_{i}

then (W^{+}, W^{−}, ⟨−, −⟩) forms a dual pair of vector spaces.

Now we apply the Propositions 2.12, 2.13 for the construction of Lie bialgebras. We choose g = W^{+}, g^{∗} = W^{−}, h = W_{n}^{+} = ∑^{+} Ce_{nk} (n ∈ lN^{+}) and h^{∗} = W_{n}^{−} = ∑^{−} Cf_{nk}. For h^{∗} ∋ b = ∑_{i=a}^{∞} b_{i}f_{in}, (f_{na} ≠ 0) we consider the 2-cocycle ω_{b}^{n} : h × h → C defined by

ω_{b}^{n}(x, y) := ⟨b, [x, y]⟩ = ∑_{i,j} n(j − i)x_{i}y_{j}b_{i+j}

where x = ∑_{i} x_{i}e_{in} and y = ∑_{i} y_{i}e_{in}. According to the identity ω_{n}^{b}(x, y) = ⟨ω_{n}^{b}(x), y⟩ we can view ω_{b}^{n} as a well-defined linear map ω_{b} : h → h^{∗} given by

ω_{b}^{n}(e_{mn}) = ∑_{k} n(k − m)b_{k+m}f_{kn}.

Let a be an odd integer and consider the infinite system of equations in the variables (c_{i})_{i≥ \frac{−a−1}{2}}

∑_{k+l+m=s} c_{k}c_{l}b_{m} = δ_{s,−1}, s = −1, 0, 1, 2, ...  (1)

i.e.,

c2 ; \frac{−a−1}{2} b_{a}  = 1
2c_{\frac{−a−1}{2}} c_{\frac{−a+1}{2}} b_{a} + c2 ; \frac{−a−1}{2} b_{a+1} = 0
... .

Lemma 3.1. (i) The equation system (1) has the two solutions ±(c_{i})_{i≥ \frac{−a−1}{2}} .

(ii) Conversely every semi-infinite sequence (c_{i})_{i≥ \frac{−a−1}{2}} is the solution of an equation system (1) for certain (b_{i}).

(iii) Every solution of the equation system (1) is also a solution of the equation system

∑_{k+l=s}(2l + k + 1)b_{k}c_{l} = 0 s ∈ ZZ .

Proof. (i) The first equation has the two solutions c_{\frac{−a−1}{2}} = ±\sqrt{\frac{1}{b_{a}}}. Both solutions admit the unique successive solution of the remaining equations. (ii) Fix (c_{i}) and consider (1) as an equation system for the (b_{i}). This triangular equation system has a unique solution. (iii) (1) corresponds to the formal power series identity b(z)c(z)^{2} = 1 where b(z) = ∑_{k} b_{k}z^{k+1} and c(z) = ∑_{k} c_{k}z^{k}. If we differentiate the identity b(z)c(z)^{2} = 1 and divide the result by c(z) we obtain the identity b′(z)c(z)+2b(z)c′(z) = 0 which in turn yields the identities.

∑_{k+l=s}(2l + k + 1)b_{k}c_{l} = 0 s ∈ ZZ .

Consider an element W′ ∋ b := b_{a}z^{a} + b_{a−1}z^{a−1} + ... ∈ W′ with b_{a} ≠ 0. We say that b is even (respectively, odd) if a is even (respectively, odd ).

Proposition 3.2. (i) If b is odd then ω_{b}^{n} is invertible and as the inverse the map r_{b}^{n} : W^{+} → W^{−} defined by

r_{b}^{n}(f_{mn}) = \frac{1}{n} ∑_{k,l} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}e_{kn}.

(ii) If b is even then ω_{b} is not invertible and we have dim Ker ω_{b}^{n} = 1.

Proof. (i) It is easy to verify that r_{b}^{n} maps W^{+} into W^{−}. Further we have

ω_{b}^{n}r_{b}^{n}(f_{mn}) = ω_{b}(\frac{1}{n} ∑_{k,l} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}e_{kn})

= n ∑_{j}(j − k)b_{j+k} \frac{1}{n} ∑_{k,l} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}f_{jn}

= ∑_{j,k,l} c_{l}c_{−k−l−m−1}b_{k+j} \frac{j − k}{2m + 2l + 1}

Because of Lemma 3.1(iii) we have

∑ ; \begin{matrix}k, l \\ j + k + l = s \end{matrix} (−2l − k − 2m − 1 + k + j)b_{k+j}c_{−k−l−m−1} = 0 for s ∈ ZZ .

