Journal of Lie Theory Volume 10 (2000) 383–397 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

Lie quasi-bialgebras with quasi-triangular decomposition

Nicolás Andruskiewitsch and Alejandro Tiraboschi^{∗}

Communicated by K. Schmüdgen

Abstract. A class of Lie bialgebras and Lie quasi-bialgebras related to a triangular decomposition of the underlying Lie algebras is discussed. New examples are presented.

1. Introduction

A Lie bialgebra is a vector space which is simultaneously a Lie algebra and a Lie coalgebra, both structures connected by a cocycle condition. This fundamental concept was introduced by Drinfeld [5] as the infinitesimal counterpart of the notion of Poisson-Lie group: a Lie group which is a Poisson manifold, both structures related by imposing the multiplication to be a Poisson manifold mapping. Poisson-Lie groups appear naturally in deformation-quantization theory. Their quantizations are the quantum groups. The subsidiary notion of Lie quasi-bialgebra was again introduced by Drinfeld in his approach to the quantization of classical solutions of the quantum Yang-Baxter equations [7]. Being more flexible that Lie bialgebras, the context of Lie quasi-bialgebras allows to use twistings, a technical tool that became very useful.

In this article, we present a unified way to endow Lie algebras with additional data (a so-called “triangular decomposition” or “quasi-triangular decomposition”, see Definition 3.1), a Lie bialgebra or Lie quasi-bialgebra cobracket. Then we provide a systematic iterative way of constructing Lie algebras with quasitriangular decomposition, analogous to a construction of Witt [14].

The paper is organized as follows: in §2, we recall the necessary definitions and results, mostly due to Drinfeld. In §3, we introduce the notion of Lie algebra with quasi-triangular decomposition and show that a Lie algebra with (quasi)- triangular decomposition is a factorizable Lie (quasi)-bialgebra. Examples of Lie algebras with triangular decomposition are given in §4: some of them were known, as Kac-Moody Lie algebras [6], extended Heisenberg algebras [4]; some of them are new, e. g. motion Lie algebras with respect to the adjoint representation.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag As a byproduct, we provide new examples of classical r-matrices. In Section §5, we discuss more examples arising form the analogue of Witt’s construction; in particular, we endow many generalized Heisenberg algebras with Lie bialgebra structures. These examples are also new, see however [12].

∗ This work was partially supported by CONICET, CONICOR, SECyT (UNC) and FAMAF (Argentina)

Parts of this paper generalize results from the unpublished preprint [1]. We thank F. Levstein for his interest in this work and B. Enriquez for many interesting conversations during his visit to FaMAF in August 1998. We are also grateful to the referee for his (her) careful reading of the typescript.

2. Preliminaries

For simplicity of the exposition, we shall work over ℂ. We collect in this section the necessary definitions and theorems, due mostly to Drinfeld [7], [5]; see [3] for further properties of Lie quasi-bialgebras. By abuse of notation, ad will mean a representation which is tensor product of copies of the adjoint representation. Definition 2.1. A Lie quasi-bialgebra is a triple (𝔤, δ, ϕ), where 𝔤 is a Lie algebra, δ : 𝔤 → ∧^{2}𝔤 ⊂ 𝔤 ⊗ 𝔤 is a 1-cocycle and ϕ ∈ ∧^{3}𝔤 ⊂ 𝔤 ⊗ 𝔤 ⊗ 𝔤 such that the following axioms hold:

Alt(δ ⊗ id)δ(x) = ad x(ϕ), x ∈ 𝔤;  (1)
Alt(δ ⊗ id ⊗ id)(ϕ) = 0,  (2)

where Alt is the alternation map and “1-cocycle” means that δ is linear and δ([x, y]) = ad x(δ(y)) − ad y(δ(x)). Examples of 1-cocycles are the 1-coboundaries: if r ∈ 𝔤 ⊗ 𝔤, then ∂r : 𝔤 → 𝔤 ⊗ 𝔤, the map given by ∂r(x) := ad x(r) − r, is called the coboundary of r. Furthermore, if ϕ = 0 we say that (𝔤, δ) is a Lie bialgebra. So that equation (1) becomes Alt(δ ⊗ id)δ(x) = 0 and equation (2) is identically satisfied. The equality Alt(δ ⊗ id)δ(x) = 0 is called the co-Jacobi identity. Definition 2.2. Let (𝔤, δ) be a Lie bialgebra. A Lie subalgebra 𝔥 ⊂ 𝔤 is a Lie subbialgebra if δ(𝔥) ⊂ 𝔥 ⊗ 𝔥. Definition 2.3. A Manin pair is a data (𝔭, 𝔭_{1}, 𝔭_{2}), where 𝔭 is a Lie algebra provided with a 𝔭-invariant, symmetric, non degenerate bilinear form < | >: 𝔭 × 𝔭 → ℂ, 𝔭_{1} is an isotropic Lie subalgebra of 𝔭 and 𝔭_{2} is an isotropic subspace of 𝔭 complementary to 𝔭_{1}. That is, 𝔭_{1} ⊕ 𝔭_{2} = 𝔭, < 𝔭_{i}|𝔭_{i} >= 0, i = 1, 2. If 𝔭_{2} is a Lie subalgebra of 𝔭, we say that (𝔭, 𝔭_{1}, 𝔭_{2}) is a Manin triple.

The terminology “Manin pair” is justified as follows: to a Manin pair (𝔭, 𝔭_{1}, 𝔭_{2}) corresponds a Lie quasi-bialgebra structure on 𝔭_{1}; and changing the complementary subspace 𝔭_{2} amounts only to a twisting of this structure, so that up to twisting only the pair (𝔭, 𝔭_{1}) counts. See below for details. We remark that Drinfeld does not fixes the isotropic complement 𝔭_{2} since he is interested in the notion of Lie quasi-bialgebra up to twisting.

We now recall the relation between Manin pairs and Lie quasi-bialgebras. Let (𝔭, 𝔭_{1}, 𝔭_{2}) be a Manin pair such that 𝔭 is finite dimensional. Then there is a Lie quasi-bialgebra structure on 𝔭_{1}. Indeed, the restriction of the bracket to 𝔭_{2} ⊗ 𝔭_{2} → 𝔭 = 𝔭_{1} ⊕ 𝔭_{2} has two components: [ , ]_{2} : 𝔭_{2} ⊗ 𝔭_{2} → 𝔭_{2} and [ , ]_{1} : 𝔭_{2} ⊗ 𝔭_{2} → 𝔭_{1}. Since 𝔭_{1} ≃ 𝔭_{2}^{∗} and 𝔭_{1} ⊗ 𝔭_{1} ≃ (𝔭_{2} ⊗ 𝔭_{2})^{∗}, the first defines by transposition a cobracket δ : 𝔭_{1} → 𝔭_{1} ⊗ 𝔭_{1}; that is, < δ(x)|u ⊗ v >=< x|[u, v] >, x ∈ 𝔭_{1}, u, v ∈ 𝔭_{2}. Similarly, the second defines an element ψ ∈ 𝔭_{1} ⊗ 𝔭_{1} ⊗ 𝔭_{1}. Put ϕ = −ψ. Then (𝔭_{1}, δ, ϕ) is a Lie quasi-bialgebra.

