Double quantization on coadjoint representations of simple Lie groups and its orbits

Joseph Donin^{∗}

1 Introduction

Quantum groups can be considered as symmetry objects of certain “quantum spaces” described by noncommutative algebra of functions. This point of view was developed, for example, in [RTF] and [Ma]. Here we study the inverse problem: given the quantum group corresponding to a Lie group G, we want to define a “quantum space” corresponding to a given classical G-manifold.

Let M be a manifold with an action of a Lie group G, 𝔤 the Lie algebra of G, and U_{h}(𝔤) the quantized universal enveloping algebra. Let A be the sheaf of function algebras on M. It may be a sheaf of smooth, analytic, or algebraic functions. For shortness, we simply call A a function algebra. The algebra A is of course invariant under the induced action of the bialgebra U(𝔤).

We consider the following two general problems.

The first problem. Does there exist a deformation quantization, A_{h}, of A, which is invariant under the action of the quantum group U_{h}(𝔤)?

The second problem. Suppose A_{t} is a U(𝔤) invariant quantization of A. Does there exist a two parameter quantization, A_{t,h}, of A such that A_{t,0} = A_{t}, which is invariant under U_{h}(𝔤)?

In this paper, we study the first and the second problems for two cases. The first case, when M is the coadjoint representation of a simple Lie group. The second case, when M is a semisimple orbit in this representation. This paper is motivated by papers [Do2] and [DGS] where we started to study these problems. In this paper we develop results of [Do2] and [DGS] and present some additional results.

The paper is organized as follows.

In Section 2 we recall some facts about quantum groups and related categories, which are essential for a strict formulation of our problems and for our approach to U_{h}(𝔤) invariant quantization of algebras. In particular, we use the Drinfeld category with nontrivial associativity constraint determined by an invariant element Φ_{h} ∈ U(𝔤)^{⊗3}[[h]] and show that the problem of U_{h}(𝔤) invariant quantization is equivalent to the problem of deforming the function algebra in such a way that the deformed algebra to be G invariant and Φ_{h} associative (see Subsection 2.3).

∗This research is partially supported by Israeli Academy Grant N 8007/99 and by the Emmy Noether Institute and the Minerva Foundation of Germany

Subsection 2.4 is very important for the paper. In this subsection we give, for all commutative algebras, a description of Poisson brackets related to one and two parameter U_{h}(𝔤) invariant quantizations. We show the following. If A_{h} is a U_{h}(𝔤) invariant quantization, the corresponding Poisson bracket, p, on M has to be a difference of two brackets, p = f − r_{M}. Here r_{M} is the so called r-matrix bracket obtained from a classical r-matrix r ∈ ∧^{2}𝔤 with the help of the action morphism 𝔤 → Vect(M). So, the Schouten bracket [[r_{M}, r_{M}]] is equal to the image φ_{M} of the invariant element φ ∈ ∧^{3}𝔤. The bracket f is U(𝔤) invariant and such that [[f, f]] = −φ_{M}. Of course, any invariant bracket, f, is compatible with r_{M}, so that [[p, p]] = 0.

We see that for existence of the family A_{h} one needs existence of an invariant bracket f on M such that

[[f, f]] = −φ_{M}.  (1.1)

Note that the manifold M endowed with the bracket p = f −r_{M} is a Poisson manifold with a Poisson action of G, where G is considered to be the Poisson-Lie group with Poisson structure defined by r. We shall not use this fact in the paper.

Similarly, given a two parameter quantization, A_{t,h}, a pair of compatible Poisson brackets is determined. These brackets are: the bracket p = f − r_{M} considered above and a U(𝔤) invariant Poisson bracket, s, the initial term of the U(𝔤) invariant quantization A_{t}. We may perceive the family A_{t,h} as a U_{h}(𝔤) invariant quantization of the Poisson bracket s.

We assume that s is given in advance and determined, for example, by a G invariant simplectic structure on M. From the compatibility of p and s (this means [[p, s]] = 0) follows that

[[f, s]] = 0.  (1.2)

So, for existence of the family A_{t,h} one needs existence of an invariant bracket f on M such the both equations (1.1) and (1.2) hold.

Thus, our problems divide into two steps. The first step is looking for invariant brackets f on M satisfying either (1.1) (in case of the first problem) or both (1.1) and (1.2) (in case of the second problem). The second step is quantizing these brackets.

In Section 3 we consider the one and two parameter quantization on M = 𝔤^{∗}, the coadjoint representation of a simple Lie algebra 𝔤. As a function algebra on 𝔤^{∗}, we take the symmetric algebra S𝔤. It turns out that the cases 𝔤 = sl(n) and 𝔤 ≠ sl(n) are quite different.

We prove that for 𝔤 ≠ sl(n) the two parameter family which is a U_{h}(𝔤) invariant quantization of the Lie bracket on S𝔤 does not exist. Moreover, as a conjecture we state that in this case even a one parameter U_{h}(𝔤) invariant quantization of S𝔤 does not exist.

In the case 𝔤 = sl(n), the two parameter quantization of S𝔤 exists. Moreover, the picture looks like in the classical case. Recall that in the classical case, the natural one parameter U(𝔤) invariant quantization of S𝔤 is given by the family (S𝔤)_{t} = T (𝔤)[t]/J_{t}, where J_{t} is the ideal generated by the elements of the form x ⊗ y − σ(x ⊗ y) − t[x, y], x, y ∈ 𝔤, σ is the permutation. By the PBW theorem, (S𝔤)_{t} is a free module over ℂ[t]. We have (S𝔤)_{0} = S𝔤, so this family of quadratic-linear algebras gives a U(𝔤) invariant quantization of S𝔤. It is obvious that the Poisson bracket, s, related to this quantization is the Lie bracket on 𝔤^{∗}.

We show that for 𝔤 = sl(n) this picture can be extended to the quantum case. Namely, there exist deformations, σ_{h} and [·, ·]_{h}, of both the mappings σ and [·, ·] such that the two parameter family of algebras (S𝔤)_{t,h} = T (𝔤)[[h]][t]/J_{t,h}, where J_{t,h} is the ideal generated by the elements of the form x ⊗ y − σ_{h}(x ⊗ y) − t[x, y]_{h}, x, y ∈ 𝔤, gives a U_{h}(𝔤) invariant quantization of the Lie bracket s on 𝔤^{∗}. In this case, the corresponding bracket f from (1.2) is a quadratic bracket which is, up to a factor, a unique nontrivial invariant map ∧^{2}𝔤 → S^{2}𝔤.

Taking t = 0 we obtain the family (S𝔤)_{h} which is a quadratic algebra over ℂ[[h]]. This algebra can be called the quantum symmetric algebra (or quantum polynomial algebra on 𝔤^{∗}). We show (Subsection 3.4) that (S𝔤)_{h} can be included in the deformed graded differential algebra (deformed de Rham complex). In Subsection 3.5 we prove that the family (S𝔤)_{t,h} can be restricted on the maximal semisimple orbits in 𝔤^{∗} to give a two parameter quantization on these orbits.

In Section 4 we study the problems of one and two parameter quantization on semisimple orbits in 𝔤^{∗} for all simple Lie algebras 𝔤. First of all, we classify all the brackets f satisfying (1.1) and both (1.1) and (1.2) for s being the Kirillov-A_{t,h} (KKS) bracket on the orbit. After that, we construct quantizations of these brackets.

Let M be a semisimple orbit. In Subsection 4.1 we prove that the brackets f satisfying (1.1) form a dim H^{2}(M)-dimensional variety. We give a description of this variety and prove (in Subsection 4.3) that almost all these brackets can be quantized. So, we obtain for M a dim H^{2}(M) parameter family of non-equivalent one parameter quantizations. Note that in [DG2] we have built one of these quantizations, the quantization of the so called Sklyanin-Drinfeld Poisson bracket.

It turns out that brackets f satisfying (1.1) and (1.2) exist not for all orbits. We call an orbit M good if there exists a bracket f satisfying (1.1) and (1.2) for the Kirillov-Kostant-Souriau (KKS) bracket s.

In Subsection 4.1 we give the following classification of the semisimple good orbits for all simple 𝔤, [DGS].

In the case 𝔤 = sl(n) all semisimple orbits are good. (Actually we prove that in this case all orbits are good.)

For 𝔤 ≠ sl(n) all symmetric orbits (which are symmetric spaces) are good. In this case φ_{M} = 0, so r_{M} itself is a Poisson bracket compatible with s.

Only in the case 𝔤 of type D_{n} and E_{6} (except of A_{n}) there are good orbits different from the symmetric ones. For such orbits φ_{M} ≠ 0.

We show that brackets f on a good orbit satisfying (1.1) and (1.2), form a one parameter family.

In Subsection 4.2 we consider cohomologies of an invariant complex with the differential given by the Schouten bracket with the bivector f. These cohomologies are needed for our construction of quantization.

In Subsection 4.3 we construct one and two parameter quantizations for semisimple orbits. According to our approach, as a first step we construct a G invariant Φ_{h} associative quantization, i.e., a quantization in the Drinfeld category with non-trivial associativity constraint given by Φ_{h}. Note that the bracket f from (1.1) can be considered as a “Poisson bracket” in that category. As a second step, we make a passage to the category with trivial associativity to obtain the associative U_{h}(𝔤) invariant quantization. We applied this method earlier for quantizing the function algebra on the highest weight orbits in irreducible representations of G, the algebra of sections of linear vector bundles over flag manifolds, and the function algebra on symmetric spaces, [DGM], [DG1], [DS1].

I put in the text some questions which naturally appeared by exposition. They are open (for me) and seem to be important.

Acknowledgment. I am very grateful to J.Bernstein, D.Gurevich, S.Shnider, and V.Ostapenko for many useful discussions.

I thank Max-Planck-Institut für Mathematik for hospitality and very stimulating working atmosphere.

2 Preliminaries

2.1 Quantum groups

We shall consider quantum groups in sense of Drinfeld, [Dr2], as deformed universal enveloping algebras. If U(𝔤) is the universal enveloping algebra of a complex Lie algebra 𝔤, then the quantum group (or quantized universal enveloping algebra) corresponding to U(𝔤) is a topological Hopf algebra, U_{h}(𝔤), over ℂ[[h]], isomorphic to U(𝔤)[[h]] as a topological ℂ[[h]] module and such that U_{h}(𝔤)/hU_{h}(𝔤) = U(𝔤) as a Hopf algebra over ℂ. In particular, the deformed comultiplication in U_{h}(𝔤) has the form

Δ_{h} = Δ + hΔ_{1} + o(h),  (2.1)

where Δ is the comultiplication in the universal enveloping algebra U(𝔤). One can prove, [Dr2], that Δ_{1} : U(𝔤) → U(𝔤) ⊗ U(𝔤) is such a map that Δ_{1} − σΔ_{1} = δ (σ is the usual permutation) being restricted on 𝔤 gives a map δ : 𝔤 → ∧^{2}𝔤 which is a 1-cocycle and defines the structure of a Lie coalgebra on 𝔤 (the structure of a Lie algebra on the dual space 𝔤^{∗}). The pair (𝔤, δ) is considered as a quasiclassical limit of U_{h}(𝔤).

In general, a pair (𝔤, δ), where 𝔤 is a Lie algebra and δ is such a 1-cocycle, is called a Lie bialgebra. It is proven, [EK], that any Lie bialgebra (𝔤, δ) can be quantized, i.e., there exists a quantum group U_{h}(𝔤) such that the pair (𝔤, δ) is its quasiclassical limit.

A Lie bialgebra (𝔤, δ) is said to be a coboundary one if there exists an element r ∈ ∧^{2}, called the classical r-matrix, such that δ(x) = [r, Δ(x)] for x ∈ 𝔤. Since δ defines a Lie coalgebra structure, r has to satisfy the so-called classical Yang-Baxter equation which can by written in the form

[[r, r]] = φ,  (2.2)

where [[·, ·]] stands for the Schouten bracket and φ ∈ ∧^{3}𝔤 is an invariant element. We denote the coboundary Lie bialgebra by (𝔤, r).

In case 𝔤 is a simple Lie algebra, the most known is the Sklyanin-Drinfeld r-matrix:

r = ∑_{α} X_{α} ∧ X_{−α},

where the sum runs over all positive roots; the root vectors X_{α} are chosen is such a way that (X_{α}, X_{−α}) = 1 for the Killing form (·, ·). This is the only r-matrix of weight zero, [SS], and its quantization is the Drinfeld-Jimbo quantum group. A classification of all r-matrices for simple Lie algebras was given in [BD].

We are interested in the case when 𝔤 is a semisimple finite dimensional Lie algebra. In this case, from results of Drinfeld and Etingof and Kazhdan one can derive the following

Proposition 2.1. Let 𝔤 be a semisimple Lie algebra. Then a) any Lie bialgebra (𝔤, δ) is a coboundary one;

b) the quantization, U_{h}(𝔤), of any coboundary Lie bialgebra (𝔤, r) exists and is isomorphic to U(𝔤)[[h]] as a topological ℂ[[h]] algebra;

c) the comultiplication in U_{h}(𝔤) has the form

Δ_{h}(x) = F_{h}Δ(x)F_{h}^{−1},  x ∈ U(𝔤),  (2.3)

where F_{h} ∈ U(𝔤)^{⊗2}[[h]] and can be chosen in the form

F_{h} = 1 ⊗ 1 + \frac{h}{2}r + o(h).  (2.4)

Proof. a) follows from the fact that H^{1}(𝔤, ∧^{2}𝔤) = 0. From the fact that H^{2}(𝔤, U(𝔤)) = 0 follows that U(𝔤) does not admit any nontrivial deformations as an algebra, (see [Dr1]), which proves b). From the fact that H^{1}(𝔤, U(𝔤)^{⊗2}) = 0 follows that any deformation of the algebra morphism Δ : U(𝔤) → U(𝔤) ⊗ U(𝔤) appears as a conjugation of Δ. In particular, the comultiplication in U_{h}(𝔤) looks like (2.3) with some F_{h} such that F_{0} = 1 ⊗ 1.