Using this identity and (1) we obtain

ω_{b}^{n}r_{b}^{n}(f_{mn}) = ∑_{j,k,l} c_{l}c_{−k−l−m−1}b_{k+j} \frac{j − k}{2m + 2l + 1} = δ_{j−m−1,−1} = δ_{j,m}.

The proof of r_{b}^{n}ω_{b}^{n} = id is analogous. (ii) For simplicity consider the case n = 1. ω_{b} has the following matrix representation with respect to the bases e_{j}, f_{j}.

( \underline{\begin{matrix}⋯ & ⋯ & ⋯ & ⋯ \\ ⋯ & 0 & 0 & 0 \\ ⋯ & 0 & 0 & 0 \\ ⋯ & 0 & 0 & 0 \end{matrix} | \begin{matrix}⋯ \\ 0 \\ 0 \\ 0 \end{matrix} | \begin{matrix}⋯ & ⋯ & ⋯ & ⋯ \\ 0 & 0 & −6b_{0} & ⋯ \\ 0 & −4b_{0} & −5b_{1} & ⋯ \\ −2b_{0} & −3b_{1} & −4b_{2} & ⋯ \end{matrix}} ; \underline{⋯  0  0  0 | 0 | −b_{1} −2b_{2} −3b_{3} ⋯} ; \begin{matrix}⋯ & 0 & 0 & 2b_{0} \\ ⋯ & 0 & 4b_{0} & 3b_{1} \\ ⋯ & 6b_{0} & 5b_{1} & 4b_{2} \\ ⋯ & ⋯ & ⋯ & ⋯ \end{matrix} | \begin{matrix}b_{1} \\ 2b_{2} \\ 3b_{3} \\ ⋯ \end{matrix} | \begin{matrix}0 & −b_{3} & −2b_{4} & ⋯ \\ b_{3} & 0 & −b_{5} & ⋯ \\ 2b_{4} & b_{5} & 0 & ⋯ \\ ⋯ & ⋯ & ⋯ & ⋯ \end{matrix} )

=: ( \begin{matrix}O & o & D_{1} \\ o^{T} & 0 & −d \\ D_{2} & d^{T} & B \end{matrix} )

Let g ∋ e = ∑_{i≤k} a_{i}e_{i} ∈ Ker ω_{b}. It follows that a_{1} = ... = a_{k} = 0, a_{0} = γ ∈C and that, for i ≤ 0 the a_{i} are determined by the equation system D_{2}a^{T} = −γd^{T} where a = (..., a_{−2}, a_{−1}). That is, Ker ω_{b} = 1.

By Lemma 3.1 all operators ω_{b}^{n} are parametrized by semi-infinite sequences (c_{i})_{i≤l} (up to multiplication with −1). By Lemma 2.8(i) we can also write

r_{b}^{n} = \frac{1}{2} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}e_{kn} ∧ e_{mn}.

As a consequence of Proposition 2.12 and 3.2 we can classify all r-matrices which arise from invertible 2-cocycles ω_{b}^{n} on W_{n}^{+}. That is the content of the following result.

Theorem 3.3. Let (c_{i})_{i≥a} for a ∈ ZZ be a semi-infinite complex sequence. Then

r_{b}^{n} = \frac{1}{2} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}e_{kn} ∧ e_{mn}

is a classical r-matrix.

Now we consider two special cases.

Examples 3.4. 1. n=1: We have

r_{b}^{1} = \frac{1}{2} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}e_{k} ∧ e_{m}.

These r-matrices correspond those in [B-M, page 27]. But in their paper, instead considering the spaces W^{±} of formal power series, Beggs and Majid consider certain spaces of analytic functions.

2. b = f_{u} = f_{−2a−1}, (c_{i} = ±δ_{i,a}): This is the simplest nontrivial case. The summation reduces to one summation parameter. We obtain

r_{u=−2a−1}^{n} = \frac{1}{2n} ∑_{m} \frac{1}{2m − u}e_{(u−m)n} ∧ e_{mn}.

3.2. Construction of Lie bialgebras and matched pairs. Next we describe the Lie bialgebra and matched pair structures which arise from the r-matrices of Theorem 3.3.

Proposition 3.5. Set

△_{b}^{n}(e_{i}) := ad_{e_{i}}^{(2)}r_{b}^{n} = \frac{1}{n} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}(mn − i)e_{mn+i} ∧ e_{kn}.

Then (W^{+}, △_{b}^{n}) is a locally convex Lie bialgebra.