Conversely, let (𝔤, δ, ϕ) be a finite dimensional Lie quasi-bialgebra. Put 𝔭 = 𝔤 ⊕ 𝔤^{∗}, 𝔭_{1} = 𝔤, 𝔭_{2} = 𝔤^{∗} and endow 𝔭 with the canonical scalar product. Let δ^{∗} : 𝔤^{∗} ⊗𝔤^{∗} → 𝔤^{∗} be the transpose of the bracket and let θ : 𝔤^{∗} ⊗𝔤^{∗} → 𝔤 be induced by ϕ. Take δ^{∗} − θ as the commutator in 𝔤^{∗}; it takes values in 𝔤 ⊕ 𝔤^{∗}. Then [x, l] can be uniquely defined for x ∈ 𝔤, l ∈ 𝔤^{∗} so that 𝔭 is a Lie algebra and the scalar product in 𝔭 is invariant. Explicitly, if {x_{i}} is a basis of 𝔤, {x^{i}} is the dual basis in 𝔤^{∗} and

[x_{i}, x_{j}] = c_{ij}^{k}x_{k},  δ(x_{i}) = d_{i}^{jk}x_{j} ⊗ x_{k}, and ϕ = ϕ^{ijl}x_{i} ⊗ x_{j} ⊗ x_{l} (3)

(here and below, summation is assumed for repeated indices), then {x_{i}} ∪ {x^{i}} is a basis of 𝔭 and

\begin{aligned}< x_{i}|x^{j} > & = δ_{i}^{j}, & < x_{i} & |x_{j} >= 0, & < x^{i}|x^{j} > & = 0, \\ [x^{i}, x^{j}] & = d_{k}^{ij}x^{k} − ϕ^{ijl}x_{l}, & [x_{i}, x^{j}] & = d_{i}^{jk}x_{k} + c_{li}^{j}x^{l}. \end{aligned} (4)

It is clear from the preceding discussion that 𝔭_{2} is a Lie subalgebra if and only if ϕ = 0. Example 2.4. We recall that, for a Lie algebra 𝔤 and a 𝔤-module V , the motion Lie algebra 𝔤 ⋉ V is the vector space 𝔤 ⊕ V with the bracket

[(x, u), (y, v)] = ([x, y], x.v − y.u), x, y ∈ 𝔤, u, v ∈ V.

If (𝔤, 0, 0) is the trivial Lie bialgebra with underlying Lie algebra 𝔤 then 𝔭 = 𝔤 ⊕ 𝔤^{∗} is the motion Lie algebra with respect to the coadjoint representation. Remark 2.5. If (𝔤, δ) is a finite dimensional Lie bialgebra and (𝔭, 𝔭_{1}, 𝔭_{2}) is the corresponding Manin triple, there is a one-to-one correspondence between subbialgebras 𝔮 ⊂ 𝔤 and subalgebras 𝔮 of 𝔭_{1} = 𝔤 such that 𝔮^{⊥} ∩ 𝔭_{2} is an ideal of 𝔭_{2}.

Let (𝔤, δ) be a finite dimensional Lie bialgebra and let (𝔭, 𝔭_{1}, 𝔭_{2}) be the corresponding Manin triple. The double of 𝔤 is the Lie bialgebra 𝔡(𝔤) whose underlying Lie algebra is 𝔭 and whose Lie cobracket is ∂r, where r is the image of the canonical element of 𝔤 ⊗ 𝔤^{∗} under the embedding 𝔤 ⊗ 𝔤^{∗} ↪ 𝔡(𝔤) ⊗ 𝔡(𝔤) (the canonical element is e_{i} ⊗ e^{i}, where e_{i} is a basis of 𝔤 and e^{i} is the dual basis in 𝔤^{∗}). Let (𝔮, 𝔮_{1}, 𝔮_{2}) be the Manin triple corresponding to the Lie bialgebra 𝔡(𝔤) and identify 𝔮_{2} with 𝔭 by means of the bilinear form ⟨ | ⟩; the Lie bracket in 𝔮_{2}, denoted [ , ]_{∗}, is

[u, v]_{∗} = [v_{1}, u_{1}] + [u_{2}, v_{2}],

where u_{i}, v_{i} belongs to 𝔭_{i} and the bracket in the right hand side is that of 𝔭. Indeed, ⟨δ(x)|u⊗v⟩ = ∑_{i}(⟨[x, e_{i}]|u⟩⟨e^{i}|v⟩+⟨e_{i}|u⟩⟨[x, e^{i}]|v⟩) = ⟨[x, ∑_{i}⟨e^{i}|v⟩e_{i}]|u⟩+ ⟨[x, ∑_{i}⟨e_{i}|u⟩e^{i}]|v⟩ = ⟨[x, v_{1}]|u⟩ + ⟨[x, u_{2}]|v⟩ = ⟨x|[v_{1}, u]⟩ + ⟨x|[u_{2}, v]⟩. Here, x ∈ 𝔮_{1}, u, v ∈ 𝔮_{2}.

Remark 2.6. Let now (𝔭, 𝔭_{1}, 𝔭_{2}) be a Manin pair with 𝔭 not necessarily finite dimensional. We decompose again [x, y] = [x, y]_{1} + [x, y]_{2} for x, y ∈ 𝔭_{2} and let δ : (𝔭_{2})^{∗} → (𝔭_{2} ⊗ 𝔭_{2})^{∗} be the transpose of [ , ]_{2}. Identifying 𝔭_{1} with a subspace of (𝔭_{2})^{∗}, the space of those x ∈ 𝔭_{1} such that δ(x) ∈ 𝔭_{1} ⊗ 𝔭_{1} is a Lie subalgebra of 𝔭_{1}. Let {x_{i} : i ∈ I} be a basis of 𝔭_{1} and assume that there exists a family {x^{i} : i ∈ I} in 𝔭_{2} such that ⟨x_{i}|x^{j}⟩ = δ_{i}^{j}. If the support of the family (d_{k}^{ij})_{i,j∈I} is finite for each k, then δ(𝔭_{1}) ⊂ 𝔭_{1} ⊗ 𝔭_{1}.

Let ϕ^{ijl} be given by [x^{i}, x^{j}]_{1} = −ϕ^{ijl}x_{l}. If the support of the family (ϕ^{ijl})_{i,j,l∈I} is finite then it defines ϕ ∈ 𝔭_{1} ⊗ 𝔭_{1} ⊗ 𝔭_{1} and (𝔭_{1}, δ, ϕ) is a Lie quasibialgebra. In fact, a weak version would be that ad x(ϕ^{ijl}) ∈ 𝔭_{1} ⊗ 𝔭_{1} ⊗ 𝔭_{1} for any x ∈ 𝔭_{1}.

We now recall the notion of twisting of Lie quasi-bialgebras [7]. If r is an element of 𝔤 ⊗ 𝔤, then set

r̂:= [r^{12}, r^{13}] + [r^{12}, r^{23}] + [r^{13}, r^{23}];  (5)

the identity r̂ = 0 is the classical Yang-Baxter equation (CYBE). If r = r^{ij}x_{i} ⊗x_{j}, then r̂ = r̂^{ijk}x_{i} ⊗ x_{j} ⊗ x_{k}, where, keeping the notation from (3),

r̂^{ijk} = r^{sj}r^{tk}c_{st}^{i} + r^{is}r^{tk}c_{st}^{j} + r^{is}r^{jt}c_{st}^{k}.  (6)

Let (𝔤, δ, ϕ) be a Lie quasi-bialgebra and let r ∈ ∧^{2}𝔤. Put

δ_{r} = δ(x) + ad xr,  ϕ_{r} = ϕ + Alt(δ ⊗ id)r − r̂.  (7)

Then (𝔤, δ_{r}, ϕ_{r}) is also a Lie quasi-bialgebra; we shall say that it is obtained from (𝔤, δ, ϕ) by twisting via r. If (𝔤, δ) is a Lie bialgebra, r̂ = 0 and Alt(δ ⊗ id)r = 0 then (𝔤, δ_{r}) is a Lie bialgebra. These hypotheses hold if r ∈ ∧^{2}𝔤_{0} where 𝔤_{0} is an abelian subalgebra of 𝔤 such that δ(𝔤_{0}) = 0.

