The quantum Frobenius for character varieties and multiplicative quiver varieties
Abstract
We develop a general mechanism for constructing sheaves of Azumaya algebras on moduli spaces obtained by Hamiltonian reduction, via their quantizations at roots of unity. We achieve this by exploiting a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of Frobenius Poisson orders. We use these tools to construct canonical central subalgebras of quantum algebras, and explicitly compute the resulting Azumaya loci we encounter, using a natural nondegeneracy assumption. As our main applications, we prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and finally we compute the Azumaya locus of the Kauffman bracket skein algebras, proving a strong form of the Unicity Conjecture of Bonahon and Wong.
in: \UseAllTwocells\SilentMatrices\SelectTipscm
1 Introduction
In this paper we study quantized character varieties, and quantized multiplicative quiver varieties, when the quantization parameter is a root of unity. Our main results describe these quantized moduli spaces as Azumaya algebras – meaning that, étalelocally, they are sheaves of matrix algebras – over an explicitly given open locus on the spectrum of their centers. In the quiver examples, this spectrum is the classical multiplicative quiver variety, while for character variety examples it is the classical character variety. As a key application, we also treat the socalled Kauffman bracket skein algebra of a surface. The ‘Unicity Conjecture’ of BonahonWong asserts that the Kauffman bracket skein algebra is Azumaya over some nonempty open subset[BonahonWongI, BonahonWongII, BonahonWongIII]; in this form it was first proved by Frohman, KaniaBartoszynska, and Lê [FrohmanUnicity], who showed that the set of Azumaya points was nonempty and open but did not determine the locus precisely. We prove that the Azumaya locus in fact coincides with the smooth locus, hence yielding the strongest possible resolution of the Unicity Conjecture. Both classes of quantizations are constructed via the process of quantum Hamiltonian reduction, and in both cases the theory of Poisson orders equips the quantization with a connection over an open symplectic leaf. Our primary techniques, therefore, exploit several remarkable compatibilities between Lusztig’s quantum Frobenius homomorphism on the quantum group, the theory of Poisson orders, and the procedure of quantum Hamiltonian reduction. To this end, we develop the frameworks of Frobenius Poisson orders and their Frobenius quantum Hamiltonian reduction. We exploit the Frobenius Poisson order to determine the Azumaya points before reduction, and then we show that Frobenius Poisson orders descend to Poisson orders on the Hamiltonian reduction, and hence conclude that the Azumaya algebras we constructed before reduction descend to Azumaya algebras on the Hamiltonian reduction. A further remarkable feature of both classes of examples is that they are nondegenerate Hamiltonian Poisson spaces, which allows us to describe the symplectic leaves very explicitly in terms of the classical multiplicative moment map. We first recall our two motivating classes of examples, then state our main results concerning them, and then discuss the general results and techniques in more detail.
1.1 Character varieties and their quantization
Given a reductive group , the character stack of a connected topological surface (possibly with boundary) is the moduli space of local systems on , equivalently of representations , modulo the conjugation action. To construct , one considers first the framed character variety, , consisting of a local system with a fixed trivialization at one point; this is an affine variety, equipped with a action by changes of framing. The framed character variety carries the Fock–Rosly Poisson structure, and admits a multiplicative moment map . The character variety is then the quotient of by the action. Given now a closed surface of genus , let us denote by the surface obtained by removing some small disk. An important observation is that the character variety of is obtained from the character variety of via a procedure of groupvalued Hamiltonian reduction [AlekseevLiegroupvalued1998], [AlekseevQuasiPoisson], [SafronovQuasiHamiltonian]. This means that carries a multiplicative moment map valued in the group , which records the holonomy around the boundary. The multiplicative Hamiltonian reduction then computes the joint effect of attaching the disk and quotienting the action. The character variety obtained in this way is complicated – it is singular in general, and may have several irreducible components. There is an important open subset of the character variety called the ‘good locus’; the good locus consists of the closed orbits whose stabilizer is the center. It is empty in genus one, but nonempty for genus greater than one (and in fact dense for ). Functorial quantizations of character varieties were introduced in [BenZviIntegratingquantumgroups2018]; it was also proved there that the framed quantizations of punctured surfaces could be described algebraically via (mild generalizations of) certain “moduli” algebras defined combinatorially by Alekseev, Grosse, and Schomerus [AlekseevCombinatorialquantizationHamiltonian1996]. In the case of closed surfaces, the resulting quantized character varieties were shown in [BenZviQuantumcharactervarieties2018] to admit a description via quantum Hamiltonian reduction of the algebras , echoing the classical construction. In particular, there was introduced in [BenZviIntegratingquantumgroups2018] a “distinguished object” – a noncommutative standin for the structure sheaf – whose endomorphism algebra gives a quantization of the affine character variety, and which is computed via quantum Hamiltonian reduction. In the case , the Kauffman bracket skein algebra provides another celebrated quantization of the character variety. The Kauffman bracket skein algebra with parameter is the vector space spanned by isotopy classes of links drawn in the cylinder over the surface, modulo the relations,
where the diagrams represent links which are as depicted in some oriented 3ball, and identical outside of it. The algebra structure on is obtained by vertically stacking links in . It was shown by Turaev [TuraevSkein] that the Kauffman bracket skein algebra provides a deformation quantization of the character variety of .
1.2 Multiplicative quiver varieties and their quantization
Let be a quiver with dimension vector . The multiplicative quiver variety is a moduli space of representations of a doubled quiver satisfying certain moment map relations, first introduced in [CrawleyBoeveyMultiplicativepreprojectivealgebras2006]. It is constructed by first recalling that the collection of representations of the doubled quiver of with dimension vector forms a product of matrix spaces. The framed representation variety is an open locus of this product of matrix spaces, defined by the nonvanishing of certain determinants, and admits a multiplicative moment map to the gauge group , where the product runs over the set of vertices of . The multiplicative quiver variety is the multiplicative GIT Hamiltonian reduction of the framed representation variety by the gauge group at a moment parameter , and with stability parameter (see [VandenBerghDoublePoissonalgebras2008, YamakawaGeometryMultiplicativePreprojective2008]):
According to [CrawleyBoeveyMultiplicativepreprojectivealgebras2006], certain special cases of multiplicative quiver varieties yield moduli spaces of connections with irregular singularities or, equivalently, moduli spaces of representations of , with prescribed monodromies around the punctures. Quantizations of multiplicative quiver varieties were introduced in [JordanQuantizedmultiplicativequiver2014]. The construction involved first quantizing via an algebra , then quantizing the moment map, and defining as a quantum Hamiltonian reduction. For a more thorough recollection about multiplicative quiver varieties and their quantization, see Section LABEL:sec:DqMat. The algebras give deformations of , which are flat over the ring . In [JordanQuantizedmultiplicativequiver2014] it was shown that the quantum Hamiltonian reductions of are formally flat (i.e., they are flat when tensored over the ring , where ) whenever the classical multiplicative moment map is flat; by [CrawleyBoeveyMultiplicativepreprojectivealgebras2006] this can be read off from the dimension vector and the moment map parameters. However, flatness of the quantum Hamiltonian reduction over remains unsettled.
1.3 Main results: examples and applications
Our main results in the context of the preceding examples are as follows:
Theorem 1.1.
Let be a connected reductive group and a primitive th root of unity, which together satisfy Assumption LABEL:EllAssumption. Let be a closed topological surface of genus , and let us denote by the surface obtained by removing some open disk from . Then:

The moduli algebra is finitely generated over its center, which is isomorphic to the coordinate ring of the classical framed character variety .

Moreover, the Azumaya locus of the moduli algebra coincides with the preimage of open cell under the classical moment map .

The quantized character variety of the closed surface is finitely generated over its center, which is isomorphic to the coordinate ring of the classical character variety. It may be constructed as a Frobenius quantum Hamiltonian reduction of .

The quantized character variety of the closed surface is Azumaya over the entire ‘good locus’ of .
We remark that the proofs of Statements 3 and 4 apply identically to the twisted character variety of , where we take , and we require that that the holonomy around the boundary of is a primitive th root of unity [Hausel]. In this case all points are ‘good’ in the above sense and the quotient is smooth; we obtain in this way an Azumaya algebra defined over the entire twisted character variety. Our techniques apply, in particular, to skein algebras such as the Kauffman bracket skein algebra. In a series of papers [BonahonWongI, BonahonWongII, BonahonWongIII] it was proved that the Kauffman bracket skein algebra has a center isomorphic to the functions on the classical character variety, and in [FrohmanUnicity] it was proved that the Azumaya locus is open and dense. The following result gives a complete description of the Azumaya locus:
Theorem 1.2.
Suppose that is an odd integer, and that is a primitive th root of unity. Then the skein algebra is Azumaya over the whole smooth locus of .
Remark 1.3.
While Theorem 1.2 fits naturally in the broader framework we develop in the paper, those readers who are only interested in the proof of Theorem 1.2 (i.e. in the precise determination of the nonsingular locus), and who are already familiar with [FrohmanUnicity] may wish to skip directly to Section LABEL:Kauffmansection, refering back to Sections 2 and LABEL:sec:frobenius only as needed.
It is remarkable that the proof of the corresponding ‘Unicity theorem’ for quantum character varieties proved in Theorem 1.1 is uniform in all gauge groups, and follows from functoriality and generalities about quantum Hamiltonian reduction, in contrast to the hands on algebraic methods in the skein literature. Precise determination of the Azumaya locus, as well as an extension of the unicity theorem to more general gauge groups in this way was a major motivation for this work. Turning now to the quiver examples, we have:
Theorem 1.4.
Let be an odd integer, and a primitive th root of unity. Then:

The algebra is finitely generated over a central subalgebra, which is isomorphic to the coordinate ring of the classical framed multiplicative quiver variety.