Proof. The assertion follows from Propostion 2.13, Theorem 3.3 and the calculation

△_{b}^{n}(e_{i}) = ad_{e_{i}} \frac{1}{n} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}e_{mn} ⊗ e_{kn}
= \frac{1}{n} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}[e_{i}, e_{mn}] ⊗ e_{kn} + e_{mn} ⊗ [e_{i}, e_{kn}]
= \frac{1}{n} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}(mn − i)e_{mn+i} ⊗ e_{kn} + (kn − i)e_{mn} ⊗ e_{kn+i}
= \frac{1}{n} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}(mn − i)e_{mn+i} ∧ e_{kn}.

Example 3.6. Let r = r_{u=−2a−1}^{n}. We obtain

△_{b}^{n}(e_{i}) = ∑_{m} \frac{1}{2m − u}e_{mn+i} ∧ e_{(u−m)n}.

Proposition 3.7. Let b ∈ h^{∗} = W_{n}^{−} = ∑^{−}Cf_{nk} even. Set

[f_{rn}, f_{sn}]_{W^{−}} = ∑_{j} c_{l}c_{−r−s+j−l−1} \frac{(r−s)(2(l+j)+1)}{(2(r−j+l)+1)(2(s−j+l)+1)}f_{jn},  (1)
[f_{rn}, f_{q}]_{W^{−}} = −\frac{1}{n} ∑_{j} c_{l}c_{−r−j−l−1} \frac{(2jn − q)}{2j + 2l + 1}f_{q−jn} if n/| q,  (2)
[f_{p}, f_{q}]_{W^{−}} = 0 if n/| p, q,  (3)

α_{e_{i}}f_{j} = (2i − j)f_{j−i},  (4)
β_{x}(e) = \{ \begin{matrix}r_{c}P_{h_{c}}′α_{r_{c}(x)}ω(e) & x ∈ h_{c}′, e ∈ h_{c} \\ P_{h_{c}_{′}}_{⊥}[r_{c}(x), e] & x ∈ h_{c}′, e ∈ h_{c}′^{⊥} \\ r_{c}(α_{e}x) & x ∈ h_{c}^{⊥}, e ∈ g_{c} . \end{matrix}  (5)

Then M_{b}^{(n)} = (W^{+}, W^{−}, α, β) is a matched pair of Lie algebras.

Proof. We calculate the matched pair structure using Lemma 2.14(i). It suffices to examine the following formula in 3 separate cases:

⟨e_{i}, [f_{p}, f_{q}]⟩ = ⟨△_{b}^{n}(e_{i}), f_{p} ⊗ f_{q}⟩ .

= \frac{1}{n} ∑_{k,l,m} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}(mn − i)(δ_{mn+i,p}δ_{kn,q} − δ_{kn,p}δ_{mn+i,q})f_{i}  (6)

Case 1 : n|p and n|q. In this case formula (1) follows upon taking p = rn, q = sn and i = jn. Case 2 : n|p but n/| q. In this case the first summand in (6) vanishes. Formula (2) follows upon taking p = rn and i = q − jn ≠ 0. Case 3 : n/| p and n/| q. In this case formula (3) follows since both summands vanish in (6).

The formula for α follows from the fact that

⟨α_{e_{i}}f_{j}, e_{k}⟩ = ⟨−ad_{e_{i}}^{T}f_{j}, e_{k}⟩ = ⟨f_{j}, [e_{k}, e_{i}]⟩ = (i − k)δ_{j,i+k}.

Now let n=1 and choose the basis g_{i} = ω_{b}(e_{i}) in W^{−}. Then the matched pairs of Proposition 3.7 take on the following self-dual form.

Proposition 3.8. Let b ∈ W^{−}. Then M_{b}^{(1)} = (W^{+}, W^{−}, α, β) is given by

[e_{i}, e_{j}] = (j − i)e_{i+j} [g_{i}, g_{j}] = (j − i)g_{i+j},

α_{e_{i}}g_{j} = ∑_{k,l,m} \frac{(m + i − j)(i − m)}{2m + 2l + 1}b_{m+i+j}c_{l}c_{−k−l−m−1}e_{k},

and

β_{g_{i}}e_{j} = ∑_{k,l,m} \frac{(m + i − j)(i − m)}{2m + 2l + 1}b_{m+i+j}c_{l}c_{−k−l−m−1}g_{k}.

Example 3.9. Consider the special case where b = f_{un} = f_{(−2a−1)n} (i.e., c_{i} = δ_{i,a}). Then

[f_{rn}, f_{sn}] = \frac{(r − s)(2r + 2s + 3u)}{(2s + u)(2r + u)}f_{(r+s+u)n},

[f_{rn}, f_{q}] = −\frac{2rn + q + 2un}{2rn + un}f_{rn+q+un} if n/| q ,

and

[f_{p}, f_{q}] = 0 if n/| p, q .