Lemma 2.8 below is stated in [7] without proof; we include one for completeness. We need the following elementary linear algebra facts: Remark 2.7. (a). Let W be a vector subspace of a vector space V . Fix a complement U of W in V , i. e. V = W ⊕ U. There is a bijection between the set of all complements of W in V and hom(U, W). Explicitly, if Z is such a complement and x ∈ U then write x = x_{W} + x_{Z}, with x_{W} ∈ W, x_{Z} ∈ Z and define φ_{Z}(x) := x_{W}. Conversely, if φ ∈ hom(U, W) then Z := the image of Φ, where Φ(x) := x − φ(x), is a complement of W.

(b). If in addition V is provided with a non-degenerate symmetric bilinear form ( , ), W is isotropic and admits an isotropic complement U then there is a bijection between the set of all isotropic complements of W in V and {φ ∈ hom(U, W) : (φ(x), y) = −(x, φ(y))}.

(c). If in addition V is finite dimensional, there is a bijection between the set of all isotropic complements of W in V and ∧^{2}W. Explicitly, let (x_{i}) be a basis of W and let x^{j} ∈ U such that (x_{i}, x^{j}) = δ_{i}^{j}. If r = r^{ij}x_{i} ⊗ x_{j} ∈ ∧^{2}W then the subspace V_{r} := ⟨x^{i} + r^{ji}x_{j}⟩ of V is an isotropic complement of W. Lemma 2.8. Let (𝔤, δ, ϕ) be a finite dimensional Lie quasi-bialgebra and let (𝔭, 𝔭_{1}, 𝔭_{2}) be the corresponding Manin pair. Then changing the isotropic complement of 𝔭_{1} amounts to twisting the corresponding Lie quasi-bialgebra. Explicitly, if r ∈ ∧^{2}𝔤 and 𝔲 = 𝔭_{−r} corresponds to −r as described in Remark 2.7 (c), then the Manin pair (𝔭, 𝔭_{1}, 𝔲) corresponds to the Lie quasi-bialgebra (𝔤, δ_{r}, ϕ_{r}). Proof. Let x_{1}, …, x_{n} be a basis of 𝔭_{1} and x^{1}, …, x^{n} be the dual basis in 𝔭_{2}; so equations (3) and (4) hold. Let u^{1}, …, u^{n} be the basis in 𝔲 given by u^{i} := x^{i} + r^{ij}x_{j}. Let r̂ have the same meaning as in (6). Then

\begin{aligned}[u^{i}, u^{j}] & = [r^{il}x_{l} + x^{i}, r^{jk} + x_{k}x^{j}] \\ & = [x^{i}, x^{j}] + r^{jk}[x^{i}, x_{k}] + r^{il}[x_{l}, x^{j}] + r^{il}r^{jk}[x_{l}, x_{k}] \\ & = d_{s}^{ij}x^{s} − ϕ^{ijs}x_{s} + r^{kj}d_{k}^{is}x_{s} + r^{kj}c_{sk}^{i}x^{s} \\ & + r^{il}d_{l}^{js}x_{s} + r^{il}c_{sl}^{j}x^{s} + r^{il}r^{jk}c_{lk}^{s}x_{s} \\ & = d_{s}^{ij}u^{s} + r^{kj}c_{sk}^{i}u^{s} + r^{il}c_{sl}^{j}u^{s} − ϕ^{ijk}x_{k} \\ & − < u^{i} ⊗ u^{j} ⊗ id | Alt(δ ⊗ id)r > +r̂^{ijk}x_{k} \end{aligned}  (8)

It follows that ϕ_{r} = ϕ + Alt(δ ⊗ id)r − r̂ and

\begin{aligned}δ_{r}(x_{t}) & = d_{t}^{ij}x_{i} ⊗ x_{j} + r^{kj}c_{tk}^{i}x_{i} ⊗ x_{j} + r^{il}c_{tl}^{j}x_{i} ⊗ x_{j} \\ & = δ(x_{t}) + ad x_{t}(r^{kj}x_{k}) ⊗ x_{j} + r^{ik}x_{i} ⊗ ad x_{t}(x_{k}) = δ(x_{t}) + ad x_{t}(r). \end{aligned}

Definition 2.9. A Lie quasi-bialgebra (𝔤, δ, ϕ) is quasitriangular if there exists r ∈ 𝔤 ⊗ 𝔤, such that:

1. the coboundary of r is the cobracket of 𝔤, i.e. ∂r = δ, and

2. r̂ = ϕ. (The definition of r̂ is given in formula (5).)

So that if 𝔤 is Lie bialgebra (i.e. ϕ = 0), then it is quasitriangular if and only if ∂r = δ and r satisfies the classical Yang-Baxter equation. The following result is also stated in [7] without proof; a proof appears in [3]. Lemma 2.10. Let (𝔤, δ, ϕ) be a finite dimensional Lie quasi-bialgebra and let (𝔭, 𝔭_{1}, 𝔭_{2}) be the corresponding Manin pair. Let 𝔲 be a subspace of 𝔭 such that 𝔭 = 𝔭_{1} ⊕ 𝔲, and let r ∈ 𝔤 ⊗ 𝔤 be the tensor associated to 𝔲 (i.e. 𝔭_{r} = 𝔲). Then

(a) [𝔭_{1}, 𝔲] ⊂ 𝔲 if and only if the coboundary of r is the cobracket of 𝔤.

(b) [𝔭, 𝔲] ⊂ 𝔲 if and only if the coboundary of r is the cobracket of 𝔤 and r̂ = ϕ. In other words, (𝔤, δ, ϕ) is quasitriangular if and only if 𝔭_{1} admits a complementary ideal in 𝔭.

Proof. (a). Let x_{1}, …, x_{n} be a basis of 𝔤 = 𝔭_{1} and x^{1}, …, x^{n} be a dual basis in 𝔤^{∗} = 𝔭_{2}. Let u^{i} = r^{ji}x_{j} + x^{i}; then [x_{l}, u^{i}] = (r^{ti}c_{lt}^{k} + d_{l}^{ik})x_{k} + c_{sl}^{i}x^{s}. Thus [x_{l}, u^{i}] ∈ 𝔲 if and only if (r^{ti}c_{lt}^{k} + d_{l}^{ik})x_{k} + c_{sl}^{i}x^{s} = α_{sl}^{i}(r^{ks}x_{k} + x^{s}), for some scalars α_{sl}^{i}. This happens if and only if c_{sl}^{i} = α_{sl}^{i} and r^{ti}c_{lt}^{k} + d_{l}^{ik} = c_{sl}^{i}r^{ks}, or d_{l}^{ik} = r^{ti}c_{tl}^{k} + r^{kt}c_{tl}^{i}, for all j, k, l. On the other hand,

∂r(x_{l}) = (r^{ti}c_{lt}^{k} + r^{kt}c_{lt}^{i})x_{k} ⊗ x_{i} = −(r^{kt}c_{tl}^{i} + r^{ti}c_{tl}^{k})x_{k} ⊗ x_{i}.  (9)

That is, [𝔭_{1}, 𝔲] ⊂ 𝔲 if and only if d_{l}^{ik} = r^{ti}c_{tl}^{k} + r^{kt}c_{tl}^{i} for all j, k, l if and only if ∂r(x_{l}) = d^{ki}x_{k} ⊗ x_{i} = δ(x_{l}).