From the coassociativity of Δ_{h} follows that F_{h} satisfies the equation

(F_{h} ⊗ 1) · (Δ ⊗ id)(F_{h}) = (1 ⊗ F_{h}) · (id ⊗ Δ)(F_{h}) · Φ_{h}  (2.5)

for some invariant element Φ_{h} ∈ U(𝔤)^{⊗3}[[h]].

The element F_{h} satisfying (2.3) and (2.4) can be obtained by correction of some F_{h} only obeying (2.3), [Dr2]. This procedure also makes use simple cohomological arguments and essentially (2.5). This proves c).

From (2.5) follows that if F_{h} has the form (2.4), then the coefficient by h for Φ_{h} vanishes. Moreover, as a coefficient by h^{2} one can take the element φ from (2.2), i.e.,

Φ_{h} = 1 ⊗ 1 ⊗ 1 + h^{2}φ + o(h^{2}).  (2.6)

In addition, from (2.5) follows that Φ_{h} satisfies the pentagon identity

(id^{⊗2} ⊗ Δ)(Φ_{h}) · (Δ ⊗ id^{⊗2})(Φ_{h}) = (1 ⊗ Φ_{h}) · (id ⊗ Δ ⊗ id)(Φ_{h}) · (Φ_{h} ⊗ 1). (2.7)

Question 2.1. Let (𝔤, r) be a coboundary Lie bialgebra. Does there exist a quantization of it, U_{h}(𝔤), such that U_{h}(𝔤) is isomorphic to U(𝔤)[[h]] as a topological ℂ[[h]] algebra and the comultiplication has the form (2.3)?

From [Dr4] follows that if [[r, r]] = 0, the answer to this question is positive.

2.2 Categorical interpretation

It is known that the elements constructed above have a nice categorical interpretation. First, recall some facts about the Drinfeld algebras and the monoidal categories determined by them.

Let A be a commutative algebra with unit, B a unitary A-algebra. The category of representations of B in A-modules, i.e. the category of B-modules, will be a monoidal category if the algebra B is equipped with an algebra morphism, Δ : B → B ⊗_{A} B, called comultiplication, and an invertible element Φ ∈ B^{⊗3} such that Δ and Φ satisfy the conditions (see [Dr2])

(id ⊗ Δ)(Δ(b)) · Φ = Φ · (Δ ⊗ id)(Δ(b)), b ∈ B,  (2.8)
(id^{⊗2} ⊗ Δ)(Φ) · (Δ ⊗ id^{⊗2})(Φ) = (1 ⊗ Φ) · (id ⊗ Δ ⊗ id)(Φ) · (Φ ⊗ 1).  (2.9)

Define a tensor product functor for the category of B modules C, denoted ⊗_{C} or simply ⊗ when there can be no confusion, in the following way: given B-modules M, N, M ⊗_{C} N = M ⊗_{A} N as an A-module. The action of B is defined by

b(m ⊗ n) = (Δb)(m ⊗ n) = b_{1}m ⊗ b_{2}n,

where Δb = b_{1} ⊗ b_{2} (we use the Sweedler convention of an implicit summation over an index). The element Φ = Φ_{1} ⊗ Φ_{2} ⊗ Φ_{3} defines the associativity constraint,

a_{M,N,P} : (M ⊗ N) ⊗ P → M ⊗ (N ⊗ P), a_{M,N,P} ((m ⊗ n) ⊗ p) = Φ_{1}m ⊗ (Φ_{2}n ⊗ Φ_{3}p).

Again the summation in the expression for Φ is understood. By virtue of (2.8) Φ induces an isomorphism of B-modules, and by virtue of (2.9) the pentagon identity for monoidal categories holds. We call the triple (B, Δ, Φ) a Drinfeld algebra. The definition is somewhat non-standard in that we do not require the existence of an antipode. The category C of B-modules for B a Drinfeld algebra becomes a monoidal category. When it becomes necessary to be more explicit we shall denote C(B, Δ, Φ).

Let (B, Δ, Φ) be a Drinfeld algebra and F ∈ B^{⊗2} an invertible element. Put

Δ̃(b) = F Δ(b)F^{−1}, b ∈ B,  (2.10)
Φ̃ = (1 ⊗ F) · (id ⊗ Δ)(F) · Φ · (Δ ⊗ id)(F^{−1}) · (F ⊗ 1)^{−1}.  (2.11)

Then Δ̃ and Φ̃ satisfy (2.8) and (2.9), therefore the triple (B, Δ̃, Φ̃) also becomes a Drinfeld algebra. We say that it is obtained by twisting from (B, Δ, Φ). It has an equivalent monoidal category of modules, C̃(B, Δ̃, Φ̃). Note that the equivalent categories C and C̃ consist of the same objects as B-modules, and the tensor products of two objects are isomorphic as A-modules. The equivalence C → C̃ is given by the pair (Id, F), where Id : C → C̃ is the identity functor of the categories (considered without the monoidal structures, but only as categories of B-modules), and F : M ⊗_{C} N → M ⊗C̃ N is defined by m ⊗ n ↦ F_{1}m ⊗ F_{2}n where F_{1} ⊗ F_{2} = F.

We are interested in the case when A = ℂ[[h]], B = U(𝔤)[[h]] where 𝔤 is a complex semisimple Lie algebra. In this case, all tensor products over ℂ[[h]] are completed in h-adic topology.

We have two nontrivial Drinfeld algebras. The first is (U(𝔤)[[h]], Δ, Φ_{h}), with the usual comultiplication and Φ_{h} from (2.5). The condition (2.8) means the invariantness of Φ_{h}, while (2.9) coincides with (2.7). The second Drinfeld algebra is (U(𝔤)[[h]], Δ_{h}, 1). It obtaines by twisting of the first one by the element F_{h} from (2.3). The equation (2.11) follows from (2.5). The pair (Id, F_{h}) defines an equivalence between the corresponding monoidal categories C(U(𝔤)[[h]], Δ, Φ_{h}) and C(U(𝔤)[[h]], Δ_{h}, 1). The last is the category of representations of the quantum group U_{h}(𝔤).

It is clear that reduction modulo h defines a functor from either of these categories to the category of representations of U(𝔤) and the equivalence just described reduces to the identity modulo h. In fact, both categories are ℂ[[h]]-linear extensions (or deformations) of the ℂ-linear category of representations of 𝔤. Ignoring the monoidal structure the extension is a trivial one, but the associator Φ_{h} in the first case and the comultiplication Δ_{h} in the second case make the extension non-trivial from the point of view of monoidal categories.

2.3 U_{h}(𝔤) ivariant quantizations of algebras

Let (B, Δ, Φ) be a Drinfeld algebra. Assume A is a B-module with a multiplication μ : A ⊗_{A} A → A which is a homomorphism of A-modules. We say that μ is Δ invariant if

bμ(x ⊗ y) = μΔ(b)(x ⊗ y) for b ∈ B, x, y ∈ A,  (2.12)

and μ is Φ associative, if

μ(Φ_{1}x ⊗ μ(Φ_{2}y ⊗ Φ_{3}z))) = μ(μ(x ⊗ y) ⊗ z) for x, y, z ∈ A.  (2.13)

Note, that a B-module A equipped with Δ invariant and Φ associative multiplication is an associative algebra in the monoidal category C(B, Δ, Φ). If (B, Δ̃, Φ̃) is a Drinfeld algebra twisted by (2.10) and (2.11), then the algebra A may be transfered into the equivalent category C̃(B, Δ̃, Φ̃): the multiplication μ̃ = μF^{−1} : M ⊗_{A} M → M is Φ̃- associative and invariant in the category C̃.

Let A be a U(𝔤) invariant associative algebra, i.e., an algebra with U(𝔤) invariant multiplication μ in sense of (2.12). A deformation (or quantization) of A is an associative algebra, A_{h}, which is isomorphic to A[[h]] = A ⊗ ℂ[[h]] (completed tensor product) as a ℂ[[h]]-module, with multiplication in A_{h} having the form μ_{h} = μ + hμ_{1} + o(h). The algebra U(𝔤)[[h]] is clearly acts on the ℂ[[h]] module A_{h}.

We will study quantizations of A which will be invariant under the comultiplication Δ_{h}. In other words, A_{h} will be an algebra in the category of representations of the quantum group U_{h}(𝔤). It is clear from the previous Subsection that if A_{h} is such a quantization, then the multiplication μ_{h}F_{h} makes the module A[[h]] into an algebra in the category C(U(𝔤)[[h]], Δ, Φ_{h}), i.e., this multiplication is U(𝔤) invariant and Φ_{h} associative.

We shall see that often it is easier to constract U(𝔤) invariant and Φ_{h} associative quantization of A. After that, the ivariant quantization with respect to any quantum group from Proposition 2.1 can be obtained by twisting by the appropriate F_{h}.

As an algebra A we may take an algebra A_{t} that is itself a U(𝔤) invariant quantization of a commutative algebra A. In this case, a U_{h}(𝔤) invariant quantization of A_{t} is an algebra A_{t,h} over ℂ[[t, h]].

2.4 Poisson brackets associated with the U_{h}(𝔤) invariant quantization

Let A be a U(𝔤) invariant commutative algebra with multiplication μ and A_{h} its quantization with multiplication μ_{h} = μ + hμ_{1} + o(h). The Poisson bracket corresponding to the quantization is given by {a, b} = μ_{1}(a, b) − μ_{1}(b, a), a, b ∈ A.

In general, we call a skew-symmetric bilinear form A ⊗ A → A a bracket, if it satisfies the Leibniz rule in either argument when the other is fixed. The term Poisson bracket indicates that the Jacobi identity is also true.

A bracket of the form

{a, b}_{r} = (r_{1}a)(r_{2}b) = μr(a ⊗ b)  a, b ∈ A,  (2.14)

where r = r_{1} ⊗ r_{2} (summation implicit) is the representation of r-matrix r, will be called an r-matrix bracket.

Assume A_{h} is a U_{h}(𝔤) invariant quantization, i.e., the multiplicatin μ_{h} is Δ_{h} invariant. We shall show that in this case the Poisson bracket {·, ·} has a special form. Suppose f and g are two brackets on A. Define their Schouten bracket [[f, g]] as

[[f, g]](a, b, c) = f(g(a, b), c) + g(f(a, b), c) + cyclic permutations of a, b, c. (2.15)

Then [[f, g]] is a skew-symmetric map A^{⊗3} → A. We call f and g compatible if [[f, g]] = 0.

Proposition 2.2. Let A be a U(𝔤) invariant commutative algebra and A_{h} a U_{h}(𝔤) invariant quantization. Then the corresponding Poisson bracket has the form

{a, b} = f(a, b) − {a, b}_{r}  (2.16)

where f(a, b) is a U(𝔤) invariant bracket.

The brackets f and {·, ·}_{r} are compatible and [[f, f]] = −φ_{A}, where φ_{A}(a, b, c) = (φ_{1}a)(φ_{2}b)(φ_{3}c) and φ_{1} ⊗ φ_{2} ⊗ φ_{3} = φ ∈ ∧^{3}𝔤 is the invariant element from (2.2).

Proof. Let the comultiplication for U_{h}(𝔤) have the form (2.1). Let A be a commutative algebra with the U(𝔤) invariant multiplication μ. Suppose A_{h} is a U_{h}(𝔤) invariant quantization of A . This means that the deformed multiplication has the form

μ_{h} = μ + hμ_{1} + o(h)  (2.17)

and satisfies the relation

xμ_{h}(a ⊗ b) = μ_{h}Δ_{h}(x)(a ⊗ b)  for x ∈ U(𝔤), a, b ∈ A.  (2.18)

Substituting (2.1) and (2.17) in (2.18) and collecting the terms by h we obtain

μ_{1}(a ⊗ b) = μΔ(x)(a ⊗ b) + mΔ_{1}(x)(a ⊗ b).

Subtracting from this equation the similar one with permuting a and b and making use that Δ is commutative and δ = Δ_{1}−σΔ_{1} is skew-commutative, we derive that the Poisson bracket p = {·, ·} has to satisfy the property

xp(a ⊗ b) = pΔ(x)(a ⊗ b) + μδ(x)(a ⊗ b),  x ∈ U(𝔤).  (2.19)

Let us prove that the bracket f(a, b) = {a, b} + {a, b}_{r} is U(𝔤) invariant. Indeed, from (2.14) we have for x ∈ U(𝔤), a, b ∈ A

xμr(a ⊗ b) = μΔ(x)r(a ⊗ b) = μrΔ(x)(a ⊗ b) − μ[r, Δ(x)](a ⊗ b).

Using this expression, (2.19), and the fact that δ(x) = [r, Δ(x)], we obtain

xf = xp + xμr = (pΔ(x) + μ[r, Δ(x)]) + (μrΔ(x) − μ[r, Δ(x)]) =
= pΔ(x) + μrΔ(x) = fΔ(x),

which proves the invariantness of f.