Now let n=1. After the basis change g_{i} := (u − 2i)f_{−i+u} (i.e., ⟨e_{i}, g_{j}⟩ = −(2j +n)δ_{i+j+n,0}) we get mutually non-equivalent structures parametrized by odd integers u in a self-dual form

[e_{i}, e_{j}] = (j − i)e_{i+j}, [g_{i}, g_{j}] = (j − i)g_{i+j},

α_{e_{i}}g_{j} = \frac{(2j − u)(2i + j − u)}{2i + 2j − u}g_{i+j}, and β_{g_{i}}e_{j} = \frac{(2j − u)(2i + j − u)}{2i + 2j − u}e_{i+j}.

α, β are not compatible with the real structure described next in section 3.3.

3.3. The real case. Consider the real structure Ie_{k} := −e_{−k}.

Remark 3.10. This involution is motivated by the function space representation e_{k} = −i exp(ikx)\frac{d}{dx}.

Proposition 3.11. (g_{IR}, △|_{IR}) and (g_{IR}, g′_{IR}, α, β) are real if and only if c_{l} = \overline{c_{−l−1}} (∀l ∈ ZZ ) or c_{l} = −\overline{c_{−l−1}} (∀l ∈ ZZ ).

Proof. According to Lemma 2.15 (g_{IR}, △_{IR}) and (g_{IR}, g′_{IR}, α, β) are real if and only if r_{b}^{(n)} is real.

(i) Let c_{l} = ±\overline{c_{−l−1}} (∀l ∈ ZZ ). Because of I^{T} f_{k} = −f_{−k} we have to show that r_{b}^{n}(f_{j}) = −\overline{r_{b}^{n}(f_{−j})} (∀j ∈ ZZ ). The relation is satisfied if n/| j. Now let j = mn. We have

r_{b}^{(n)}(f_{mn}) = \frac{1}{n} ∑_{k,l} \frac{c_{l}c_{−k−l−m−1}}{2m + 2l + 1}e_{kn} = \frac{1}{n} ∑_{k,l} \overline{\frac{c_{−l−1}c_{k+l+m}}{2m + 2l + 1}}e_{kn}

= \frac{1}{n} ∑_{k,l} \overline{\frac{c_{l}c_{−k−l−m−1}}{−2m − 2l − 1}}e_{−kn} = −\overline{r_{b}^{n}(f_{−mn})}.

(ii) Let r(f_{j}) = −\overline{r(f_{−j})}. It follows that

∑_{l} \frac{c_{l}c_{−k−l−m−1} − \overline{c_{−l−1}}\overline{c_{k+m+l}}}{2m + 2l + 1} = 0 ∀m.

Because the last equation is true for all m we obtain

c_{l}c_{−k−l−m−1} − \overline{c_{−l−1}}\overline{c_{k+m+l}} = 0 ∀k + m, l.  (1)

In other words there is a constant γ ∈C with

c_{l} = γ\overline{c_{−l−1}} ∀l.  (2)

¿From (1) and (2) we find that γ^{2} = 1, i.e., c_{l} = ±\overline{c_{−l−1}} ∀l.

Remark 3.12. (i) It follows from c_{l} = ±\overline{c_{−l−1}} that c is a finite power series. (ii) Let r_{b}^{n} be real. Then the operator ω_{b}^{n} = r_{b}^{−1} is complex, since b = \frac{1}{c^{2}} is an infinite power series for finite c.

Example 3.13. n = 1, c_{0} = c_{−1} = 1. W_{IR}^{+} has the real base u_{k} = \frac{i}{2}(e_{k} + e_{−k}), v_{k} = \frac{1}{2}(e_{k} − e_{−k}). We obtain r = ∑_{m} \frac{1}{2m+1}(v_{m} − iu_{m}) ⊗ (v_{−m−1} − iu_{−m−1}) + \frac{1}{2m−1}(v_{m} − iu_{m}) ⊗ (v_{−m+1} − iu_{−m+1}) +(\frac{1}{2m+1} + \frac{1}{2m−1})(v_{m} − iu_{m}) ⊗ (v_{−m} − iu_{−m}).