For (b), we can assume that ∂r = δ by (a). We have [x^{l}, u^{k}] = (r^{sk}d_{s}^{tl} − ϕ^{lkt})x_{t} + (r^{sk}c_{sp}^{l} + d_{p}^{lk})x^{p}. Thus [x^{l}, u^{k}] ∈ 𝔲 if and only if [x^{l}, u^{k}] = (r^{sk}c_{sp}^{l} + d_{p}^{lk})(r^{tp}x_{t} + x^{p}), i.e. r^{sk}d_{s}^{tl} − ϕ^{lkt} = r^{sk}c_{sw}^{l}r^{tw} + d_{s}^{lk}r^{ts}. Thus

r^{sk}r^{wl}c_{sw}^{t} + r^{sk}r^{tw}c_{sw}^{l} + r^{sk}r^{tw}c_{ws}^{l} + r^{wl}r^{ts}c_{sw}^{k} + r^{kw}r^{ts}c_{sw}^{l} = ϕ^{lkt}.

The second and the third term cancel because of the antisymmetry of the bracket. Performing some permutations in the others terms, we have

r^{sk}r^{wl}c_{sw}^{t} + r^{ts}r^{wl}c_{sw}^{k} + r^{ts}r^{kw}c_{sw}^{l} = ϕ^{lkt}.

That is, 𝔲 is an ideal if and only if r̂ = ϕ and ∂r = δ.

Definition 2.11. [13]. A quasitriangular Lie quasi-bialgebra (𝔤, r) is factorizable if the map 𝔤^{∗} → 𝔤, α ↦ ⟨α ⊗ id, r + τ(r)⟩, is a bijection, where τ is the usual transposition.

3. Lie algebras with quasi-triangular decomposition

In this section we introduce the notions of “Lie algebra with quasi-triangular decomposition” and “Lie algebra with triangular decomposition”; these definitions are inspired by [6, Ex. 3.2] and are related to but not the same as the notion discussed in [11]. We show that such a Lie algebra has a canonical structure of quasitriangular Lie quasi-bialgebra. We give two proofs of this fact; the second one uses the double and suggests a method of constructing Lie algebras with triangular decomposition. Definition 3.1. Let 𝔤 be a Lie algebra. We shall say that the collection (𝔤_{0}, 𝔤_{+}, 𝔤_{−}, ( |)) is a quasi-triangular decomposition (QTD) of 𝔤 if 𝔤_{0} is a subalgebra of 𝔤, 𝔤_{−}, 𝔤_{+} are subspaces of 𝔤 such that 𝔤 = 𝔤_{−}⊕𝔤_{0}⊕𝔤_{+}, and ( |) : 𝔤×𝔤 → ℂ is a 𝔤-invariant, non degenerate, symmetric, bilinear form such that

0 = (𝔤_{+}|𝔤_{+}) = (𝔤_{−}|𝔤_{−}) = (𝔤_{+}|𝔤_{0}) = (𝔤_{0}|𝔤_{−}).

Furthermore, we shall say that (𝔤_{0}, 𝔤_{+}, 𝔤_{−}, ( |)) is a triangular decomposition (TD) if 𝔤_{0} is abelian, 𝔤_{−}, 𝔤_{+} are subalgebras of 𝔤 and [𝔤_{±}, 𝔤_{0}] ⊂ 𝔤_{±}.

In what follows, we shall simply say “𝔤 is a Lie algebra with quasi-triangular decomposition or triangular decomposition”, without mentioning the data defining it. We shall use the notation x = x_{+} +x_{0} +x_{−} for x ∈ 𝔤, if x_{j} ∈ 𝔤_{j}, j ∈ {+, 0, −}. Theorem 3.1. Let 𝔤 be a finite dimensional Lie algebra with QTD (respectively TD). Then 𝔤 admits a canonical structure of Lie quasi-bialgebra (resp. Lie bialgebra), which is quasitriangular. If 𝔤 has another structure of QTD with the same non-degenerate invariant form then the corresponding structures of Lie quasibialgebra are related by a twist. Proof. Let 𝔭 = 𝔤 × 𝔤 with the product Lie algebra structure, 𝔭_{1} = {(a, a) : a ∈ 𝔤} and 𝔭_{2} = {(a_{−} + a_{0}, a_{+} − a_{0}) : a_{−} ∈ 𝔤_{−}, a_{+} ∈ 𝔤_{+}, a_{0} ∈ 𝔤_{0}}. Then 𝔭_{1} is a Lie subalgebra of 𝔭 and 𝔭_{2} ⊂ 𝔭 is a subspace complementary to 𝔭_{1}. Let < | >: 𝔭×𝔭 → ℂ be the bilinear form defined by < (x, y)|(u, v) >= (x|u) − (y|v). Then < | > is 𝔭-invariant, non degenerate and < 𝔭_{1}|𝔭_{1} >= 0. If (x, y) = (x_{−} + x_{0}, x_{+} − x_{0}) and (u, v) = (u_{−} + u_{0}, u_{+} − u_{0}) belong to 𝔭_{2}, then

< (x, y)|(u, v) >= (x, u) − (y|v) = (x_{0}|u_{0}) − (−x_{0}| − u_{0}) = 0;

that is < 𝔭_{2}|𝔭_{2} >= 0. Hence (𝔭, 𝔭_{1}, 𝔭_{2}) is a Manin pair and 𝔭_{1} ≃ 𝔤 has a structure of Lie quasi-bialgebra. If 𝔤 has a TD, then 𝔭_{2} is a Lie subalgebra of 𝔭 and 𝔤 has a structure of Lie bialgebra. Since 𝔲 = {(0, x) : x ∈ 𝔤} is an ideal complementary to 𝔭_{1}, 𝔤 is quasitriangular by Lemma 2.10.

Let (𝔤′_{0}, 𝔤′_{+}, 𝔤′_{−}, ( |)) be another QTD of 𝔤; its Manin pair is (𝔭, 𝔭_{1}, 𝔭′_{2}), where 𝔭′_{2} = {(a_{−} + a_{0}, a_{+} − a_{0}) : a_{−} ∈ 𝔤′_{−}, a_{+} ∈ 𝔤′_{+}, a′_{0} ∈ 𝔤′_{0}}. The corresponding bilinear form is again < (x, y)|(u, v) >= (x|u) − (y|v); we have two Manin pairs that only differ in the complement of 𝔭_{1}, so Lemma 2.8 applies.

Remark 3.2. δ = 0 if and only if [𝔤_{+}, 𝔤_{+}] = [𝔤_{−}, 𝔤_{−}] = [𝔤, 𝔤_{0}] = 0.

Let 𝔤 = 𝔤_{+} ⊕ 𝔤_{0} ⊕ 𝔤_{−} be a finite dimensional Lie algebra with TD, and consider on 𝔤 the structure of Lie bialgebra provided by Theorem 3.1. Lemma 3.3. (a) 𝔟_{+} = 𝔤_{0} ⊕ 𝔤_{+} and 𝔟_{−} = 𝔤_{0} ⊕ 𝔤_{−} are Lie subbialgebras of 𝔤.

As Lie algebras, 𝔟_{±}^{∗} ≅ 𝔟_{∓}.