So, we have {a, b} = f(a, b) − {a, b}_{r}, as required.

It is easy to check that any bracket of the form {a, b} = (X_{1}a)(X_{2}b) = μ(X_{1}a, X_{2}b), for X_{1} ⊗ X_{2} ∈ 𝔤 ∧ 𝔤, is compatible with any invariant bracket. In particular, an r-matrix bracket is compatible with f. In addition, {·, ·} is a Poisson bracket, so its Schouten bracket with itself is equal to zero. Using this and the fact that the Schouten bracket of r-matrix bracket with itself is equal to φ_{A}, we obtain from (2.16) that [[f, f]] = −φ_{A}.

Remark 2.1. Let A be the function algebra on a G-manifold M, where the Lie group G corresponds to the Lie algebra 𝔤. It is easy to see that condition (2.19) with δ(x) = [r, Δ(x)] is equivalent to the condition that the pair (M, p) becomes a (G, r̃)-Poisson manifold, where r̃ is the Poisson structure on G defined by the r-matrix r: r̃ = r′ − r′′, where r′ and r′′ are the left- and right-invariant bivector fields on G corresponding to r. It is known that r̃ makes G into a Poisson-Lie group. So Proposition 2.2 gives a description of Poisson structures p on M making (M, p) into a (G, r̃)-Poisson manifold.

We shall also consider two parameter quantizations of algebras. A two parameter quantization of an algebra A is an algebra A_{t,h} isomorphic to A[[t, h]] as a ℂ[[t, h]] module and having a multiplication in the form

μ_{t,h} = μ + tμ′_{1} + hμ′′_{1} + o(t, h).

With such a quantization, one associates two Poisson brackets: the bracket s(a, b) = μ′_{1}(a, b) − μ′_{1}(b, a) along t, and the bracket p(a, b) = μ′′_{1}(a, b) − μ′′_{1}(b, a) along h. It is easy to check that p and s are compatible Poisson brackets, i.e., the Schouten bracket [[p, s]] = 0.

A pair of compatible Poisson brackets we call a Poisson pencil.

Corollary 2.1. Let A_{t,h} be a two parameter U_{h}(𝔤) invariant quantization of a commutative algebra A such that A_{t,0} is a one parameter U(𝔤) invariant quantization of A with Poisson bracket s. Then the U_{h}(𝔤) invariant quantization A_{0,h} has a Poisson bracket p of the form (2.16): p = f − {·, ·}_{r}, where f is an invariant bracket such that [[f, f]] = −φ_{A} and compatible with s, i.e.,

[[f, s]] = 0.  (2.20)

Proof. For the two parameter quantization, the Poisson brackets p and s form a Poisson pencil, hence must be compatible. Also, s is a U(𝔤) invariant bracket, so that s is compatible with the r-matrix bracket {·, ·}_{r}. It follows from (2.16) that s has to be compatible with f.

In what follows, we shall often call A_{t,h} a U_{h}(𝔤) invariant quantization (or double quantization) of the invariant Poisson bracket s, or of the Poisson pencil s and p.

Remark 2.2. As we have seen in Subsection 2.3, to construct a U_{h}(𝔤) invariant quantization of A is the same that to construct a U(𝔤) invariant Φ_{h} associative quantization of A. We shall see that the last problem often turns out to be simpler (see Subsection 4.3). We observe that if p = f − {·, ·}_{r} is an admissible Poisson bracket for U_{h}(𝔤) invariant quantization, then the invariant bracket f with the property [[f, f]] = −φ_{A} may be considered as a “Poisson bracket” of quantization in the category with Φ_{h} defining the associativity constraint. Also, the pair f, s is a Poisson pencil in that category.

3 Double quantization on coadjoint representations

In this section we study a two parameter (or double) quantization on coadjoint representations of simple Lie algebras.

Let 𝔤 be a complex Lie algebra. Then, the symmetric algebra S𝔤 can be considered as a function algebra on 𝔤^{∗}. The algebra U(𝔤) is included in the family of algebras (S𝔤)_{t} = T (𝔤)[t]/J_{t}, where J_{t} is the ideal generated by the elements of the form x ⊗ y − σ(x ⊗ y) − t[x, y], x, y ∈ 𝔤, σ is the permutation. By the PBW theorem, (S𝔤)_{t} is a free module over ℂ[t]. We have (S𝔤)_{0} = S𝔤, so this family of quadratic-linear algebras gives a U(𝔤) invariant quantization of S𝔤 by the Lie bracket s.

It turns out that for 𝔤 = sl(n) this picture can be extended to the quantum case, [Do2]. Namely, there exist deformations, σ_{h} and [·, ·]_{h}, of both the mappings σ and [·, ·] such that the two parameter family of algebras (S𝔤)_{t,h} = T (𝔤)[[h]][t]/J_{t,h}, where J_{t,h} is the ideal generated by the elements of the form x ⊗ y − σ_{h}(x ⊗ y) − t[x, y]_{h}, x, y ∈ 𝔤, gives a U_{h}(𝔤) invariant quantization of the Lie bracket s on 𝔤^{∗}. In this case, the corresponding bracket f from (2.20) is a quadratic bracket which is, up to a factor, a unique nontrivial invariant map ∧^{2}𝔤 → S^{2}𝔤.

We shall show that for other simple Lie algebras, double quantizations of the Lie brackets do not exist.

We give two constructions of the algebra (S𝔤)_{t,h}. The first construction uses an idea from the paper [LS] on a quantum analog of Lie algebra for sl(n). The second construction using the so called reflection equations (RE), [KS], [Maj], is presented in Remark 3.4..

3.1 Quantum Lie algebra for U_{h}(sl(n))

Let U_{h}(𝔤) be a quantized universal enveloping algebra for a Lie algebra 𝔤. We consider U_{h}(𝔤) as a U_{h}(𝔤) module with respect to the left adjoint action: ad(x)y = x_{1}yγ(x_{2}), where x, y ∈ U_{h}(𝔤), Δ_{h}(x) = x_{1} ⊗ x_{2} (summation implicit).

There were attemptions to define quantum Lie algebras as deformed standard classical embeddings of 𝔤 into U_{h}(𝔤) obeying some additional properties, [DG], [LS].

In the classical case, there is probably the following way (not using comultiplication) to distinguish the standard embedding 𝔤 → U(𝔤) from other invariant embeddings: with respect to this embedding U(𝔤) is a quadratic-linear algebra. So, we give the following (working) definition of quantum Lie algebras.

Definition 3.1. Let 𝔤_{h} be a subrepresentation of U_{h}(𝔤), which is a deformation of the standard embedding of 𝔤 in U(𝔤). We call 𝔤_{h} a quantum Lie algebra, if the kernel of the induced homomorphism T (𝔤_{h}) → U_{h}(𝔤) is defined by (deformed) quadratic-linear relations.

We are going to show that the quantum Lie algebra exists in case 𝔤 = sl(n). On the other hand, if such an algebra exists for some Lie algebra 𝔤, then a double quantization of the Lie bracket on 𝔤^{∗} also exists. But, as we shall see, no double quantization exists for simple 𝔤 ≠ sl(n). So, among simple finite dimensional Lie algebras, only sl(n) has a quantum Lie algebra in our sense.

Our construction is the following. Let R = R′_{i} ⊗R′′_{i} ∈ U_{h}(𝔤)⊗U_{h}(𝔤) (completed tensor product) be the R-matrix (summation by i is assumed). It satisfies the properties [Dr2]

Δ′_{h}(x) = RΔ_{h}(x)R^{−1}, x ∈ U_{h}(𝔤),  (3.1)

where Δ_{h} is the comultiplication in U_{h}(𝔤) and Δ′_{h} is the opposite one,

(Δ_{h} ⊗ 1)R = R^{13}R^{23} = R′_{i} ⊗ R′_{j} ⊗ R′′_{i}R′′_{j}
(1 ⊗ Δ_{h})R = R^{13}R^{12} = R′_{i}R′_{j} ⊗ R′′_{j} ⊗ R′′_{i},  (3.2)

and

(1 ⊗ ε)R = (ε ⊗ 1)R = 1 ⊗ 1,  (3.3)

where ε is the counit in U_{h}(𝔤).

Consider the element Q = Q′_{i} ⊗ Q′′_{i} = R^{21}R. It follows from (3.1) that Q commutes with elements from U_{h}(𝔤) ⊗ U_{h}(𝔤) of the form Δ_{h}(x). This is equivalent for Q to be invariant under the adjoint action of U_{h}(𝔤) on U_{h}(𝔤) ⊗ U_{h}(𝔤).

Let V be an irreducible finite dimensional representation of U_{h}(𝔤) and ρ : U_{h}(𝔤) → End(V) the corresponding map of algebras. Consider the dual space End(V)^{∗} as a left U_{h}(𝔤) module setting

(xφ)(a) = φ(γ(x_{(1)})ax_{(2)}),

where φ ∈ End(V)^{∗}, a ∈ End(V), Δ_{h}(x) = x_{(1)} ⊗ x_{(2)} in Sweedler notions, and γ denotes the antipode in U_{h}(𝔤).

Consider the map

f : End(V)^{∗} → U_{h}(𝔤)  (3.4)

defined as φ ↦ φ(ρ(Q′_{i})Q′′_{i}. From the invariance of Q it follows that f is a U_{h}(𝔤) equivariant map, so \overline{L} = Im(f) is a U_{h}(𝔤) submodule.

It follows from (3.2) that \overline{L} is a left coideal in U_{h}(𝔤), i.e., Δ(x) ∈ U_{h}(𝔤) ⊗ \overline{L} for any x ∈ \overline{L}. Indeed, Q = R′′_{i}R′_{j} ⊗ R′_{i}R′′_{j}. Applying (3.2) we obtain

(1 ⊗ Δ_{h})R^{21}R = R′′_{i}R′′_{j}R′_{k}R′_{l} ⊗ R′_{i}R′′_{l} ⊗ R′_{j}R′′_{k}  (3.5)

Let φ ∈ End(V)^{∗}. Define ψ_{il} ∈ End(V)^{∗} setting ψ_{il}(a) = φ(R′′_{i}aR′_{l}) for a ∈ End(V). Then Δφ(R′′_{i}R′_{j})R′_{i}R′′_{j} = R′_{i}R′′_{l} ⊗ ψ_{il}(R′′_{j}R′_{k})R′_{j}R′′_{k}, which obviously belongs to U_{h}(𝔤) ⊗ \overline{L}.

Recall, [Dr2], that R = F_{h}^{21}e^{\frac{h}{2} t}F_{h}^{−1}. Here t = ∑_{i} t_{i} ⊗ t_{i} is the split Casimir, where t_{i} form an orthonormal basis in 𝔤 with respect to the Killing form, F = 1 ⊗ 1 + \frac{h}{2}r + o(h) (see (2.4)), and r is a classical r-matrix. Therefore,

Q = R^{21}R = F e^{ht}F^{−1} = 1 ⊗ 1 + ht + \frac{h^{2}}{2} (t^{2} + [r, t]) + o(h^{2}).  (3.6)

Denote by Tr the unique (up to a factor) invariant element in End(V)^{∗}. Let Z_{0} = ρ_{0}(𝔤), and denote by Z_{h} some U_{h}(𝔤) invariant deformation of Z_{0} in End(V). Then we have a decomposition End(V) = I ⊕ Z_{h} ⊕ W, where I is the one dimensional invariant subspace generated by the identity map, W is a complement to I ⊕ Z_{h} invariant subspace. This gives a decomposition End(V)^{∗} = I^{∗} ⊕ Z_{h}^{∗} ⊕ W^{∗} where W^{∗} consists of all the elements which are equal to zero on I ⊕ Z_{h}. The space I^{∗} is generated by Tr, and after normalizing in such a way that Tr(id) = 1, we obtain that C_{V} = f(Tr) is of the form

C_{V} = Tr(ρ(Q′_{i}))Q′′_{i} = 1 + h^{2}c + o(h^{2}),  (3.7)

where c is an invariant element of U(𝔤). It follows from (3.3) that ε(C) = 1. From (3.6) follows that the elements of f(Z_{h}^{∗}) have the form

z = hx + o(h), x ∈ 𝔤,  (3.8)

hence the subspace L_{1} = h^{−1}f(Z_{h}^{∗}) forms a subrepresentation of U_{h}(𝔤) with respect to the left adjoint action of U_{h}(𝔤) on itself, which is a deformation of the standard embedding of 𝔤 into U(𝔤). It follows from (3.3) that ε(L_{1}) = 0.

The elements from f(W^{∗}) have the form w = h^{2}b + o(h^{2}) and ε(W^{∗}) = 0. Denote L_{2} = h^{−2}f(W^{∗}).