4. The case of smooth functions

In this section we apply Proposition 2.3 and 2.4 to obtain real Lie bialgebras for Lie algebras of smooth functions with fixed zeros on the circle. In [B-M] the authors obtain a class of complex matched pairs for certain Lie algebras of analytic vector fields and indicate an obstruction for the case of real structures. These structures are parametrisized by certain moments b ∈ g′. We obtain additional structures by restriction to subalgebras g_{c} of g and by an enlargement of g′ to g_{c}′ (cf. 4.1) and secondly by application of the construction of proposition 2.4 (cf. 4.2). 4.1. Matched pairs M_{c}. First we introduce the function class D. We identify 2π(respectively, 4π)-periodic functions with functions which are defined on S^{1} (respectively, on S^{1(2)}, the double cover of S^{1}).

Definition 4.1. Let D be the class of real C^{∞}-functions c(x), which are defined on the double cover S^{1(2)} of S^{1} and which have the following properties:

(i) c(x + 2π) = −c(x).

(ii) c(x) has zeros x_{1}, ..., x_{2k} of orders p_{1}, ..., p_{2k} (p_{i} ∈ lN)

with p_{1} + p_{2} + ... + p_{2k} = 2u (u odd). (We remark that because of (i) we have x_{j+k} = x_{j} + 2π, p_{j+k} = p_{j}.)

Example 4.2. c(x) = sin \frac{nx}{2} (n odd). We have k = n, x_{j} = (j − 1)\frac{2π}{n}, p_{j} = 1 (j = 1, ..., 2n).

For the following we fix an element c ∈ D. We consider the points x_{j} also as elements of S^{1} (i.e., x_{j} = x_{j+k}).

Definition 4.3. (i) For c ∈ D, g_{c} is the subspace of C^{∞}(S^{1}) generated by all real C^{∞}-vector fields f for which f(x_{j}) = f′(x_{j}) = ... = f^{(p_{j}−1)}(x_{j}) = 0 ∀j ∈ {1, ..., k}.

(ii) g_{c}′ is the vector space of C^{∞}-functions on S^{1} with poles at x_{j} of order not greater than p_{j}.

Remark 4.4. g_{c} is a Lie subalgebra with odd codimension of the Lie algebra g of all real C^{∞}-vector fields (s. Lemma 4.6 (i)). g_{c} corresponds the Lie subgroup G_{c} of Diff_{+}(S^{1}) with

G_{c} = {ϕ ∈ Diff_{+}(S^{1}) |ϕ(x_{i}) = x_{i}, ϕ′(x_{j}) = ... = ϕ^{(p_{k}−1)}(x_{j}) = 0} .

Now consider the linear maps ω_{c} : g_{c} → g_{c}′ and r_{c} : g_{c}′ → g_{c} defined, respectively, by

ω_{c}(f) = \frac{−2fc′}{c^{3}} + \frac{2f′}{c^{2}} ,

and by

r_{c}(p) = −\frac{1}{4}c(x) ∫_{x}^{x+2π} c(y)p(y)dy.

Lemma 4.5. (i) ω_{c} is a well-defined linear map.

(ii) r_{c} is a well-defined linear map.

(iii) ω_{c}r_{c} = id_{g′}, r_{c}ω_{c} = id_{g}.

Proof. (i) ω_{c} changes the zeros at x_{j} of order ≥ p_{j} into poles of order ≤ p_{j}. The only nontrivial case is a p_{j} -zero at x_{j}. Without loss of generality let x_{j} = 0 and f(x) = a_{1}(x)x^{p_{j}}, c(x) = a_{2}(x)x^{p_{j}} (a_{i}(0) ≠ 0). We obtain

ω_{c}(f) = \frac{−2}{c^{3}}(fc′ − f′c)

= \frac{−2}{x^{3p_{j}}a_{2}^{3}} (p_{j}x^{2p_{j}−1}a_{1}a_{2} + x^{2p_{j}}a_{1}a′_{2} − p_{j}x^{2p_{j}−1}a_{1}a_{2} − x^{2p_{j}}a′_{1}a_{2})

= \frac{1}{x^{p_{j}}} (−\frac{2a_{1}(x)a_{2}′(x)}{a_{2}(x)^{3}} + \frac{2a_{1}′(x)}{a_{2}(x)^{2}}) .  (1)

In the second line of (1) the (p_{j} + 1)-poles for the first and the third summand cancel and so we have a p_{j} -pole. Consequently ω_{c} is well-defined on g_{c}. (ii) Let p ∈ g_{c}′. r_{c}(p) is smooth and 2π-periodic because of Def. 4.1(i). Obviously r_{c}(p) has at least a p_{j} -zero at x_{j}. That is r_{c}(p) ∈ g_{c}. (iii)