(b) 𝔡(𝔟_{+}) is isomorphic as a Lie algebra to the direct product 𝔤 × 𝔤_{0}. Proof. (a). Keep the notation of the proof of Theorem 3.1. The subspace orthogonal to 𝔟_{+} (resp., 𝔟_{−}) in 𝔭_{2} is 0 × 𝔤_{+} (resp., 𝔤_{−} × 0) which is clearly an ideal of 𝔭_{2}, and clearly 𝔭_{2}/(0 × 𝔤_{+}) (resp., 𝔭_{2}/𝔤_{−} × 0)) is isomorphic to 𝔟_{−} (resp., 𝔟_{+}), as Lie algebras. Notice that the pairing ⟨ , ⟩ between 𝔟_{+} and 𝔟_{−} is

⟨x, y⟩ = (x_{0}|y_{0}) + (x|y).  (10)

(b). Let Υ : 𝔡(𝔟_{+}) → 𝔤×𝔤_{0} be the linear isomorphism Υ(x_{+}+x_{0}, y_{0}+y_{−}) = (x_{+} + x_{0} + y_{0} + y_{−}, x_{0} − y_{0}). We want to show that Υ([u, v]) = [Υ(u)Υ(v)]; it suffices to consider u = x ∈ 𝔟_{+}, v = y ∈ 𝔟_{−}. Let us write [x, y] = [x, y]_{1} + [x, y]_{2}, where [x, y]_{1} ∈ 𝔟_{+}, [x, y]_{2} ∈ 𝔟_{−}. We deduce easily from (10) that

[x, y]_{1} = [x, y]_{+} + \frac{1}{2}[x, y]_{0}, [x, y]_{2} = [x, y]_{−} + \frac{1}{2}[x, y]_{0}.  (11)

Indeed, if u ∈ 𝔟_{−}, ⟨[x, y]_{1}, u⟩ = ⟨x, [y, u]⟩ = (x|[y, u]) = ⟨[x, y]_{+} + \frac{1}{2}[x, y]_{0}, u⟩. Now (11) implies our claim.

Let {x_{j} : j ∈ J} be a basis of 𝔤_{+}, {y_{j}} be its dual basis in 𝔤_{−}, {h_{i} : i ∈ I} is an orthonormal basis of 𝔤_{0}. Then the dual basis of B = {x_{j}} ∪ {y_{j}} ∪ {h_{i}} in 𝔭_{2} is constituted by the vectors

x_{j}^{∗} = (y_{j}, 0), y_{j}^{∗} = (0, −x_{j}) and h_{i}^{∗} = \frac{1}{2}(h_{i}, −h_{i}),  j ∈ J, i ∈ I. (12)

Corollary 3.4. Let 𝔤 be a finite dimensional Lie algebra with TD. Then the Lie cobracket on 𝔤 provided by Theorem 3.1 is ∂r_{0}, where, in the notation above,

r_{0} = ∑_{j∈J} x_{j} ⊗ y_{j} + \frac{1}{2} ∑_{i∈I} h_{i} ⊗ h_{i}.  (13)

This gives a new proof of the quasitriangularity of 𝔤.

Proof. Preserve the notation of the preceding proof. The orthogonal of the ideal Υ^{−1}(0 × 𝔤_{0}) is {(u, v) ∈ 𝔡(𝔟_{+}) : u_{0} = v_{0}}, clearly a Lie subalgebra of the dual of 𝔡(𝔟_{+}). Then 𝔡(𝔟_{+})/Υ^{−1}(0 × 𝔤_{0}) ≃ 𝔤 inherits a Lie bialgebra structure and the canonical projection is a morphism of Lie bialgebras. We claim that this Lie bialgebra structure coincides with the structure defined in Theorem 3.1. Let {(u, v) ∈ 𝔡(𝔟_{+}) : u_{0} = v_{0}} → 𝔭_{2} be the application (u, v) ↦ (v, −u); it is easy to check that it is an isomorphism of Lie algebras. Since the introduced isomorphisms preserve the corresponding dualities, the claim follows. Let r be the canonical element of 𝔡(𝔟_{+}). It is easy to see that the image of r under the above projection is r_{0}; the latter satisfies CYBE because the former does.

Corollary 3.5. δ(𝔤_{0}) = 0.

Proof. If H ∈ 𝔤_{0}, then write [H, x_{j}] = ∑_{i} φ_{ji}(H)x_{i}. It follows from the invariance of the bilinear form that [H, y_{j}] = − ∑_{i} φ_{ij}(H)y_{i}. Hence δ(H) = ∑_{j∈J}[H, x_{j}] ⊗ y_{j} + ∑_{j∈J} x_{j} ⊗ [H, y_{j}] = 0.

Corollary 3.6. A finite dimensional Lie bialgebra with TD is factorizable.

Lemma 3.3 and Corollary 3.4 suggest the following method of constructing Lie algebras with TD.

Theorem 3.2. Let 𝔟 be a finite dimensional Lie bialgebra. Consider 𝔟 ⊂ 𝔡(𝔟) the double of 𝔟. Assume that

(a) there exists an abelian subalgebra 𝔥 such that, as vector spaces, 𝔟 = 𝔥⊕[𝔟, 𝔟];

(b) 𝔥^{⊥} = [𝔟^{∗}, 𝔟^{∗}]; there exists an abelian subalgebra 𝔥̃ such that, as vector spaces, 𝔟^{∗} = 𝔥̃ ⊕ [𝔟^{∗}, 𝔟^{∗}], and 𝔥̃^{⊥} = [𝔟, 𝔟];

(c) for any x ∈ 𝔥, there exists a unique x̃ ∈ 𝔥̃ such that ad x̃ coincides with ad x on 𝔟^{∗} ⊂ 𝔡(𝔟).

Given h in 𝔥, let ĥ be the unique element of 𝔥̃ such that ⟨x|ĥ⟩ = ⟨x̃|h⟩, for all x in 𝔥.

Let 𝔤_{+} = [𝔟, 𝔟], 𝔤_{−} = [𝔟^{∗}, 𝔟^{∗}], 𝔤_{0} = {h + ĥ : h ∈ 𝔥}. Then

𝔤 =: 𝔤_{−} ⊕ 𝔤_{0} ⊕ 𝔤_{+}

is a Lie subalgebra of 𝔡(𝔟) with TD. The non-degenerate invariant bilinear form is the one inherited from 𝔡(𝔟).

Proof. First, we remark that if x ∈ 𝔥, u ∈ 𝔟 and w ∈ 𝔟^{∗}, then

< ad(x)u|w >=< [x, u]|w >= − < u|[x, w] >= − < u| ad(x)w > .

By (c) we know that ad(x)w = ad(x̃)w, hence

− < u| ad(x)w >= − < u| ad(x̃)w >= − < u|[x̃, w] >=< ad(x̃)u|w > .

Thus, ad(x̃) coincides with ad(x) on 𝔟.

Let 𝔯 = {x − x̃ : x ∈ 𝔥}, then 𝔯 is an ideal of 𝔡(𝔟). In fact if u ∈ 𝔟, then [u, x − x̃] = [u, x] − [u, x̃] = 0 (see above). In an analogous way, for (c), we obtain that if w ∈ 𝔟^{∗}, then [w, x − x̃] = 0. So, [𝔯, 𝔡(𝔟)] = 0 and clearly 𝔯 is an ideal.

Let z ∈ 𝔯^{⊥}, so z = u+w, u ∈ 𝔟, w ∈ 𝔟^{∗} and < u|x−x̃ > + < w|x−x̃ >= 0 for all x ∈ 𝔥. For (a) and (b) we have u = h_{1} + u_{1} and w = s_{1} + w_{1}, with h_{1} ∈ 𝔥, u_{1} ∈ [𝔟, 𝔟], s_{1} ∈ 𝔥̃ and w_{1} ∈ [𝔟^{∗}, 𝔟^{∗}]. So, 0 =< z|x − x̃ >=< h_{1} + u_{1}| − x̃ > + < s_{1} + w_{1}|x >=< h_{1}| − x̃ > + < s_{1}|x > . Thus, < h_{1}|x̃ >=< s_{1}|x > for all x ∈ 𝔥, then by hypothesis s_{1} = ĥ_{1}. This implies that

𝔯^{⊥} = 𝔤_{−} ⊕ 𝔤_{0} ⊕ 𝔤_{+} = 𝔤.