So, \overline{L} = ℂC_{V} ⊕ hL_{1} ⊕ h^{2}L_{2} = ℂC_{V} + hL, where L = L_{1} ⊕ hL_{2}. Since \overline{L} is a left coideal in U_{h}(𝔤), for any x ∈ \overline{L} we have

Δ_{h}(x) = x_{(1)} ⊗ x_{(2)} = z ⊗ C_{V} + v ⊗ x′,

where z, v ∈ U_{h}(𝔤), x′ ∈ L. Applying to the both hand sides (1 ⊗ ε) and multiplying we obtain x = x_{(1)}ε(x_{(2)}) = zε(C_{V}) + vε(x′) = z. So, z has to be equal to x. and we obtain

Δ_{h}(x) = x_{(1)} ⊗ x_{(2)} = x ⊗ C_{V} + v ⊗ x′, x, x′ ∈ L.  (3.9)

From (3.9) we have for any y ∈ L

xy = x_{(1)}yγ(x_{(2)})x_{(3)} = x_{(1)}yγ(x_{(2)})C_{V} + v_{(1)}yγ(v_{(2)})x′.  (3.10)

Introduce the following maps:

σ′_{h} : L ⊗ L → L ⊗ L, x ⊗ y ↦ v_{(1)}yγ(v_{(2)}) ⊗ x′,
[·, ·]′_{h} : L ⊗ L → L,  x ⊗ y ↦ x_{(1)}yγ(x_{(2)}).  (3.11)

We may rewrite (3.10) in the form

m(x ⊗ y − σ′_{h}(x ⊗ y)) − [x, y]′_{h}C_{V} = 0.  (3.12)

Observe now that, as follows from (3.7), C_{V} is an invertible element in U_{h}(𝔤). Put P = C_{V}^{−1}. Transfer the maps (3.11) to the space P · L, i.e., define

σ_{h}(P x, P y) = (P ⊗ P)σ′_{h}(x, y),
[P x, P y]_{h} = P [x, y]′_{h}.

From (3.9) we obtain

P_{(1)}x_{(1)} ⊗ P_{(2)}x_{(2)}) = P_{(1)}x ⊗ P_{(2)}C_{V} + P_{(1)}v ⊗ P_{(2)}x′.  (3.13)

Using this relation and taking into account that P commutes with all elements from U_{h}(𝔤), we obtain as in (3.10)

P xP y = P_{(1)}x_{(1)}P yγ(x_{(2)})γ(P_{(2)})P_{(3)}x_{(3)} =  (3.14)
P_{(1)}x_{(1)}P yγ(x_{(2)})γ(P_{(2)})P_{(3)}C_{V} + P_{(1)}v_{(1)}P yγ(v_{(2)})γ(P_{(2)})P_{(3)}x′ = (3.15)
P [x, y]′_{h} + P^{2}mσ′_{h}(x ⊗ y) = [P x, P y]_{h} + mσ_{h}(P x ⊗ P y)).  (3.16)

This equality may be written as

m(x ⊗ y − σ_{h}(x ⊗ y)) − [x, y]_{h} = 0, x, y ∈ C_{V}^{−1}L.  (3.17)

Define L_{V} = C_{V}^{−1}L. Let T (L_{V}) = ⊕_{k=0}^{∞}L_{V}^{⊗k} be the tensor algebra over L_{V}. Notice, that T (L_{V}) is not supposed to be completed in h-adic topology. Let J be the ideal in T (L_{V}) generated by the relations

(x ⊗ y − σ_{h}(x ⊗ y)) − [x, y]_{h}, x, y ∈ L_{V} .  (3.18)

Due to (3.17) we have a homomorphism of algebras over ℂ[[h]]

ψ_{h} : T (L_{V})/J → U_{h}(𝔤),  (3.19)

extending the natural embedding L_{V} → U_{h}(𝔤) of U_{h}(𝔤) modules..

Now we can prove

Proposition 3.1. For 𝔤 = sl(n) the quantum Lie algebra exists.

Proof. Apply the above construction to V = ℂ^{n}[[h]], the deformed basic representation of 𝔤. In this case End(V) = I ⊕ Z_{h}, where Z_{h} is a deformed adjoint representation. So, 𝔤_{h} = L_{V} = h^{−1}C_{V}^{−1}f(Z_{h}^{∗}) is a deformation of the standard embedding of 𝔤 in U(𝔤). It is easy to see that in this case σ_{h} is a deformation of the usual permutation: σ_{0}(x⊗y) = y⊗x, and [·, ·]_{h} is a deformation of the Lie bracket on 𝔤: [x, y]_{0} = [x, y], x, y ∈ 𝔤 ⊂ U(𝔤). Hence, at h = 0, the quadratic-linear relations (3.18) are exactly the defining relations for U(𝔤), therefore the map (3.19) is an isomorphism at h = 0. It follows that (3.19) is an embedding. (Actually, (3.19) is essentially an isomorphism, i.e., it is an isomorphism after completion of T (L_{V}) in h-adic topology.) So, the kernel of the map T (L_{h}) → U_{h}(𝔤) is defined by the quadratic-linear relations (3.18).

Remark 3.1. Quadratic-linear relations (3.18) can be obtained in another way. Note that equation (3.5) may be rewritten as

(1 ⊗ Δ_{h})Q = R_{21}Q_{13}R_{12}.  (3.20)

Since Q commutes with all elements of the form Δ_{h}(x), x ∈ U_{h}(𝔤), one derives from (3.20):

Q_{23}R_{21}Q_{13}R_{12} = R_{21}Q_{13}R_{12}Q_{23}.  (3.21)

Consider the element Q_{ρ} = ρ(Q_{1}) ⊗ Q_{2} as a dim(V) × dim(V) matrix with the entries from U_{h}(𝔤). Applying to (3.21) operator ρ ⊗ ρ ⊗ 1, we obtain the following relation for Q_{ρ}:

(Q_{ρ})_{2}\overline{R}_{21}(Q_{ρ})_{1}\overline{R} = \overline{R}_{21}(Q_{ρ})_{1}\overline{R}(Q_{ρ})_{2},  (3.22)

where \overline{R} = (ρ⊗ρ)R is a number matrix, the Yang-Baxter operator in V ⊗V. Replacing in this equation \overline{R} by S = σ\overline{R}, we obtain that the matrix Q_{ρ} satisfies the following reflection equation (RE):

(Q_{ρ})_{2}S(Q_{ρ})_{2}S = S(Q_{ρ})_{2}S(Q_{ρ})_{2}.  (3.23)

It is clear that the entries of the matrix Q_{ρ} generate the image of the map (3.4). From (3.7) follows that Q_{ρ} has the form

Q_{ρ} = Id_{V} C_{V} + hB′,  (3.24)

where B′ has the form B′ = ∑ D_{i} ⊗ b_{i}, D_{i} belong to the complement to ℂ Id_{V} submodule in End(V) and b_{i} ∈ U_{h}(𝔤). Note that the entries of the matrix B′ form the subspace L, whereas the entries of B = C_{V}^{−1}B′ form the subspace L_{V} from (3.17). From (3.24) we obtain

C_{V}^{−1}Q_{ρ} = Id +hB.  (3.25)

Since the element C_{V}^{−1} belongs to the center of U_{h}(𝔤), the matrix C_{V}^{−1}Q_{ρ} obeys the RE (3.23) as well. So, B satisfies the relation

(Id +hB)_{2}S(Id +hB)_{2}S = S(Id +hB)_{2}S(Id +hB)_{2}.  (3.26)

One checks that (3.26), considered as a qudratic-linear relations for indetermined entries of B, is equivalent to (3.18) in the case 𝔤 = sl(n).

3.2 Double quantization on sl(n)^{∗}

Introduce a new variable, t, and consider a homomorphism of algebras, T (L_{V})[t] → U_{h}(𝔤)[t], which extends the embedding t · ı : L_{V} [t] → U_{h}(𝔤)[t], where ı stands for the standard embedding L_{V} → U_{h}(𝔤). From (3.17) follows that t · ı factors through the homomorphism of algebras over ℂ[[h]][t]

ϕ_{t,h} : T (L_{V})[t]/J_{t} → U_{h}(𝔤)[t],  (3.27)

where J_{t} is the ideal generated by the relations

(x ⊗ y − σ_{h}(x ⊗ y)) − t[x, y]_{h}, x, y ∈ L_{V} .  (3.28)

Proposition 3.2. For 𝔤 = sl(n) the algebra (S𝔤)_{t,h} = T (L_{V})[t]/J_{t} is a double quantization of the Lie bracket on S𝔤.

Proof. Since in this case L_{V} = 𝔤_{h}, from Proposition 3.1 follows that (3.27) is a monomorphism at t = 1. Due to the PBW theorem the algebra Im(ϕ_{t,h}) at the point h = 0 is a free ℂ[t]-module and is equal to

(S𝔤)_{t} = T (𝔤)/{x ⊗ y − y ⊗ x − t[x, y]}.  (3.29)

For t = 0 this algebra is the symmetric algebra S𝔤, the algebra of algebraic functions on 𝔤^{∗}. For t ≠ 0 this algebra is isomorphic to U(𝔤). Since U_{h}(𝔤) is a free ℂ[[h]]-module, it follows that ϕ_{t,h} in (3.27) is a monomorphism of algebras over ℂ[[h]][t] and Im(ϕ_{t,h}) is a free ℂ[[h]][t]-module isomorphic to

(S𝔤)_{t,h} = T (𝔤_{h})[t]/{x ⊗ y − σ_{h}(x ⊗ y) − t[x, y]_{h}}.  (3.30)

It is clear that (S𝔤)_{t} = (S𝔤)_{t,0} is the standard quantization of the Lie bracket on 𝔤^{∗}.

Call the algebra

(S𝔤)_{h} = (S𝔤)_{0,h} = T (𝔤_{h})/{x ⊗ y − σ_{h}(x ⊗ y)}  (3.31)

a quantum symmetric algebra (or quantum polynomial algebra on 𝔤^{∗}). It is a free ℂ[[h]] module and a quadratic algebra equal to S𝔤 at h = 0.

Remark 3.2. Up to now, all our constructions were considered for the quantum group in sense of Drinfeld, U_{h}(𝔤), defined over ℂ[[h]]. But one can deduce the results above for the quantum group in sense of Lusztig, U_{q}(𝔤), defined over the algebra ℂ[q, q^{−1}]. We show, for example, how to obtain the quantum symmetric algebra over 𝔤. Let E be a Grassmannian consisting of subspaces in 𝔤⊗𝔤 of dimension equal to dim(∧^{2}𝔤), and Z the closed algebraic subset of E consisting of subspaces J such that dim(E ⊗ J ∩ J ⊗ E) ≥ dim(∧^{3}𝔤). Let X be the algebraic subset in Z × (ℂ \ 0) consisting of points (J, q) such that J is invariant under the action of U_{q}(𝔤). The projection π : X → ℂ \ 0 is a proper map. It is clear that the fiber of this projection over q = 1 contains the point corresponding to the symmetric algebra S𝔤 as an isolated point, because there are no quadratic U(𝔤) invariant Poisson brackets on S𝔤.

As follows from the existence of (S𝔤)_{h} (completed situation at q = 1), the dimension of X is equal to 1. Hence, the projection π : X → ℂ \ 0 is a covering. For x ∈ X let J_{x} be the corresponding subspace in 𝔤 ⊗ 𝔤 and (S𝔤)_{x} = T (𝔤)/{J_{x}} the corresponding quadratic algebra. Due to the projection π, the family (S𝔤)_{x}, x ∈ X, is a module over ℂ[q, q^{−1}]. Since J_{x} is U_{p(x)}(𝔤) invariant, (S𝔤)_{x} is a U_{p(x)}(𝔤) invariant algebra. Hence, after possible deleting from X some countable set of points, we obtain a family of quadratic algebras with the same dimensions of graded components as S𝔤. So, the family (S𝔤)_{x}, x ∈ X can be considered as a quantum symmetric algebra over U_{q}(𝔤).

Note also that the family (S𝔤)_{h} can be thought of as a formal section of the map π : X → (ℂ \ 0) over the formal neighborhood of point q = 1. It follows that there is also an analytic section of π over some neighborhood, U, of the point q = 1. If (S𝔤)_{h} is a quantization with Poisson bracket f−{·, ·}_{r} (see Proposition 2.2), then a quantization with Poisson bracket −f − {·, ·}_{r} gives another section of π over U. Hence, in a neighborhood of the “classical” point x_{0} ∈ X, π(x_{0}) = 1, the space X has a singularity of type “cross”.

3.3 Poisson pencil corresponding to (S𝔤)_{t,h}

Let 𝔤 = sl(n) and (S𝔤)_{t,h} be the double quantization from Proposition 3.2.

Proposition 3.3. The Poisson pencil corresponding to the quantization (S𝔤)_{t,h} consists of two compatible Poisson brackets:

s (along t) is the Lie bracket;

p (along h) is a quadratic Poisson bracket of the form p = f − {·, ·}_{r}, where f is an invariant quadratic bracket which is a unique up to a factor invariant map f : ∧^{2}𝔤 → S^{2}𝔤, and \{·, ·\}_{r} is the r-matrix bracket. Moreover, [[s, f]] = 0 and [[f, f]] = −\overline{φ}, where \overline{φ} has the form \overline{φ}(a, b, c) = [φ_{1}, a][φ_{2}, b][φ_{3}, c], and φ = φ_{1} ∧ φ_{2} ∧ φ_{3} = [[r, r]]. Recall that φ is a unique up to a factor invariant element of ∧^{3}𝔤.

Proof. That s coincides with the Lie bracket is obvious from (3.29). From Corollary 2.1 we have p = f − {·, ·}_{r}. Since (S𝔤)_{h} is a quadratic algebra over ℂ[[h]], p must be a quadratic bracket. But the r-matrix bracket {·, ·}_{r} is quadratic, too. Hence, f must be a quadratic invariant bracket. There is only one possibility for such a bracket: it must be a unique (up to a factor) nontrivial invariant map f : ∧^{2}𝔤 → S^{2}𝔤. Now apply Proposition 2.2 and Corollary 2.1.