ω_{c}r_{c}(f) = −\frac{1}{4}ω_{c} (c(x) ∫_{x}^{x+2π}c(y)p(y)dy) = −\frac{1}{4} \frac{−2c′(x)c(x) ∫_{x}^{x+2π} c(y)p(y)dy}{c(x)^{3}}

−\frac{1}{4} \frac{2c(x)c′(x) ∫_{x}^{x+2π} c(y)p(y)dy − c(x)(c(x + 2π)p(x + 2π) − c(x)p(x))}{c(x)^{2}}

= −\frac{1}{4} \frac{−2 · 2c(x)p(x)}{c(x)^{2}} = p(x)

and

r_{c}ω_{c}(f) = r_{c}(\frac{−2fc′}{c^{3}} + \frac{2f′}{c^{2}} ) = −\frac{1}{4}c(x) ∫_{x}^{x+2π} \frac{−2fc′}{c^{2}} + \frac{2f′}{c} dy

= −\frac{1}{4}c(x) ∫_{x}^{x+2π} 2(\frac{f}{c})′dy = −\frac{1}{4}2c(x) (\frac{f(x + 2π)}{c(x + 2π)} − \frac{f(x)}{c(x)}) = f(x) .

Lemma 4.6. (i) g_{c} is a Lie subalgebra of the Lie algebra g of all real C^{∞}- vector fields on S^{1}.

(ii) g_{c}′ is a dual of g_{c} (cf. Def.2.1) with respect to the bilinear form

⟨f(x), p(x)⟩ = ∫S^{1} f(x)p(x)dx .

(iii) ω_{c} is a 2-cocycle (cf. 2.1).

Proof. (i) One has to show that [f, g] has a zero of order p_{j} at x_{j}. This follows from the formula

[f, g]^{(n)} = fg^{(n+1)} + ∑_{i=0}^{n−1} [\binom{n}{i} − \binom{n}{i + 1}] f^{(i+1)}g^{(n−i)} − f^{(n+1)}g.

(ii) Clearly g_{c}, (respectively, g_{c}′) separates the points of g_{c}′, (respectively, g_{c}). Then K(f)p = fp′ + 2f′p acts invariantly on g because if f has an r^{th}-order zero at x_{j} and if p has an s^{th}-order pol at x_{j} then K(f)p has at most a pole of order s − r − 1 ≥ −1 ≥ −p_{j} at x_{j}. (iii) We have

⟨ω_{c}(f), g⟩ = ∫S^{1} \frac{−2fgc′}{c^{3}} + \frac{2f′g}{c^{2}} dy = ∫S^{1} \frac{−1}{c^{2}}(fg)′ + \frac{2f′g}{c^{2}} dy

= − ∫S^{1} \frac{1}{c^{2}} (fg′ − f′g)dy = − ⟨\frac{1}{c^{2}} , [f, g]⟩ .

The assertion follows from the Jacobi identity.

Remark 4.7. We have ⟨ω_{b}(f), g⟩ = −⟨b, [f, g]⟩ (cf. the proof of Lemma 4.6(iii)). Because of the behavior of c at the poles x_{i}, b = \frac{1}{c^{2}} ∉ g_{c}′ and therefore b is not a moment of g_{c} but we can consider b as a moment of [g_{c}, g_{c}].

Proposition 4.8. Let c(x) ∈ D, and let

[p, q]_{g_{c}}′ = \frac{2c′p + cp′}{4} ∫_{x}^{x+2π} cqdy − \frac{2c′q + cq′}{4} ∫_{x}^{x+2π} cpdy,

α_{f}p = fp′ + 2f′p,

and

β_{p}f = c ∫_{x}^{x+2π} \frac{f(cp′ + 2c′p)}{4}dy + \frac{1}{2}c^{2}fp + \frac{1}{4}(c′f − f′c) ∫_{x}^{x+2π} cpdy.

Then M_{c} = (g_{c}, g_{c}′, α, β) is a real matched pair of Lie algebras.

Proof. The assumptions of Proposition 2.3 are satisfied by Lemmas 4.5 and 4.6. One calculates the above formulas by Proposition 2.3.

Because of [p, q]_{g_{c}}′ = ω_{c}([r_{c}(p), r_{c}(q)]) we can identify g_{c} and g_{c}′. Applying Proposition 2.1.a we obtain a self-dual representation for M_{c}.

Theorem 4.9. Set

α′_{f}g = \frac{1}{2}c(x) ∫_{x}^{x+2π} (\frac{3fg′c′}{c^{2}} + \frac{fgc′′}{c^{2}} − \frac{3fg(c′)^{2}}{c^{3}} + \frac{2f′gc′}{c^{2}} − \frac{2f′g′}{c} − \frac{fg′′}{c} ) dy.