As 𝔯 is an ideal, 𝔤 is also an ideal. In particular, 𝔤 is a subalgebra of 𝔡(𝔟).

4. Examples.

Example 4.1. A Lie algebra 𝔤 with TD such that 𝔤_{0} = 0 is equivalent to a Manin triple (𝔤, 𝔤_{+}, 𝔤_{−}). Example 4.2. Let 𝔤 be a Lie bialgebra with TD. If we twist via r ∈ ∧^{2}𝔤_{0} then (𝔤, δ_{r}) is a Lie bialgebra (use Corollary 3.5). Example 4.3. Let 𝔤 be a Lie algebra and let < | > be a non degenerate invariant bilinear form on 𝔤. Then (𝔤, 0, 0, < | >) is a QTD. It is a TD if and only if 𝔤 is abelian. Example 4.4. Let 𝔤 = 𝔨 ⊕ 𝔭 be the complexification of a decomposition of a real simple Lie algebra. It is known that the representation of 𝔨 on 𝔭 is either irreducible or a direct sum of two irreducible components; in the latter case, the corresponding symmetric space is hermitian. See [9]. Assume we are in the hermitian case, i. e. that 𝔭 = 𝔭_{1}⊕𝔭_{2} as 𝔨-modules, with 𝔭_{1} and 𝔭_{2} irreducible. Then (𝔨, 𝔭_{1}, 𝔭_{2}, ( |)), where ( |) is the Killing form, is a QTD. It is seldom a TD; only when 𝔤 = sl(2, ℂ). Example 4.5. Let A be a symmetrizable complex matrix of size n× and let D = (d_{1}, … d_{n}) be an invertible diagonal matrix such that DA = A^{t}D. Let 𝔤̃ = 𝔤̃(A) be the Lie algebra defined in [10, §1.2] and let 𝔤 = 𝔤(A) be the corresponding contragradient Lie algebra [10, Ch. 1]. We preserve the notation 𝔥, 𝔫̃^{±}, e_{i}, f_{i}, α_{i}^{∨}, etc. from loc cit. Let h_{i} = d_{i}α_{i}^{∨}. Let 𝔯 be the unique maximal ideal among the ideals intersecting 𝔥 trivially; then 𝔤 ≃ 𝔤̃/𝔯. Then 𝔤 has a well-known triangular decomposition, cf [10, 1.2, 2.2], which gives rise to a Lie bialgebra structure by the method of Proposition 3.1. It is well-known [6] that the corresponding cobracket is given by

δ(h_{i}) = 0, δ(e_{i}) = \frac{1}{2}(e_{i} ⊗ h_{i} − h_{i} ⊗ e_{i}) and δ(f_{i}) = \frac{1}{2}(f_{i} ⊗ h_{i} − h_{i} ⊗ f_{i}).(14)

Alternatively, it is not difficult to see that formula (14) determines a Lie bialgebra structure on 𝔤̃ or 𝔤. Indeed, [2, Ch. II §2 Prop. 8] allows to define the 1-cocycle on 𝔤̃ or 𝔤. The co-Jacobi identity is also easy to check; it suffices to verify it on generators.

Example 4.6. Let 𝔤 be a Lie algebra and (𝔥, 𝔫_{+}, 𝔫_{−}, ( |)) a triangular decomposition of 𝔤. Let 𝔩 = 𝔤 × 𝔤 be the motion Lie algebra with respect to the adjoint representation, cf. Example 2.4. Take

𝔩_{0} = 𝔥 × 𝔥, 𝔩_{+} = 𝔫_{+} × 𝔫_{+}, and 𝔩_{−} = 𝔫_{−} × 𝔫_{−}.

Thus 𝔩 = 𝔩_{+} ⊕ 𝔩_{0} ⊕ 𝔩_{−}. Let k_{𝔩}( |) : 𝔩 × 𝔩 → ℂ be defined by

k_{𝔩}((x, y)|(u, v)) = (x|u) + (y|u) + (x|v).

Then (𝔩_{0}, 𝔩_{+}, 𝔩_{−}, k_{𝔩}(|)) is a TD of 𝔩.

Now, we assume that 𝔤 is a simple Lie algebra and 𝔤 = 𝔫_{+} ⊕ 𝔥 ⊕ 𝔫_{−} is the usual decomposition, where 𝔥 is a Cartan subalgebra, 𝔫_{±} is the span of the positive, resp. negative, root vectors. In this context, (|) will be the Killing form. Let A be the Cartan matrix of 𝔤, Φ be the root system of 𝔤, Φ^{+} be the set of positive roots and Π = {α_{1}, …, α_{n}} be the set of simple roots. We choose a_{α} ∈ 𝔤_{α} − {0}, α ∈ Φ, (𝔤_{α} is the root space) and H_{i}, m_{i} ∈ 𝔥 such that:

(H_{i}|H) = α_{i}(H), ∀H ∈ 𝔥, [a_{α_{i}}, a_{−α_{i}}] = H_{i}, (a_{α}|a_{−α}) = 1, (H_{i}|m_{j}) = δ_{ij}.

Let us consider the following elements of 𝔩:

\begin{aligned}x_{α} & = (a_{α}, 0), & y_{α} & = (0, a_{−α}), & u_{α} & = (0, a_{α}), & v_{α} & = (a_{−α}, −a_{−α}), α ∈ Φ, \\ h_{i} & = (H_{i}, 0), & l_{i} & = (m_{i}, 0), & r_{i} & = (0, H_{i}), & s_{i} & = (m_{i}, −m_{i}) i = 1, …, n. \end{aligned}

Then it is clear that {x_{α}, u_{α}}_{α∈Φ^{+}} (resp. {h_{i}, r_{i}}_{1≤i≤n}) is a basis of 𝔩_{+} (resp.𝔩_{0}), whose dual basis is {y_{α}, v_{α}}_{α∈Φ^{+}} (resp. {l_{i}, s_{i}}_{1≤i≤n}).

Applying (4) and (12), we obtain the cobracket δ:

δ(x_{±α_{i}}) = \frac{1}{2}x_{±α_{i}} ∧ r_{i} + \frac{1}{2}u_{±α_{i}} ∧ (h_{i} − r_{i}) δ(u_{±α_{i}}) = \frac{1}{2}u_{±α_{i}} ∧ r_{i} δ(h_{i}) = δ(r_{i}) = 0.

The corresponding r-matrix is given by

r_{0} = ∑_{α∈Φ^{+}} x_{α} ⊗ y_{α} + u_{α} ⊗ v_{α} + \frac{1}{2} ∑_{i,t}(m_{t}|m_{i})(h_{i} ⊗ h_{t} − r_{i} ⊗ r_{t})  (15)

Note that (15) is a new example of a classical r-matrix.

Remark 4.7. Let V be a 𝔤-module and consider the motion Lie algebra 𝔤⊕V , i. e. with the Lie bracket given by [(x, y), (u, v)] = ([x, u], xv − uy). Suppose that 𝔤 ⊕ V admits a non degenerate invariant bilinear form ( |). Then (V |𝔤V) = 0. If V is irreducible and non trivial, (V |V) = 0 and we obtain a monomorphism of 𝔤-modules V ↪ 𝔤^{∗}. Assume that 𝔤 is simple: then V ≃ 𝔤^{∗}. If in addition 𝔤 is finite dimensional, identify 𝔤^{∗} with 𝔤 via the Killing form. Then any invariant non-degenerate bilinear form on 𝔤 ⊕ V is a(x|u) + b(y|u) + b(x|v), for some scalars a, b. Let c be a scalar and let T_{c} be the Lie algebra automorphism of 𝔤 ⊕ V , T_{c}((x, v)) = (x, cv). By using an appropriate T_{c}, we may assume that an invariant non-degenerate bilinear form on 𝔤 ⊕ V is a multiple of the one considered in Example 4.6.