Consider now the quadratic bracket f in more detail.

We say that a k-vector field, g, on a manifold M is strongly restricted on a submanifold N ⊂ M if at any point of N the polyvector g can be presented as an exterior power of tangent vectors to N.

Consider the coadjoint action of the Lie group G = SL(n) on 𝔤^{∗} = sl(n)^{∗}. We want to prove that the bracket f is strongly restricted on any orbit of G in 𝔤. It turns out that there is the following general fact.

Proposition 3.4. Let G be a semisimple Lie group with its Lie algebra 𝔤, s = [·, ·] the Lie bracket on 𝔤^{∗}. Let f = {·, ·} be an invariant bracket on 𝔤^{∗} such that the Schouten bracket [[s, f]] is a three-vector field, ψ, strongly restricted on an orbit O of G. Then f is strongly restricted on O.

Proof. Let x, y, z ∈ 𝔤. The invariance condition for {·, ·} means:

[x, {y, z}] = {[x, y], z} + {y, [x, z]}.  (3.32)

The Schouten bracket [[s, f]] is:

[x, {y, z}] + [y, {z, x}] + [z, {x, y}] +
{x, [y, z]} + {y, [z, x]} + {z, [x, y]} = ψ(x, y, z).

In the left hand side of this expression, the 1-st, 5-th, and 6-th terms are canceled due to (3.32), and we have

[y, {z, x}] + [z, {x, y}] + {x, [y, z]} = ψ(x, y, z).

Putting in this equation instead of [y, {z, x}] its expression from (3.32), i.e., {[y, z], x} + {z, [y, x]}, we obtain, since the term {x, [y, z]} is canceled:

{z, [x, y]} = [z, {x, y}] + ψ(x, y, z).  (3.33)

Now observe that, due to the Leibniz rule, equation (3.33) is valid for any z ∈ S𝔤. To prove the proposition, it is sufficient to show that if z belongs to the ideal I_{O} defining the orbit O, then {z, u} also belongs to this ideal. Again, due to the Leibniz rule, it is sufficient to show this for u ∈ 𝔤. Since 𝔤 is semisimple, there are elements x, y ∈ 𝔤 such that [x, y] = u. We have from (3.33)

{z, u} = {z, [x, y]} = [z, {x, y}] + ψ(x, y, z).

But [z, {x, y}] ∈ I_{O}, since the Lie bracket is restricted on any orbit, ψ(x, y, z) ∈ I_{O} by hypothesis of the proposition. So, {z, u} ∈ I_{O}.

As a consequence we obtain

Proposition 3.5. Let 𝔤 = sl(n). Then the bracket f from Proposition 3.3 is strongly restricted on any orbit of SL(n).

Proof. Follows from Propositions 3.3 and 3.4.

Remark 3.3. According to Remark 2.1, this Proposition shows that in case G = SL(n) any orbit in coadjoint representation has a Poisson bracket p = f − r_{M} such that the pair (M, p) becomes a (G, r̃)-Poisson manifold.

Remark 3.4. Recall that in case 𝔤 = sl(n) the tensor square 𝔤 ⊗ 𝔤, considered as a representation of 𝔤, has a decomposition into irreducible components which are contained in 𝔤 ⊗ 𝔤 with multiplicity one, except of the component isomorphic to 𝔤 having multiplicity two. Moreover, both the symmetric and skew-symmetric parts of 𝔤 ⊗ 𝔤 contain components, 𝔤^{1} and 𝔤^{2}, isomorphic to 𝔤. Hence, the bracket f takes 𝔤^{2} onto 𝔤^{1} and all the other components to zero.

For 𝔤 simple not equal to sl(n), the decomposition of 𝔤 ⊗ 𝔤 is multiplicity free, hence non-trivial invariant maps ∧^{2}𝔤 → S^{2}𝔤 do not exist at all. It follows that for 𝔤 ≠ sl(n), there do not exist quadratic algebras (S𝔤)_{h} which are U_{h}(𝔤) invariant quantizations of S𝔤.

Question 3.1. Prove that there exist no one parameter U_{h}(𝔤) invariant quantizations of S𝔤 (not necessarily in the class of quadratic algebras) for all simple Lie algebras 𝔤 ≠ sl(n).

Now we prove that for simple 𝔤 ≠ sl(n), the double quantization does not exist (not necessarily in the class of quadratic-linear algebras).

Proposition 3.6. Let 𝔤 be a simple finite dimensional Lie algebra not equal to sl(n). Then a U_{h}(𝔤) invariant quantization of the Lie bracket on 𝔤^{∗} does not exist.

Proof. If such a quantization exists, then from Corollary 2.1 follows that there exists an invariant bracket f on 𝔤^{∗} such that [[s, f]] = 0 and [[f, f]] = −\overline{φ}. Here s is the Lie bracket and \overline{φ} is the three-vector field induced by φ (see Proposition 3.3). We show that such f does not exist. Observe that \overline{φ} has type (3, 3), i.e., is a sum of terms of the view b ∂_{x} ∧ ∂_{y} ∧ ∂_{z}, where b is a homogeneous polynomial of degree 3. Observe also that the Schouten bracket of two polyvector fields of degrees (i, j) and (k, l) is a polyvector field of degree (i + k − 1, j + l − 1). We shall write i for degree (i, j) when the second number, j, is clear from context.

It is obvious that on 𝔤 there are no invariant bivector fields of degree 0 and, up to a factor, there is a unique invariant bivector field of degree 1, the Lie bracket s itself. Since 𝔤 ≠ sl(n), there are no bivector fields of degree 2 (see Remark 3.4). Therefore, f must be of the form: f = s + f_{1}, where f_{1} is a bracket of degree ≥ 3. Since f is compatible with s and [[f, f]] = −\overline{φ}, it must be [[f_{1}, f_{1}]] = −\overline{φ}. But it is impossible, because [[f_{1}, f_{1}]] has at least degree 5.

3.4 Quantum de Rham complex on (sl(n))^{∗}

Consider the algebra Ω^{•} of differencial forms on 𝔤^{∗} with polynomial coefficients. This is a graded differential algebra with differential d of degree 1 which forms the de Rham complex

(3.34)

where Ω^{k} is the space of k-forms with polynomial coefficients.

We call a complex over ℂ[[h]]

Ω_{h}^{•} : (S𝔤)_{h} ⟶^{d_{h}} Ω_{h}^{1} ⟶^{d_{h}} Ω_{h}^{2} ⟶^{d_{h}} ⋯  (3.35)

a quantum (deformed) de Rham complex if it consists of U_{h}(𝔤) invariant topologically free modules over ℂ[[h]] and coincides with (3.34) at h = 0.

Proposition 3.7. Let 𝔤 = sl(n). Then the quantized polynomial algebra (S𝔤)_{h} from (3.31) can be included in a U_{h}(𝔤) invariant graded differential algebra, Ω_{h}^{•}, which form a quantum de Rham complex (3.35).

Proof. First of all, define a quantum exterior algebra, (Λ𝔤)_{h}, an algebra of differential forms with constant coefficients. Let us modify the operator σ_{h} from (3.31). Since the representation 𝔤_{h}^{∗} is isomorphic to 𝔤_{h}, there exists a U_{h}(𝔤) invariant bilinear form on 𝔤_{h}, deformed Killing form. This form can be naturally extended to all tensor degrees 𝔤_{h}^{⊗k}. Let W_{h}^{2} be the ℂ[[h]] submodule in 𝔤_{h} ⊗ 𝔤_{h} orthogonal to V_{h}^{2} = Im(id ⊗ id −σ_{h}). Define an operator σ̄_{h} on 𝔤_{h} ⊗ 𝔤_{h} in such a way that it has the eigenvalues −1 on V_{h}^{2} and 1 on W_{h}^{2}. It is clear that V_{h}^{2} and W_{h}^{2} are deformed skew symmetric and symmetric subspaces of 𝔤 ⊗ 𝔤.

Now observe that the third graded component in the quadratic algebra (S𝔤)_{h} is the quotient of 𝔤_{h}^{⊗3} by the submodule V_{h}^{2} ⊗𝔤_{h} +𝔤_{h} ⊗V_{h}^{2}, hence this submodule and, therefore, the submodule V_{h}^{2} ⊗𝔤_{h} ∩𝔤_{h} ⊗V_{h}^{2} are direct submodules in 𝔤_{h}^{⊗3}, i.e., they have complement submodules. As the complement submodules one can choose the submodules W_{h}^{2} ⊗ 𝔤_{h} ∩ 𝔤_{h} ⊗ W_{h}^{2} and W_{h}^{2} ⊗ 𝔤_{h} + 𝔤_{h} ⊗ W_{h}^{2}, respectively, since they are complement at the point h = 0 and W_{h}^{2} is orthogonal to V_{h}^{2} with respect to the Killing form extended to 𝔤_{h} ⊗ 𝔤_{h}. Hence, W_{h}^{2} ⊗ 𝔤_{h} + 𝔤_{h} ⊗ W_{h}^{2} is a direct submodule. Also, the symmetric algebra S𝔤 is Koszul. From a result of Drinfeld, [Dr3] (see also [DM]), follows that the quadratic algebra (Λ𝔤)_{h} = T (𝔤_{h})/{W_{h}^{2}} is a free ℂ[[h]] module, i.e., is a U_{h}(𝔤)-invariant deformation of the exterior algebra Λ𝔤.

Call (Λ𝔤)_{h} a quantum exterior algebra over 𝔤.

Define a quantum algebra of differential forms over 𝔤^{∗} as the tensor product Ω_{h}^{•} = (S𝔤)_{h} ⊗ (Λ𝔤)_{h} in the tensor category of representations of the quantum group U_{h}(𝔤). The multiplication of two elements a⊗α and b⊗β looks like ab_{1}⊗α_{1}β, where b_{1}⊗α_{1} = S(α⊗b) for S = σR being the permutation in that category. So, Ω_{h}^{k} = (S𝔤)_{h} ⊗ (Λ^{k}𝔤)_{h}.

As in the classical case, the algebras (S𝔤)_{h} and (Λ𝔤)_{h} can be embedded in T (𝔤_{h}) as a graded submodules in the following way. Call the submodule W_{h}^{k} = (W_{h}^{2} ⊗ 𝔤_{h} ⊗ ⋯ ⊗ 𝔤_{h}) ∩ (𝔤_{h} ⊗ W_{h}^{2} ⊗ 𝔤_{h} ⊗ ⋯ ⊗ 𝔤_{h}) ∩ ⋯ ∩ (𝔤_{h} ⊗ 𝔤_{h} ⊗ ⋯ ⊗ W_{h}^{2}) of T^{k}(𝔤_{h}) a k-th symmetric part of T (𝔤_{h}). It is clear that the natural map π_{W} : T (𝔤_{h}) → (S𝔤)_{h} restricted to W_{h}^{k} is a bijection onto the k-degree component (S^{k}𝔤)_{h} of (S𝔤)_{h}. Denote by π′_{W} : (S^{k}𝔤)_{h} → W_{h}^{k} the inverse bijection. Similarly we define V_{h}^{k}, the k-th skew symmetric part of T (𝔤_{h}), and the bijection π′_{V} : (Λ^{k}𝔤)_{h} → V_{h}^{k}.

Now, define a differential d_{h} in Ω_{h}^{•} as a homogeneous operator of degree (−1, 1). It acts on an element, a ⊗ ω, of degree (k, m) in the following way. Let a ⊗ ω = (a_{1} ⊗ ⋯ ⊗ a_{k}) ⊗ (ω_{1} ⊗ ⋯ ⊗ ω_{m}) be its realization as an element from W_{h}^{k} ⊗ V_{h}^{m}. Then the formula

d_{h}(a ⊗ ω) = (a_{1} ⊗ ⋯ ⊗ a_{k−1} ⊗ π′_{V} π_{V} (a_{k} ⊗ ω_{1} ⊗ ⋯ ⊗ ω_{m})  (3.36)

presents the element d_{h}(a ⊗ ω) through its realization in W_{h}^{k−1} ⊗ V_{h}^{m+1}. It is obvious that d_{h}^{2} = 0.

So, the graded differential algebra Ω_{h}^{•} is constructed. It is easy to see that at the point h = 0 this algebra coincides with Ω^{•}.

Note that the quantum de Rham complex is exact, because it is exact at h = 0.

3.5 Restriction of (S𝔤)_{t,h} on orbits

In this section G = SL(n), 𝔤 = sl(n).

Let M be an invariant closed algebraic subset in 𝔤^{∗} and A the algebra of algebraic functions on M. The algebra A can be presented as a quotient of S𝔤 by some ideal, S𝔤 → A → 0.

We say that the quantization (S𝔤)_{t,h} can be restricted on M if there exists a U_{h}(𝔤) invariant quantization, A_{t,h}, of A, which can be presented as a quotient of (S𝔤)_{t,h} by some ideal, (S𝔤)_{t,h} → A_{t,h} → 0.

Note that, on the infinitesimal level, there are no obstructions for (S𝔤)_{t,h} to be restricted on M. Indeed, the Lie bracket on 𝔤^{∗} is strongly restricted on any orbit of G and induces the Kirillov-Kostant-Souriau bracket on M. Also , by Proposition 3.5, the bracket f involved in the quantization along h is also strongly restricted on any orbit.