Then (g_{c}, g_{c}, α′, α′) and M_{c} are isomorphic matched pairs of Lie algebras.

Proof. One calculates the above expression for α′ by α′ = rαω.

Example 4.10. Let c(x) = sin \frac{x}{2} . Then g_{c} turns out to be the space of real analytic functions with f(0) = 0 while g_{c}′ is the space of real analytic functions possibly having a simple pole at 0. We obtain r_{c}(p)(x) = sin \frac{x}{2} ∫_{x}^{x+2π} sin \frac{y}{2}f(y)dy and

α′_{f}g = \frac{1}{2}c(x) ∫_{x}^{x+2π} \frac{(3fg′ + 2f′g) cos y}{sin^{2} y} − \frac{fg(1 + 2 cos^{2} y)}{sin^{3} y} − \frac{2f′g′ + g′′f}{sin y} dy.

Remark 4.11. With respect to some weaker topology of the Lie algebra g_{c} and a certain completion g_{c}\overline{⊗}g_{c} of the algebraic tensor product g_{c} ⊗ g_{c}, the matched pairs M_{c} correspond to r-matrices

g_{c}\overline{⊗}g_{c} ∋ r = r(x, y) = −\frac{1}{4}χ(x, y)c(x)c(y), χ(x, y) = \{ \begin{matrix}−1 if y ∈ [0, x) \\ +1 if y ∈ [x, 2π) \end{matrix}

and Lie bialgebras (g_{c}, △ : g_{c} → g_{c}\overline{⊗}g_{c}) with

△(f) = \frac{1}{4}χ(x, y)(c(x)c(y)(f′(x) + f′(y)) − f(x)c′(x)c(y) − f(y)c(x)c′(y)).

4.2. Matched pairs M_{c}^{(n)}. Finally we generalize Proposition 4.8 in the spirit of Proposition 2.4. Fix an element c(x) ∈ D and consider also c_{n}(x) := c(nx) as an element of D. Let c_{n} have zeros x_{j} of order p_{j} (j = 1, ..., 2nk) with p_{1} + ... + p_{2nk} = 2nu and u odd. We have x_{j+k} = x_{j} + \frac{2π}{n} and p_{j} = p_{j+k}. We can consider the elements x_{j} ∈ S^{1(2)} as elements of S^{1} (i.e., x_{j+nk} = x_{j} + 2π).

Definition 4.12. (i) g_{c}^{(n)} is the vector space of C^{∞}-vector fields on S^{1} with zeros x_{j} of order at least p_{j},

(ii) g_{c}^{(n)}′ is the vector space of C^{∞}-functions on S^{1} with poles x_{j} of order at most p_{j},

(iii) h_{c}^{(n)} is the subspace of g_{c}^{(n)} of \frac{2π}{n} -periodic vector fields,

(iv) h_{c}^{(n)}′ is the subspace of g_{c}^{(n)}′ of \frac{2π}{n} -periodic functions.

Lemma 4.13. (i) g_{c}^{(n)}, h_{c}^{(n)} are subalgebras of g.

(ii) With respect to the pairing ⟨f(x), p(x)⟩ = ∫S^{1} f(x)g(x)dx the spaces g_{c}^{(n)}′ , (respectively, h_{c}^{(n)}′) are duals of g_{c}^{(n)} (respectively h_{c}^{(n)}) (cf. Def. 2.1).

(iii) We have the direct decompositions g_{c}^{(n)} = h_{c}^{(n)} +^{.} h_{c}^{(n)⊥}′ and g_{c}^{(n)}′ = h_{c}^{(n)}′ +^{.} h_{c}^{(n)⊥}.

Proof. (i), (ii) See the proof of Lemma 4.6(i),(ii). (iii) Let f ∈ g_{c}^{(n)}. The first identity follows from the decomposition

f(x) = \frac{1}{n} ∑_{j=1}^{n} f(x + \frac{2πj}{n}) + (f(x) − \frac{1}{n} ∑_{j=1}^{n} f(x + \frac{2πj}{n})) .

The proof of the second identity is analogous.

Define linear maps ω_{c}^{(n)} : h_{c}^{(n)} → h_{c}^{(n)}′ and r_{c}^{(n)} : h_{c}^{(n)}′ → h_{c}^{(n)} by setting
ω_{c}^{(n)}(f) = \frac{−2fc_{n}′}{c_{n}^{3}} + \frac{2f′}{c_{n}^{2}} ,

and

r_{c}^{(n)}(p) = −\frac{1}{4}c_{n}(x) ∫_{x}^{x+2π} c_{n}(y)p(y)dy.