Example 4.8. The extended Heisenberg algebras have a quasitriangular Lie bialgebra structure considered in [4] as well as their quantizations. It is easy to see that the Lie bialgebra structure arises from a TD; see Example 5.6 below.

Example 4.9. Let L = ℂ[t, t^{−1}] be the algebra of Laurent polynomials in t. Recall that the residue of a Laurent polynomial P is defined by Res P = the coefficient of P at degree −1. Let ϕ : L × L → ℂ be defined by ϕ(P, Q) = Res \frac{dP}{dt} Q. Then

ϕ(P, Q) = −ϕ(Q, P),  (16)

ϕ(P Q, R) + ϕ(QR, P) + ϕ(RP, Q) = 0 (P, Q, R ∈ L).  (17)

Let 𝔤 be a Lie algebra with QTD. As in [10], consider the loop algebra L(𝔤) := L ⊗ 𝔤, with the bracket [ , ]_{0} given by [P ⊗ x, Q ⊗ y]_{0} = P Q ⊗ [x, y], P, Q ∈ L, x, y ∈ 𝔤. Let ψ : L(𝔤) × L(𝔤) → ℂ, ψ(P ⊗ x, Q ⊗ y) = (x|y)ϕ(P, Q). It is easy to check, using (16), (17) and the symmetry and invariance of ( |)), that ψ is a 2-cocycle on L(𝔤):

ψ(a, b) = −ψ(b, a),
ψ([a, b]_{0}, c) + ψ([b, c]_{0}, a) + ψ([c, a]_{0}, b) = 0, a, b, c ∈ L(𝔤).

Denote by L̃(𝔤) the extension of the Lie algebra L(𝔤) by a 1-dimensional center, associated to the cocycle ψ. Explicity, L̃(𝔤) = L(𝔤)⊕ℂK and the bracket is given by

[a + λ_{1}K, b + λ_{2}K]_{1} = [a, b]_{0} + ψ(a, b)K, a, b ∈ L(𝔤); λ_{1}, λ_{2} ∈ ℂ.

The derivation t\frac{d}{dt} : L → L extends to a derivation of L(𝔤) by t\frac{d}{dt}(x ⊗ P) = x ⊗ t\frac{d}{dt}P. Let L̂(𝔤) be the Lie algebra obtained by adjoining to L̃(𝔤) a derivation D which acts on L(𝔤) as t\frac{d}{dt} and which kills K. In other words, L̂(𝔤) is L̂(𝔤) = L(𝔤) ⊕ ℂK ⊕ ℂD with the bracket

[a + λ_{1}K + μ_{1}D, b + λ_{2}K + μ_{2}D] = [a, b]_{0} + ψ(a, b)K + μ_{1}D(b) − μ_{2}D(a),

a, b ∈ L(𝔤),λ_{1}, λ_{2}, μ_{1}, μ_{2} ∈ ℂ. Even more explicity

[x ⊗ t^{m} + λ_{1}K + μ_{1}D, y ⊗ t^{n} + λ_{2}K + μ_{2}D] = [x, y] ⊗ t^{m+n} + mδ_{m,−n}(x|y)K + μ_{1}ny ⊗ t^{n} − μ_{2}mx ⊗ t^{m},

x, y ∈ 𝔤, m, n ∈ ℤ, λ_{1}, λ_{2}, μ_{1}, μ_{2} ∈ ℂ. We extend the form ( |) to a form ( |)_{t} on L(𝔤) by:

(P ⊗ x|Q ⊗ y)_{t} = Res(t^{−1}P Q)(x|y).

Then we extend further ( |)_{t} to a bilinear symmetric form ( |) on L̂(𝔤) imposing (K|D) = 1, (K|L(𝔤) ⊕ K) = 0 and (D|L(𝔤) ⊕ D) = 0. It is easy to see that ( |) is non degenerate and L̂(𝔤)-invariant (see [10, p. 102]).

We see, with all these coventions, that here are two QTD of L̂(𝔤), namely (G_{0}, G_{+}, G_{−}, ( |)) and (G_{0}, L_{+}, L_{−}, ( |)), where

G_{+} = 𝔤_{+} ⊗ ℂ[t, t^{−1}] ⊕ 𝔤_{0} ⊗ tℂ[t],
G_{−} = 𝔤_{−} ⊗ ℂ[t, t^{−1}] ⊕ 𝔤_{0} ⊗ t^{−1}ℂ[t^{−1}],
G_{0} = 𝔤_{0} ⊗ 1 ⊕ ℂK ⊕ ℂD,
L_{+} = (𝔤_{−} + 𝔤_{0}) ⊗ tℂ[t] ⊕ 𝔤_{+} ⊗ ℂ[t],
L_{−} = (𝔤_{+} + 𝔤_{0}) ⊗ t^{−1}ℂ[t^{−1}] ⊕ 𝔤_{−} ⊗ ℂ[t^{−1}].

If 𝔤 is a Lie algebra with TD then these are TD of L̂(𝔤). It can be shown that these two QTD give rise to Lie quasi-bialgebra structures which are topological twistings of each other.

5. A variation of Witt’s construction

We now discuss a family of examples arising from a construction due to Witt [14], see also [8]. Let 𝔤 be a Lie algebra, let V be a vector space and V^{∗} be the dual vector space of V . Let π : 𝔤 → End(V) and ρ : 𝔤 → End(V^{∗}) be representations of 𝔤 and denote 𝔤̃ = V^{∗} ⊕ 𝔤 ⊕ V . Lemma 5.1. Let β : V × V^{∗} → 𝔤 be a bilinear form. Then the bracket

[(λ, x, v), (λ′, x′, v′)] = (ρ(x)λ′ − ρ(x′)λ, [x, x]′ + β(v, λ′) − β(v′, λ), π(x)v′ − π(x′)v), (18)

λ, λ′ ∈ V^{∗}, x, x′ ∈ 𝔤 and v, v′ ∈ V , defines a Lie algebra structure on 𝔤̃ if and only if for all x ∈ 𝔤, v, v′ ∈ V and λ, λ′ ∈ V^{∗}

[x, β(v, λ)] = β(π(x)v, λ) + β(v, ρ(x)λ),  (19)

π(β(v, λ))v′ = π(β(v′, λ))v and ρ(β(v, λ))λ′ = ρ(β(v, λ′))λ.  (20)

Proof. The antisymmetry of the bracket (18) is evident. A straightforward computation shows that the Jacobi identity is equivalent to (19), (20).

Lemma 5.2. Suppose that 𝔤 is provided with a 𝔤-invariant nondegenerate symmetric bilinear form < | >. Let < | >: V × V^{∗} → ℂ be the canonical bilinear form. Extend these forms to 𝔤̃ in the following way

< λ + x + v|λ′ + x′ + v′ >=< v′|λ > + < x|x′ > + < v|λ′ >,  (21)

x, x′ ∈ 𝔤, v, v′ ∈ V , λ, λ′ ∈ V^{∗}.