From [DS1], one can derive that the problem of restriction of (S𝔤)_{t,h} is solved positively in case M is a minimal semisimple orbit, i.e., M is a hermitian symmetric space.

We are going to show here that the problem also has a positive solution for M being a maximal semisimple orbit, i.e., if M can be defined as a set of zeros of invariant functions from S𝔤. Such orbits are the orbits of diagonal matrices with distinct elements on diagonal.

Proposition 3.8. Let 𝔤 = sl(n). Then the family (S𝔤)_{t,h} can be restricted on any maximal semisimple orbit in 𝔤^{∗}.

Proof. There exists an isomorphism of U_{h}(𝔤) modules (S𝔤)_{h} → W_{h}, where W_{h} = ⊕_{k}W_{h}^{k}, the direct sum of the k-th symmetric parts of T (𝔤_{h}) (see previous Subsection). Consider the composition W_{h}[t] → T (𝔤_{h})[t] → (S𝔤)_{t,h}, where the last map appears from (3.30). It is an isomorphism, since it is an isomorphism at the point h = 0. It follows that (S𝔤)_{t,h} is isomorphic to W_{h}[t] as a U_{h}(𝔤)-module,

Denote by I_{t,h} the submodule of U_{h}(𝔤) invariant elements in (S𝔤)_{t,h}. It is obvious that I_{t,h} is isomorphic to ⊕_{k}I_{h}^{k}[t], where I_{h}^{k} is the invariant submodule in W_{h}^{k}. Hence, I_{t,h} is a direct free ℂ[[h]][t] submodule in (S𝔤)_{t,h}. Moreover, I_{t,h} is a central subalgebra in (S𝔤)_{t,h}. Indeed, for a generic t the algebra (S𝔤)_{t,h} can be invariantly embedded in U_{h}(𝔤). But ad(U_{h}(𝔤)) invariant elements in U_{h}(𝔤) form the center of U_{h}(𝔤). Also, I_{t,h} as an algebra is isomorphic to I[[h]][t] with the trivial action of U_{h}(𝔤), where I = I_{0,0}, the algebra of invariant elements in S𝔤. This follows from the fact that I is a polynomial algebra, [Dix], and, therefore, admits no nontrivial commutative deformations.

By the Kostant theorem, [Dix], U(𝔤) is a free module over its center. It follows that at the point h = 0 the module (S𝔤)_{t,0} is a free module over the algebra I_{t,0}. One can easily derive from this that (S𝔤)_{t,h} is a free module over I_{t,h}.

Now, let the maximal semisimple orbit M be defined by invariant elements from I. Consider a character defined by M, the algebra homomorphism λ : I → ℂ which takes each element from I to its value on M. Then, ℂ may be considered as an I-module, and the function algebra A on M is equal to S𝔤/ Ker(λ)S𝔤 = S𝔤 ⊗_{I} ℂ. Extend the character λ up to a character λ_{t,h} : I_{h,t} → ℂ[[h]][t] in the trivial way and consider ℂ[[h]][t] as a I_{h,t}-module. The tensor product over I_{t,h}

A_{t,h} = (S𝔤)_{t,h} ⊗ ℂ[[h]][t]

is a ℂ[[h]][t]-algebra. It is a free ℂ[[h]][t]-module, since (S𝔤)_{t,h} is a free one over I_{t,h}.

It is obvious, that A_{0,0} = A, A_{t,0} gives a quantization of the KKS bracket on M, and A_{t,h} is a quotient algebra of (S𝔤)_{t,h}.

In a next paper we shall prove that the quantization (S𝔤)_{t,h} can be restricted on all semisimple orbits.

Question 3.2. Can be the quantization (S𝔤)_{t,h} restricted on all orbits (not necessarily semisimple)?

As we have seen, the corresponding Poisson brackets are strongly restricted on all the orbits.

In next Section we consider the U_{h}(𝔤) invariant quantizations on semisimple orbits in 𝔤^{∗} for all simple Lie algebras 𝔤. It turns out that in general, on a given orbit there are many nonequivalent quantizations which are not restrictions from a quantization on 𝔤^{∗}. From this point of view, the quantization on maximal orbits described by Proposition (3.8) is a distinguished one.

4 The one and two parameter quantization on semisimple orbits in 𝔤^{∗}

4.1 Pairs of brackets on semisimple orbits

Let 𝔤 be a simple complex Lie algebra, 𝔥 a fixed Cartan subalgebra. Let Ω ⊂ 𝔥^{∗} be the system of roots corresponding to 𝔥. Select a system of positive roots, Ω^{+}, and denote by Π ⊂ Ω the subset of simple roots. Fix an element E_{α} ∈ 𝔤 of weight α for each α ∈ Ω^{+} and choose E_{−α} such that

(E_{α}, E_{−α}) = 1  (4.1)

for the Killing form (·, ·) on 𝔤.

Let Γ be a subset of Π. Denote by 𝔥_{Γ}^{∗} the subspace in 𝔥^{∗} generated by Γ. Note, that 𝔥^{∗} = 𝔥_{Γ}^{∗} ⊕ 𝔥_{Π\Γ}^{∗}, and one can identify 𝔥_{Π\Γ}^{∗} and 𝔥^{∗}/𝔥_{Γ}^{∗} via the projection 𝔥^{∗} → 𝔥^{∗}/𝔥_{Γ}^{∗}.

Let Ω_{Γ} ⊂ 𝔥_{Γ}^{∗} be the subsystem of roots in Ω generated by Γ, i.e., Ω_{Γ} = Ω ∩ 𝔥_{Γ}^{∗}. Denote by 𝔤_{Γ} the subalgebra of 𝔤 generated by the elements {E_{α}, E_{−α}}, α ∈ Γ, and 𝔥. Such a subalgebra is called the Levi subalgebra.

Let G be a complex connected Lie group with Lie algebra 𝔤 and G_{Γ} a subgroup with Lie algebra 𝔤_{Γ}. Such a subgroup is called the Levi subgroup. It is known that G_{Γ} is a connected subgroup. Let M be a homogeneous space of G and G_{Γ} be the stabilizer of a point o ∈ M. We can identify M and the coset space G/G_{Γ}. It is known, that such M is isomorphic to a semisimple orbit in 𝔤^{∗}. This orbit goes through an element λ ∈ 𝔤^{∗} which is just the trivial extension to all of 𝔤^{∗} (identifying 𝔤 and 𝔤^{∗} via the Killing form) of a map λ : 𝔥_{Π\Γ} → ℂ such that λ(α) ≠ 0 for all α ∈ Π \ Γ. Conversely, it is easy to show that any semisimple orbit in 𝔤^{∗} is isomorphic to the quotient of G by a Levi subgroup.

The projection π : G → M induces the map π_{∗} : 𝔤 → T_{o}, where T_{o} is the tangent space to M at the point o. Since the ad-action of 𝔤_{Γ} on 𝔤 is semisimple, there exists an ad(𝔤_{Γ})-invariant subspace, 𝔪 = 𝔪_{Γ}, of 𝔤 complementary to 𝔤_{Γ}, and one can identify T_{o} and 𝔪 by means of π_{∗}. It is easy to see that subspace 𝔪 is uniquely defined and has a basis formed by the elements E_{γ}, E_{−γ}, γ ∈ Ω^{+} \ Ω_{Γ}.

Let v ∈ 𝔤^{⊗m} be a tensor over 𝔤. Using the right and the left actions of G on itself, one can associate with v right and left invariant tensor fields on G denoted by v^{r} and v^{l}.

We say that a tensor field, t, on G is right G_{Γ} invariant, if t is invariant under the right action of G_{Γ}. The G equivariant diffeomorphism between M and G/G_{Γ} implies that any right G_{Γ} invariant tensor field t on G induces tensor field π_{∗}(t) on M. The field π_{∗}(t) will be invariant on M if, in addition, t is left invariant on G, and any invariant tensor field on M can be obtained in such a way. Let v ∈ 𝔤^{⊗m}. For v^{l} to be right G_{Γ} invariant it is necessary and sufficient that v to be ad(𝔤_{Γ}) invariant. Denote π^{r}(v) = π_{∗}(v^{r}) for any tensor v on 𝔤 and π^{l}(v) = π_{∗}(v^{l}) for any ad(𝔤_{Γ}) invariant tensor v on 𝔤. Note, that tensor π^{r}(v) coincides with the image of v by the map 𝔤^{⊗m} → Vect(M)^{⊗m} induced by the action map 𝔤 → Vect(M). Any G invariant tensor on M has the form π^{l}(v). Moreover, v clearly can be uniquely chosen from 𝔪^{⊗m}.

Denote by [[v, w]] ∈ ∧^{k+l−1}𝔤 the Schouten bracket of the polyvectors v ∈ ∧^{k}𝔤, w ∈ ∧^{l}𝔤, defined by the formula

[[X_{1} ∧ ⋯ ∧ X_{k}, Y_{1} ∧ ⋯ ∧ Y_{l}]] = ∑(−1)^{i+j}[X_{i}, Y_{j}] ∧ X_{1} ∧ ⋯ X̂_{i} ⋯ Ŷ_{j} ⋯ ∧ Y_{l},

where [·, ·] is the bracket in 𝔤. The Schouten bracket is defined in the same way for polyvector fields on a manifold, but instead of [·, ·] one uses the Lie bracket of vector fields. We will use the same notation for the Schouten bracket on manifolds. It is easy to see that π^{r}([[v, w]]) = [[π^{r}(v), π^{r}(w)]], and the same relation is valid for π^{l}.

Denote by \overline{Ω}_{Γ} the image of Ω in 𝔥_{Π\Γ}^{∗} without zero. It is clear that Ω_{Π\Γ} can be identified with a subset of \overline{Ω}_{Γ} and each element from \overline{Ω}_{Γ} is a linear combination of elements from Π \ Γ with integer coefficients which all are either positive or negative. Thus, the subset \overline{Ω}_{Γ}^{+} ⊂ \overline{Ω}_{Γ} of the elements with positive coefficients is exactly the image of Ω^{+}. We call elements of \overline{Ω}_{Γ} quasiroots and the images of Π \ Γ simple quasiroots.

Proposition 4.1. The space 𝔪 considered as a 𝔤_{Γ} representation space decomposes into the direct sum of subrepresentations 𝔪β̄, β̄ ∈ \overline{Ω}_{Γ}, where 𝔪β̄ is generated by all the elements E_{β}, β ∈ Ω, such that the projection of β is equal to β̄. This decomposition have the following properties:

a) all 𝔪β̄ are irreducible;

b) for β̄_{1}, β̄_{2} ∈ \overline{Ω}_{Γ} such that β̄_{1} + β̄_{2} ∈ \overline{Ω}_{Γ} one has [𝔪β̄_{1}, 𝔪β̄_{2}] = 𝔪_{β̄_{1}+β̄_{2}};

c) for any pair β̄_{1}, β̄_{2} ∈ \overline{Ω}_{Γ} the representation 𝔪β̄_{1} ⊗ 𝔪β̄_{2} is multiplicity free.

Proof. Statements a) and b) are proven in [DGS]. Statement c) follows from the fact that all the weight subspaces for all 𝔪β̄ have the dimension one (see N.Bourbaki, Groupes et algèbres de Lie, Chap. 8.9, Ex. 14).

Since 𝔤_{Γ} contains the Cartan subalgebra 𝔥, each 𝔤_{Γ} invariant tensor over 𝔪 has to be of weight zero. It follows that there are no invariant vectors in 𝔪. Hence, there are no invariant vector fields on M.

Consider the invariant bivector fields on M. From the above, such fields correspond to the 𝔤_{Γ} invariant bivectors from ∧^{2}𝔪. Note, that any 𝔥 invariant bivector from ∧^{2}𝔪 has to be of the form ∑ c(α)E_{α} ∧ E_{−α}.

Proposition 4.2. A bivector v ∈ ∧^{2}𝔪 is 𝔤_{Γ} invariant if and only if it has the form v = ∑ c(α)E_{α} ∧ E_{−α} where the sum runs over α ∈ Ω^{+} \ Ω_{Γ}, and for two roots α, β which give the same element in 𝔥^{∗}/𝔥_{Γ}^{∗} one has c(α) = c(β).

Proof. Follows from Proposition 4.1 and condition (4.1)

This proposition shows, that coefficients of an invariant element v = ∑ c(α)E_{α} ∧ E_{−α} depend only of the image of α in \overline{Ω}_{Γ}^{+}, denoted ᾱ, so v can be written in the form v = ∑ c(ᾱ)E_{α} ∧ E_{−α}. Let v ∈ ∧^{2}𝔪 be of the form v = ∑ c(ᾱ)E_{α} ∧ E_{−α}, where the sum runs over α ∈ Ω^{+} \Ω_{Γ}. Denote by θ the Cartan automorphism of 𝔤. Then, v is θ anti-invariant, i.e., θv = −v. Hence, any 𝔤_{Γ} invariant bivector is θ anti-invariant. If v, w ∈ ∧^{2}𝔪 are 𝔤_{Γ} invariant, then [[v, w]] is θ invariant and is of the form [[v, w]] = ∑ e(ᾱ, β̄)E_{α+β} ∧E_{−α} ∧E_{−β} where roots α, β are both negative or both positive and e(ᾱ, β̄) = −e(−ᾱ, −β̄). Hence, to calculate [[v, w]] for such v and w it is sufficient to calculate coefficients e(ᾱ, β̄) for positive ᾱ and β̄.