Lemma 4.14. ω_{c}^{(n)} is a 2-cocycle and r_{c}^{(n)},ω_{c}^{(n)} are well-defined linear maps with r_{c}^{(n)}ω_{c}^{(n)} = idh_{c}^{(n)} , ω_{c}^{(n)}r_{c}^{(n)} = idh_{c}^{(n)}_{′}.

Proof. See the proof of Lemma 4.5.

Theorem 4.15. Let c ∈ D and set

[p, q] = \frac{2c_{n}′p + c_{n}p′}{4} ∫_{x}^{x+2π} c_{n}(y)q(y)dy − \frac{2c_{n}′q + c_{n}q′}{4} ∫_{x}^{x+2π} c_{n}(y)p(y)dy

if p, q ∈ h_{c}^{(n)}′;

[p, q] = c_{n}^{2}pq − \frac{1}{4}(c_{n}q′ + 2c_{n}′q) ∫_{x}^{x+2π} c_{n}pdy

if p ∈ h_{c}^{(n)}′, q ∈ h_{c}^{(n)⊥};
[p, q] = 0
if p, q ∈ h_{c}^{(n)⊥};
α_{f}p = fp′ + 2f′p ;

and

β_{x}(e) = \{ \begin{matrix}r_{c}^{(n)}P_{h′}αr_{c}^{(n)}(x)ω_{c}^{(n)}(e) & x ∈ h_{c}^{(n)}′, e ∈ h_{c}^{(n)} \\ P_{h′}_{⊥}[r_{c}^{(n)}(x), e] & x ∈ h_{c}^{(n)}′, e ∈ h_{c}^{(n)}′^{⊥} \\ r_{c}^{(n)}(α_{e}x) & x ∈ h_{c}^{(n)}^{⊥}, e ∈ g_{c}^{(n)}. \end{matrix}

Then M_{c}^{(n)} = (g_{c}^{(n)}, g_{c}^{(n)}′, α, β) is a real matched pair of Lie algebras.

Proof. The proof is analogous to Proposition 4.8 and follows from Lemma 4.5, 4.6 and Proposition 2.4.

5. Some problems

Possibly additional structures can be classified by the consideration of additional classes of subalgebras. Our method does not work in the case of subalgebras of odd dimension (e.g. sl(2,C)) and for quasi-triangular r-matrices of subalgebras.

The next step in the direction of a quantization is the integration to matched pairs of Lie groups and Poisson Lie groups.

Another question is whether there are extensions of matched pairs for g_{c} to matched pairs of the Lie algebra of all smooth vector fields on the circle.

References

[B-M] Beggs, E. and S. Majid, Matched pairs of topological Lie algebras corresponding to Lie bialgebra structures on diff(S^{1}) and diff(IR), Ann. Inst. Henri Poincare, 53 no. 1 (1990), 15-34.

[D] Drinfeld, V.G., Quantum Groups, in: “Proceedings of the International Congress of Mathematicians Berkeley 1986”, 1987.

[K1] Köthe, G., “Topologische lineare Räume”, Band 1, (Grundlehren Math. Wiss. Bd. 107), Springer Verlag Berlin Heidelberg New York, 1960.

[K2] Köthe, G., “Topological vector spaces”, Part 2, (Grundlehren Math. Wiss. Bd. 237), Springer Verlag Berlin Heidelberg New York, 1979.

[M] Michaelis, W., A class of infinite-dimensional Lie bialgebras containing the Virasoro algebra, Adv. in Math. 107 (1994), 365-392.

[P-S] Pressley, A., Segal, Gr., “Loop groups”, Oxford University Press, New York 1986.

[T] Taft, E.J., Witt and Virasoro algebras as Lie bialgebras, Journal of Pure and Applied Algebra 87(1993), 301-312.

[Ta] Takhtajan, L.A., Introduction to Quantum groups, in: “Quantum groups”, Proceedings of the 8th Int. Workshop on Math. Phys. in Clausthal, ed. by Doebner, H.-D., Lecture Notes in Physics 370, Springer Verlag Berlin Heidelberg New York, 1990.

[W1] Witten, E., Quantum fields, Grassmannians and algebraic curves, Commun. Math. Phys. 113(1989), 529.

[W2] Witten, E., Coadjoint Orbits of the Virasoro-Algebra, Commun. Math. Phys. 114(1989), 1-53.

Fachbereich Mathematik/Informatik Universität Leipzig Augustusplatz 10 04109 Leipzig Federal Republic of Germany

Received November 1, 1994