Suppose that ρ = π^{∗} with respect to the form on V × V^{∗} and define β : V × V^{∗} → 𝔤 by

< β(v, λ)|x >=< π(x)v|λ >, (x ∈ 𝔤, v ∈ V, λ ∈ V^{∗}).  (22)

Then the bracket (18) defines a Lie algebra structure on 𝔤̃ if and only if the equations (20) hold. In such case, the form < | > on 𝔤̃ is 𝔤̃-invariant.

Proof. It is clear that if v ∈ V and λ ∈ V^{∗}, then [v, λ] ∈ 𝔤 is the unique element such that < x|[v, λ] >=< π(x)v|λ > for all x ∈ 𝔤. Hence, (22) implies (19). Thus the bracket (18) define a Lie algebra structure if and only if (20) hold. Let us check that the 𝔤̃-invariance of (21): let x ∈ 𝔤, v ∈ V and λ ∈ V^{∗}, then

< x|[v, λ] >=< β(v, λ)|x >=< π(x)v|λ >=< [x, v]|λ >

and

< v|[x, λ] >=< v|ρ(x)λ >= − < β(v, λ)|x >= − < π(x)v|λ >=< [v, x]|λ > .

We can deduce the other cases from the definition of < | > and the invariance of the form on 𝔤.

Remark 5.3. Instead of defining β by (22), we could define π by the formula (22); then we should check that π is a representation of 𝔤. Corollary 5.4. Let 𝔤 be a Lie algebra with QTD (respectively, with TD). Let π : 𝔤 → V be a representation and let ρ, β be as in Lemma 5.2. Then (𝔤_{0}, 𝔤_{+} ⊕ V^{∗}, 𝔤_{−} ⊕ V, < | >) is a QTD (respectively, TD) of 𝔤̃ if and only if the equations (20) hold. In such case, the motion Lie algebra 𝔤 ⊕ V is a subbialgebra of 𝔤̃. Proof. We leave the first part to the reader. Let (𝔭, 𝔭_{1}, 𝔭_{2}) be the Manin triple associated to (𝔤̃, δ) as in Theorem 3.1. Clearly, 𝔮 = {(t, t) : t ∈ 𝔤 ⊕ V } is a subalgebra of 𝔭_{1} and 𝔮^{⊥} ∩ 𝔭_{2} = {(v, 0) : v ∈ V } is an ideal of 𝔭_{2}, so 𝔤 ⊕ V is a subbialgebra of 𝔤̃.

Example 5.5. We preserve the notation above. We assume that 𝔤 is a finite dimensional semisimple Lie algebra, the invariant bilinear form is the Killing form and V a finite dimensional representation of 𝔤. Let C_{𝔤} be the value of the action of the Casimir element on the adjoint representation and assume that the action of the Casimir element on V has a single eigenvalue C_{V} . Then equations (20) hold whenever

\frac{2 dim V}{dim 𝔤} + \frac{C_{𝔤}}{C_{V}} = 2.

Indeed, let M = V ⊕ V^{∗} and let ψ be the symmetric bilinear form on M which restricted to V ×V^{∗} is the usual evaluation and such that V and V^{∗} are isotropic. It is clearly 𝔤-invariant. On the other hand, it is clear that the Casimir element acts on V^{∗} and a fortiori on M with a single eigenvalue C_{V} . The claim then follows from [8, Th. 12.1]. Example 5.6. We now consider the opposite situation to the example above. Let 𝔤 be a finite dimensional Lie algebra and let π : 𝔤 → End(V) be a finite dimensional representation. Let 𝔩 = 𝔤⊕𝔤^{∗} be the motion Lie algebra corresponding to the coadjoint representation. We extend π to a representation of 𝔩 of the same name by letting 𝔤^{∗} act by 0. The bilinear form on 𝔩 given by evaluation between 𝔤 and 𝔤^{∗}, and such that 𝔤 and 𝔤^{∗} are isotropic, is invariant (e. g. by Example 2.4). Let β : V × V^{∗} → 𝔤^{∗} be the bilinear map given by < β(v, λ)|x >=< π(x)v|λ >, v ∈ V , λ ∈ V^{∗}, x ∈ 𝔤. Then equations (20) hold because 𝔤^{∗} ⊂ ker π; therefore 𝔩̃:= V^{∗} ⊕ 𝔩 ⊕ V has a Lie algebra structure by Lemma 5.2. Furthermore, if 𝔤 has a TD then 𝔩 also does by Example 4.6 (note that the bilinear form considered in Example 4.6 is not the same as the one coming from Example 2.4; however (22) holds for both). By Corollary 5.4, 𝔩̃ also has a TD.

Lemma 5.7. The Lie subalgebra 𝔥 = V^{∗} ⊕ 𝔤^{∗} ⊕ V ; it is a of 𝔩̃ is a Lie subbialgebra.

Proof. Let (𝔭, 𝔭_{1}, 𝔭_{2}) be the Manin triple corresponding to 𝔩̃ as constructed in Theorem 3.1. By Remark 2.5, it is enough to show that 𝔥 ∩ 𝔭_{2} is an ideal of 𝔭_{2}. This is not difficult to see using the definitions.

Notice that 𝔥 is a two-step nilpotent Lie algebra, or Heisenberg-type Lie algebra since Heisenberg Lie algebras correspond to the case dim 𝔤 = 1. Hence the procedure just described allows to obtain many new Lie bialgebras with underlying Lie algebra of Heisenberg-type and to provide many new examples of factorizable Lie bialgebras.

References

[1] Andruskiewitsch, N., F. Levstein and A. Tiraboschi, Lie bialgebras with triangular decomposition, Preprint ICTP IC/92/117.

[2] Bourbaki, N., “ Groupes et algèbres de Lie,” Chapitres 7 et 8. Hermann, Paris, 1975.

[3] Bangoura, M. and Kosmann-Schwarzbach, Y., The double of a Jacobian quasi-bialgebra, Lett. Math. Phys. 28 (1993), 13–29.

[4] Celeghini, E. , R. Giachetti, E. Sorace and M.Tarlini, Three-dimensional quantum groups from contraction of SU(2)_{q}, J. Math. Phys. 31 (1990), 2548–2551. The quantum Heisenberg group H(1)_{q}, J. Math. Phys. 32 (1991), 1155–1158.

[5] Drinfeld, V. G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983), 68–71.

[6]  —, Quantum groups, Proc. of the ICM, Berkeley (1986), 798–820.
[7]  —, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419–1457.

[8] Hannabuss, K. C., A Quantum Witt Construction, J. Alg. 205 (1998), 141–167.

[9] Helgason S., “Differential geometry, Lie groups and symmetric spaces, ” Academic Press, New York, San Francisco, London, (1978).

[10] Kac V., “Infinite dimensional Lie algebras,” Prog. in Math. 44, Birkhauser, Boston, (1983).

[11] Moody, R., and A. Pianzola, “Lie algebras with triangular decomposition,” Wiley-Interscience, New York, 1995.

[12] Rieffel, M., Some solvable quantum groups, Arveson, W. B. (ed.) et al., Operator algebras and topology. Proceedings of the OATE 2 conference, Bucharest, Romania, 1989. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 270 (1992), 146–159.

[13] Reshetikhin, N., and M. Semenov-Tian-Shansky, Quantum R-matrices and factorization problems, J. Geom. Phys. 5 (1988), 533–550.

[14] Witt, E., Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe, Abh. Math. Sem. Univ. Hamburg 14 (1941), 289–337.

FAMAF Universidad Nacional de Córdoba Ciudad Universitaria (5000) Córdoba Argentina andrus@famaf.unc.edu.ar, tirabo@famaf.unc.edu.ar

Received June 15, 1999 and in final form February 10, 2000