Define by φ_{M} the invariant three-vector field on M determined by the invariant element φ ∈ ∧^{3}𝔤. A direct computation shows (see [DGS]) that the Schouten bracket of bivector v = ∑ c(ᾱ)E_{α} ∧ E_{−α} with itself is equal to K^{2}φ_{M} for a complex number K, if and only if the following equations hold

c(ᾱ + β̄)(c(ᾱ) + c(β̄)) = c(ᾱ)c(β̄) + K^{2}  (4.2)

for all the pairs of positive quasiroots ᾱ, β̄ such that ᾱ + β̄ is a quasiroot. So, if c(ᾱ) and c(β̄) are given and c(ᾱ) + c(β̄) ≠ 0,

c(ᾱ + β̄) = \frac{c(ᾱ)c(β̄) + K^{2}}{c(ᾱ) + c(β̄)} .  (4.3)

Proposition 4.3. Let (ᾱ_{1}, ⋯, ᾱ_{k}) be the k-tuple of all simple quasiroots. Given a k-tuple of complex numbers (c_{1}, ..., c_{k}), assign to each ᾱ_{i} the number c_{i}. Then

a) for almost all k-tuples of complex numbers (except an algebraic subset in ℂ^{k} of lesser dimension) equations (4.3) uniquely define numbers c(ᾱ) for all positive quasiroots ᾱ = ∑ ᾱ_{i} such that the bivector f = ∑ c(ᾱ)E_{α} ∧ E_{−α} satisfies the condition

[[f, f]] = K^{2}φ_{M};

b) when K = 0, the solution described in part a) defines a Poisson bracket on M. Numbers c(ᾱ) give a solution of (4.2) if and only if there exists a linear form λ ∈ 𝔥_{Π\Γ}^{∗} such that

c(ᾱ) = \frac{1}{λ(ᾱ)}  (4.4)

for all quasiroots ᾱ.

Proof. See [DGS].

Remark 4.1. This proposition shows that invariant brackets f on M defined by part a) of the proposition form a k-dimensional variety, X_{K}, where k is the number of simple quasiroots. On the other hand, k = dim H^{2}(M), [Bo]. If K is regarded as indeterminate, then f forms a k + 1 dimensional variety, X ⊂ ℂ^{k} × ℂ, (component ℂ corresponds to K). Subvariety X_{0} corresponds to K = 0, i.e., consists of Poisson brackets. It is easy to see that all the Poisson brackets with c(ᾱ) = 1/λ(ᾱ) ≠ 0 are nondegenerate. Since X is connected, it follows that almost all brackets f (except an algebraic subset in X of lesser dimension) are nondegenerate as well.

Remark 4.2. Equations (4.3) show that when c(ᾱ) + c(β̄) = 0, there appears a harm for determining c(ᾱ + β̄) from given c(ᾱ) and c(β̄). Nevertheless, it is easy to derive from equations (4.2) that

(*) If c(ᾱ) + c(β̄) = 0 then necessarily c(ᾱ) = ±K, c(β̄) = ∓K. So it is naturally to consider the quasiroots ᾱ where c(ᾱ) are equal to ±K or not separately.

Let c(ᾱ), ᾱ ∈ \overline{Ω}_{Γ}, be a solution of equations (4.2) (we assume c(−ᾱ) = −c(ᾱ)). It is easy to derive from equations (4.2) the following properties.

(**) If c(ᾱ) = ±K and c(β̄) ≠ ±K, then c(ᾱ + β̄) = ±K and c(ᾱ − β̄) = ±K; (***) If c(ᾱ) = ±K and c(β̄) = ±K, then c(ᾱ + β̄) = ±K.

Let \overline{Ω}′_{Γ} ⊂ \overline{Ω}_{Γ} be the subset of quasiroots ᾱ such that c(ᾱ) ≠ ±K. From (**) follows that \overline{Ω}′_{Γ} is a linear subset, i.e., \overline{Ω}′_{Γ} = \overline{Ω}_{Γ} ∩ span(\overline{Ω}′_{Γ}), where span(\overline{Ω}′_{Γ}) is the vector subspace of 𝔥^{∗}/𝔥_{Γ}^{∗} generated by \overline{Ω}′_{Γ}. Let (ᾱ_{1}, ⋯, ᾱ_{k}) be a k-tuple of elements from \overline{Ω}′_{Γ} that form a basis of span(\overline{Ω}′_{Γ}). Since by (*) c(ᾱ) + c(β̄) ≠ 0 for any ᾱ, β̄ ∈ \overline{Ω}′_{Γ}, all c(ᾱ), ᾱ ∈ \overline{Ω}′_{Γ}, can be found from (4.3) using the initial values c_{i} = c(ᾱ_{i}), as in Proposition 4.3.

Note that since c_{i} ≠ ±K, there are uniquely defined complex numbers λ_{i} ≠ 0, 1 such that c(ᾱ_{i}) = c_{i} = Kψ(λ_{j}), where

ψ(x) = \frac{x + 1}{x − 1}.

Using the formula

ψ(xy) = \frac{ψ(x)ψ(y) + 1}{ψ(x) + ψ(y)},

it is easy to derive that if λ : \overline{Ω}′_{Γ} → ℂ^{∗} is the multiplicative map (such that if ᾱ, β̄, ᾱ+β̄ ∈ \overline{Ω}′_{Γ} then λ(ᾱ + β̄) = λ(ᾱ)λ(β̄) ) defined by c(ᾱ_{i}) = λ_{i}, then the solution of (4.3) is given by the formula

c(ᾱ) = Kψ(λ(ᾱ)),  ᾱ ∈ \overline{Ω}′_{Γ}.  (4.5)

For correctness of this formula, one needs that the map λ to be regular, i.e., that λ to satisfy the condition: if ᾱ, β̄, ᾱ + β̄ ∈ \overline{Ω}′_{Γ} then λ(ᾱ)λ(β̄) = 1 only when ᾱ = −β̄.

From property (**) follows that the numbers c(ᾱ) define a function on the set π(\overline{Ω}_{Γ}), where π is the natural map 𝔥^{∗}/𝔥_{Γ}^{∗} → (𝔥^{∗}/𝔥_{Γ}^{∗})/span(\overline{Ω}′_{Γ}). This function has values ±K. Let X ⊂ π(\overline{Ω}_{Γ}) be the subset where this function has value K. From property (***) follows that X is a semilinear subset. It means that if x_{1}, x_{2} ∈ X and x_{1} + x_{2} ∈ π(\overline{Ω}_{Γ}) then x_{1} + x_{2} ∈ X, and X ∩ (−X) = ∅, X ∪ (−X) = π(\overline{Ω}_{Γ}).

The arguments above lead to the following description of the variety Z_{K} of all solutions of (4.2) (or, what is the same, the variety of invariant brackets f on M such that [[f, f]] = K^{2}φ_{M}).

Proposition 4.4. Variety Z_{K} splits into stratas. Each strata is defined by choosing a linear subset \overline{Ω}′_{Γ} of \overline{Ω}_{Γ} and a semilinear subset X of π(\overline{Ω}_{Γ}). Points of this strata are parameterized by the multiplicative regular maps λ : \overline{Ω}′_{Γ} → ℂ^{∗}.

Let the data (\overline{Ω}′_{Γ}, X, λ) corresponds to a point of Z_{K}. Then the coefficients c(ᾱ) of f are determined in the following way. If ᾱ ∈ \overline{Ω}′_{Γ} then c(ᾱ) is found by (4.5). If π(ᾱ) ∈ X then c(ᾱ) = K. If π(ᾱ) ∈ −X then c(ᾱ) = −K.

Of course, in case K = 0 the choose of X does not matter: a strata of Z_{0} is determined only by choosing \overline{Ω}′_{Γ}.

Note also that the description of Z_{K} given in the proposition does not depend on choosing a basis in \overline{Ω}_{Γ}. The variety X_{K} from the previous remark forms an open everywhere dense subset of Z_{K} and does depend on choosing a basis. According to Remark 2.1 this proposition describes all the (G, r̃)-Poisson structures on semisimple orbits.

Now we fix a Poisson bracket s = ∑(1/λ(ᾱ))E_{α} ∧ E_{−α}, where λ is a fixed linear form, and describe the invariant brackets f = ∑ c(ᾱ)E_{α} ∧ E_{−α} which satisfy the conditions

[[f, f]] = K^{2}φ_{M}  for K ≠ 0,  (4.6)
[[f, s]] = 0.

Direct computation shows that the condition [[f, s]] = 0 is equivalent to the system of equations for the coefficients c(ᾱ) of f

c(ᾱ)λ(ᾱ)^{2} + c(β̄)λ(β̄)^{2} = c(ᾱ + β̄)λ(ᾱ + β̄)^{2}  (4.7)

for all the pairs of positive quasiroots ᾱ, β̄ such that ᾱ + β̄ is a quasiroot.

Definition 4.1. Let M be an orbit in 𝔤^{∗} (not necessarily semisimple). We call M a good orbit, if there exists an invariant bracket, f, on M satisfying the conditions (4.6) for s the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M.

So, a semisimple orbit M is a good orbit if and only if equations (4.2) and (4.7) are compatible, i.e., have a common solution.

Proposition 4.5. The good semisimple orbits are the following:

a) For 𝔤 of type A_{n} all semisimple orbits are good.

b) For all other 𝔤, the orbit M is good if and only if the set Π \ Γ consists of one or two roots which appear in representation of the maximal root with coefficient 1.

c) The brackets f on good orbits form a one-dimensional variety: all such brackets have the form

±f_{0} + ts,

where t ∈ ℂ and f_{0} is a fixed bracket satisfying (4.6).

Proof. See [DGS].

Remark 4.3. From Proposition (3.5) follows that for 𝔤 = sl(n) all orbits (not only semisimple) are good ones. In addition, if an orbit, M, is such that φ_{M} = 0, then M is good: one can take f = 0. In [GP] there is a classification of orbits for all simple 𝔤, for which φ_{M} = 0.

Question 4.1. Let 𝔤 be a simple Lie algebra. Are all orbits in 𝔤^{∗} good? If not, what is a classification of good orbits?

4.2 Cohomologies defined by invariant brackets

In the next subsection we prove the existence of a U_{h}(𝔤) invariant quantiztion of the Poisson brackets described above using the methods of [DS1]. This requires us to consider the 3-cohomology of the complex (Λ^{•}(𝔤/𝔤_{Γ}))^{𝔤_{Γ}} = (Λ^{•}𝔪)^{𝔤_{Γ}} of 𝔤_{Γ} invariants with differential given by the Schouten bracket with the bivector f ∈ (Λ^{2}𝔪)^{𝔤_{Γ}} from Proposition 4.3 a),

δ_{f} : u ↦ [[f, u]] for u ∈ (Λ^{•}𝔪)^{𝔤_{Γ}}.

The condition δ_{f}^{2} = 0 follows from the Jacobi identity for the Schouten bracket together with the fact that [[f, f]] = K^{2}φ_{M}. Denote these cohomologies by H^{k}(M, δ_{f}), whereas the usual de Rham cohomologies are denoted by H^{k}(M).

Recall (see Remark 4.1) that the brackets f satisfying [[f, f]] = K^{2}φ form a connected variety X which contains a submanifold X_{0} of Poisson brackets.

Proposition 4.6. For almost all f ∈ X (except an algebraic subset of lesser dimension) one has

H^{k}(M, δ_{f}) = H^{k}(M)

for all k. In particular, H^{k}(M, δ_{f}) = 0 for odd k.

Proof. First, let v be a Poisson bracket, i.e., v ∈ X_{0}. Then the complex of polyvector fields on M, Θ^{•}, with the differential δ_{v} is well defined. Denote by Ω^{•} the de Rham complex on M. Since none of the coefficients c(ᾱ) of v are zero, v is a nondegenerate bivector field, and therefore it defines an A-linear isomorphism ṽ : Ω^{1} → Θ^{1}, ω ↦ v(ω, ·), which can be extended up to the isomorphism ṽ : Ω^{k} → Θ^{k} of k-forms onto k-vector fields for all k. Using Jacobi identity for v and invariance of v, one can show that ṽ gives a G invariant isomorphism of these complexes, so their cohomologies are the same.

Since 𝔤 is simple, the subcomplex of 𝔤 invariants, (Ω^{•})^{𝔤}, splits off as a subcomplex of Ω^{•}. In addition, 𝔤 acts trivially on cohomologies, since for any g ∈ G the map M → M, x ↦ gx, is homotopic to the identity map, (G is a connected Lie group corresponding to 𝔤). It follows that cohomologies of complexes (Ω^{•})^{𝔤} and Ω^{•} coincide.

But ṽ gives an isomorphism of complexes (Ω^{•})^{𝔤} and (Θ^{•})^{𝔤} = ((Λ^{•}𝔪)^{𝔤_{Γ}}, δ_{v}). So, cohomologies of the latter complex coincide with de Rham cohomologies, which proves the proposition for v being Poisson brackets.

Now, consider the family of complexes ((Λ^{•}𝔪)^{𝔤_{Γ}}, δ_{v}), v ∈ X. It is clear that δ_{v} depends algebraicly on v. It follows from the uppersemicontinuity of dim H^{k}(M, δ_{v}) and the fact that H^{k}(M) = 0 for odd k, [Bo], that H^{k}(M, δ_{v}) = 0 for odd k and almost all v ∈ X. Using the uppersemicontinuity again and the fact that the number ∑_{k}(−1)^{k} dim H^{k}(M, δ_{v}) is the same for all v ∈ X, we conclude that dim H^{k}(M, δ_{v}) = dim H^{k}(M) for even k and almost all v.

Remark 4.4. Call f ∈ X admissible, if it satisfies Proposition 4.6. From the proof of the proposition follows that the subset D such that X \D consists of admissible brackets does not intersect with the subset X_{0} consisting of Poisson brackets. It follows from this fact that for each good orbit there are admissible f compatible with the KKS bracket. Indeed, let M be a good orbit and f_{0} + ts the family from Proposition 4.5 c) satisfying (4.6) for a fixed K. Then for almost all numbers t this bracket is admissible. In fact, this family is contained in the two parameter family uf_{0} + ts. By u = 0, t ≠ 0 we obtain admissible brackets. So, there exist u_{0} ≠ 0 and t_{0} such that the bracket u_{0}f_{0} + t_{0}s is admissible. It follows that the bracket f_{0} + (t_{0}/u_{0})w is admissible, too. So, in the family f_{0} + ts there is an admissible bracket, and we conclude that almost all brackets in this family (except a finitely many) are admissible.

For the proof of existence of two parameter quantization for the cases D_{n} and E_{6} in the next subsection, we will use the following result on invariant three-vector fields.

Denote by θ the Cartan automorphism of 𝔤.

Lemma 4.1. For either D_{n} or E_{6} and one of the subsets, Γ, of simple roots such that G_{Γ} defines a good orbit, any 𝔤_{Γ} and θ invariant element v in Λ^{3}𝔪 is a multiple of φ_{M}, that is,

(Λ^{3}(𝔪)^{𝔤_{Γ}} ≅ ⟨φ_{M}⟩.

Proof. In this case the system of positive quasiroots consists of ᾱ, β̄, and ᾱ + β̄, where ᾱ, β̄ are the simple quasiroots. From Proposition 4.1 follows that invariant elements in 𝔪ᾱ ⊗ 𝔪β̄ ⊗ 𝔪_{−ᾱ−β̄} and 𝔪_{−ᾱ} ⊗ 𝔪_{−β̄} ⊗ 𝔪_{ᾱ+β̄} form subspaces of dimension one, I_{1} and I_{2}. Moreover, all the ivariant elements of Λ^{3}𝔪 are lying in I_{1} + I_{2}. Since θ takes I_{1} onto I_{2}, there is only one-dimansional θ invariant subspace in I_{1}+I_{2}, which is necessarily generated by φ_{M}.

4.3 U_{h}(𝔤) invariant quantizations in one and two parameters

In this subsection we prove the existence of one and two parameter U_{h}(𝔤) invariant quantization of the function algebras A on semisimple orbits, M, in 𝔤^{∗}. By Proposition 2.2, the one parameter quantization has the Poisson bracket of the form

f(a, b) − {a, b}_{r},  [[f, f]] = −φ_{M}.  (4.8)

We show that the one parameter quantization exists for all semisimple orbits and all f constructed in Proposition 4.3 a) and satisfying Proposition 4.6.

For two parameter quantization, there are two compatible Poisson brackets: the KKS bracket s and the bracket of the form (4.8) with the additional condition

[[f, s]] = 0.  (4.9)

We show that the two parameter quantization exists for good orbits in cases D_{n} and E_{6} and for almost all f satisfying (4.8) and (4.9).

Note that in subsection 3.5 we have proven that in case A_{n} the two parameter quantization exists for maximal semisimple orbits. In a next paper we shall prove the same for all semisimple orbits.

We remind the reader of the method in [DS1]. The first step is to construct a U(𝔤) invariant quantization in the category C(U(𝔤)[[h]], Δ, Φ_{h}). Then we use the equivalence given by the pair (Id, F_{h}) between the monoidal categories C(U(𝔤)[[h]], Δ, Φ_{h}) and C(U(𝔤)[[h]], Δ_{h}, 1) to define a U_{h}(𝔤) invariant quantization, either μ_{h}F_{h}^{−1} in the one parameter case or μ_{t,h}F_{h}^{−1} in the two parameter case (see Subsections 2.2 and 2.3). In the following we often write Φ for Φ_{h}.

Proposition 4.7. Let 𝔤 be a simple Lie algebra, M a semisimple orbit in 𝔤^{∗}. Then, for almost all (in sense of Proposition 4.6) 𝔤 invariant brackets f satisfying [[f, f]] = −φ_{M}, there exists a multiplication μ_{h} on A

μ_{h}(a, b) = ab + (h/2)f(a, b) + ∑_{n≥2} h^{n}μ_{n}(a, b)

which is U(𝔤) invariant (equation 2.12)) and Φ associative (equation (2.13)).

Proof. To begin, consider the multiplication μ^{(1)}(a, b) = ab + (h/2)f(a, b). The corresponding obstruction cocycle is given by

obs_{2} = \frac{1}{h^{2}} (μ^{(1)}(μ^{(1)} ⊗ id) − μ^{(1)}(id ⊗ μ^{(1)})Φ)

considered modulo terms of order h. No \frac{1}{h} terms appear because f is a biderivation and, therefore, a Hochschild cocycle. The fact that the presence of Φ does not interfere with the cocyle condition and that this equation defines a Hochschild 3-cocycle was proven in [DS1]. It is well known that if we restrict to the subcomplex of cochains given by differential operators, the differential Hochschild cohomology of A in dimension p is the space of p-polyvector fields on M. Since 𝔤 is reductive, the subspace of 𝔤 invariants splits off as a subcomplex and has cohomology given by (Λ^{p}𝔪)^{𝔤_{Γ}}. The complete antisymmetrization of a p-tensor projects the space of invariant differential p-cocycles onto the subspace (Λ^{p}𝔪)^{𝔤_{Γ}} representing the cohomology. The equation [[f, f]] + φ_{M} = 0 implies that obstruction cocycle is a coboundary, and we can find a 2-cochain μ_{2}, so that μ^{(2)} = μ^{(1)} + h^{2}μ_{2} satisfies

μ^{(2)}(μ^{(2)} ⊗ id) − μ^{(2)}(id ⊗ μ^{(2)})Φ = 0 mod h^{2}.

Assume we have defined the deformation μ^{(n)} to order h^{n} such that Φ associativity holds modulo h^{n}, then we define the (n + 1)st obstruction cocycle by

obs_{n+1} = \frac{1}{h^{n+1}} (μ^{(n)}(μ^{(n)} ⊗ id) − μ^{(n)}(id ⊗ μ^{(n)})Φ) mod h.

In [DS1] (Proposition 4) we showed that the usual proof that the obstruction cochain satisfies the cocycle condition carries through to the Φ associative case. The coboundary of obs_{n+1} appears as the h^{n+1} coefficient of the signed sum of the compositions of μ^{(n+1)} with obs_{n+1}. The fact that Φ = 1 mod h^{2} together with the pentagon identity implies that the sum vanishes identically, and thus all coefficients vanish, including the coboundary in question. Let obs′_{n+1} ∈ (Λ^{3}𝔪)^{𝔤_{Γ}} be the projection of obs_{n+1} on the totally skew symmetric part, which represents the cohomology class of the obstruction cocycle. The coefficient of h^{n+2} in the same signed sum, when projected on the skew symmetric part, is [[f, obs′_{n+1}]] which is the coboundary of obs′_{n+1} in the complex (Λ^{•}𝔪)^{𝔤_{Γ}}, δ_{f} = [[f, .]]). Thus obs′_{n+1} is a δ_{f} cocycle. By Proposition 4.6, this complex has zero cohomology. Now we modify μ^{(n+1)} by adding a term h^{n}μ_{n} with μ_{n} ∈ (Λ^{2}𝔪)^{𝔤_{Γ}} and consider the (n + 1)st obstruction cocycle for μ^{′(n+1)} = μ^{(n+1)} + h^{n}μ_{n}. Since the term we added at degree h^{n} is a Hochschild cocyle, we do not introduce a h^{n} term in the calculation of μ^{(n)}(μ^{(n)} ⊗ id) − μ^{(n)}(id ⊗ μ^{(n)})Φ and the totally skew symmetric projection h^{n+1} term has been modified by [[f, μ_{n}]]. By choosing μ_{n} appropriately, we can make the (n+1)st obstruction cocycle represent the zero cohomology class, and we are able to continue the recursive construction of the desired deformation.

Now we prove the existence of a two parameter deformation for good orbits in the cases D_{n} and E_{6}.

Proposition 4.8. Given a pair of 𝔤 invariant brackets, f, v, on a good orbit in D_{n} or E_{6} satisfying [[f, f]] = −φ_{M}, [[f, v]] = [[v, v]] = 0, there exists a multiplication μ_{h,t} on A

μ_{t,h}(a, b) = ab + (h/2)f(a, b) + (t/2)v(a, b) + ∑_{k,l≥1} h^{k}t^{l}μ_{k,l}(a, b)

which is U(𝔤) invariant and Φ associative.

Proof. The existence of a multiplication which is Φ associative up to and including h^{2} terms is nearly identical to the previous proof. Both f and v are anti-invariant under the Cartan involution θ. We shall look for a multiplication μ_{t,h} such that μ_{k,l} is θ anti-invariant and skew-symmetric for odd k + l and θ invariant and symmetric for even k + l.

So, suppose we have a multiplication defined to order n,

μ_{t,h}(a, b) = ab + hμ_{1}(a, b) + tμ′_{1}(a, b) + ∑_{k+l≤n} h^{k}t^{l}μ_{k,l}(a, b),

with mentioned above invariance properties and Φ associative to order h^{n}.

Further we shall suppose that Φ has the properties: It is invariant under the Cartan involution θ and Φ^{−1} = Φ_{321}. Such Φ always can be choosen, [DS2]. Using these properties for Φ, direct computation shows that the obstruction cochain,

obs_{n+1} = ∑_{k=0,... ,n+1} h^{k}t^{n+1−k}β_{k},

has the following invariance properties: For odd n, obs_{n+1} is θ invariant and obs_{n+1}(a, b, c) = −obs_{n+1}(c, b, a), and for even n, and obs_{n+1} is θ anti-invariant and obs_{n+1}(a, b, c) = obs_{n+1}(c, b, a).

Hence, the projection of obs_{n+1} on (Λ^{3}𝔪)^{𝔤_{Γ}} is equal to zero for even n. It follows that all the β_{k} are Hochschild coboundaries, and the standard argument implies that the multiplication can be extended up to order n + 1 with the required properties.

For odd n, Lemma 4.1 shows that the projection on (Λ^{3}𝔪)^{𝔤_{Γ}} has the form

obs_{n+1} = ( ∑_{k=0,... ,n+1} a_{k}h^{k}t^{n+1−k}) φ_{M}.

The KKS bracket is given by the two-vector

v = ∑_{α∈Ω^{+}\Ω_{Γ}} \frac{1}{λ(ᾱ)}E_{α} ∧ E_{−α}.

Setting

w = ∑_{α∈Ω^{+}\Ω_{Γ}} λ(ᾱ)E_{α} ∧ E_{−α},

gives

[[v, w]] = −3φ_{M}.

Defining

μ^{′(n)} = μ^{(n)} + \frac{a_{0}}{3} t^{n}w,

the new obstruction cohomology class is

obs′_{n+1} = ( ∑_{k=1,... ,n+1} a_{k}h^{k}t^{n+1−k})φ_{M}.

Finally we define

μ^{′′(n)} = μ^{′(n)} + ∑_{k=1,... ,n+1} a_{k}h^{k−1}t^{n+1−k})f

and get an obstruction cocycle which is zero in cohomology. Now the standard argument implies that the deformation can be extended to give a Φ associative invariant multiplication with the required properties of order n + 1.

So, we are able to continue the recursive construction of the desired multiplication.

Using the Φ_{h} associative multiplications μ_{h} and μ_{t,h} from Propositions 4.7 and 4.8 and the equivalence between the monoidal categories C(U(𝔤)[[h]], Δ, Φ_{h}) and C(U(𝔤)[[h]], Δ̃, 1) given by the pair (Id, F_{h}) (see Section 2), one can define U_{h}(𝔤) invariant multiplications, either μ_{h}F_{h}^{−1} in the one parameter case or μ_{t,h}F_{h}^{−1} in the two parameter case.

Remark 4.5. After [Ko], the philosophy is that there are no obstructions for quantizations of Poisson brackets on manifolds. In this connection, the following question arises:

Question 4.2. Let M be a G-manifold on which there exists an invariant connection. Given a G invariant Poisson bracket, v, on M, does there exist a G invariant quantization of v?

In case M is a homogeneous manifold the bracket v has a constant rank, and such a quantization can be obtained by Fedosov’s method, [Fed], [Do1].

Another question which relates to the topic of this paper is the following.

Question 4.3. Let M be a G-manifold on which there exists an invariant connection, U(𝔤) the corresponding to G universal enveloping algebra, and Φ_{h} ∈ (U(𝔤))^{⊗3}[[h]] an invariant element of the form (2.6) obeying the pentagon identity (2.7). Let f be an invariant bracket on M satisfying [[f, f]] = −φ_{M}. Does there exist a U(𝔤) (or G) invariant and Φ_{h} associative quantization of f (as in Proposition 4.7)?

Note that if the answer to this Question is positive, then the answer to Question 2.1 is also positive: we take for M the group G itself and consider it as a G-manifold by left multiplication.

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