Moreover, is Azumaya over the preimage in of the big cell under the multiplicative moment map .

Frobenius quantum Hamiltonian reduction defines a coherent sheaf of algebras over the classical multiplicative quiver variety , which is Azumaya over the locus of stable representations.
In particular, these theorems identify the center of the algebra (resp. ) with functions on (resp. ), and likewise identify the center of the quantized multiplicative quiver variety (resp. quantum character variety) with the affinization of (resp. of the classical character variety). It is already difficult to compute such centers directly, and our results imply, by the flatness in , that the center is trivial away from roots of unity, a fact which is again not easy to see directly. The Azumaya property asserts moreover that each sheaf is étalelocally the endomorphism algebra of a vector bundle on the classical variety, or equivalently, that the fiber of the algebra at each point is a matrix algebra over . In each statement, the second assertion is derived from the first via the process of Frobenius quantum Hamiltonian reduction, which is described in the next subsection.
Remark 1.5.
Modules over an Azumaya algebra form an invertible sheaf of categories which is locally trivial for the étale topology, i.e., a gerbe. It would be interesting to understand this gerbe more conceptually. In a related direction, we expect that modules over the quantum group at a root of unity form a factorizable category relative to its Müger center. Using the results of Gwilliam–Scheimbauer [GwilliamScheimbauerMorita], one may prove that this implies that this category defines an invertible object in the Morita category of braided monoidal categories relative to the classical representation category of the group, i.e., it might be interpreted as a “higher Azumaya algebra”. In particular, this will formally imply that its factorization homology over a topological surface forms an invertible sheaf of categories over the classical character variety .
1.4 Main results: methods and general results
The proofs of our main results are rooted in a collection of beautiful ideas emerging from the literature on quantum groups and geometric representation theory, most notably the seminal paper [BezrukavnikovCherednikalgebrasHilbert2006] where the Hamiltonian reduction of Azumaya algebras in characteristic was first carried out for differential operators with applications to Cherednik algebras, and [VaragnoloDoubleaffineHecke2010], where a analog was developed to study difference operators and double affine Hecke algebras at roots of unity. Similar techniques were used in the study of hypertoric varieties in positive characteristic [STADNIK2013], and of their analogs, quantum multiplicative hypertoric varieties [GanevQuantizationsmultiplicativehypertoric2018a]. A major thrust of the paper is to develop a framework in which to generalize these examples. We expect that this package of ideas will apply in greater generality than the present work. Let us therefore review some of the ingredients here.
Integral forms of the quantum group
Let be a connected reductive group, with its Lie algebra, is universal enveloping algebra. For the remainder of the paper we will reserve the letter to denote a complex root of unity, and to denote its order. We will assume and satisfy a number of mild assumptions (see Assumption LABEL:EllAssumption). We shall require several related forms of the quantum group associated to . Our basic reference is [ChariGuideQuantumGroups1995]. For us, the quantum group at generic parameters refers to the Drinfeld–Jimbo rational form of the quantum group, defined over the base ring of rational functions in a variable with coefficients in ; we will denote this rational form of the quantum group by . It is generated by the quantum Cartan subalgebra, isomorphic to the group algebra of the coweight lattice of , and by the Serre generators and for each positive simple root . We do not recall the relations in detail for general (see instead [ChariGuideQuantumGroups1995, Chapter 9]), because we will only use some essential functorial properties, which we detail later in this section. In addition to the rational form of the quantum group, we consider the socalled divided powers integral from of the quantum group, introduced by Lusztig [LusztigQuantumdeformationscertain1988], and defined as follows. Let denote the subring of consisting of Laurent polynomials in , and consider the subalgebra of generated by the quantum Cartan generators and an integer family of divided powers for each Serre generator. Here denotes the quantum integer, and the quantum factorial, in the variable . We reserve the notation for the basechanged algebra,
via the algebra homomorphism from to given by , our chosen root of unity. Finally, we denote by the small quantum group, which we regard as a subalgebra of generated by the “undivided powers” , , together with the quantum Cartan generators.
Definition 1.6.
Let , , , , and denote the categories of locally finitedimensional modules, respectively, of the rational form , Lusztig’s integral form , its specialization , the small quantum group , and the classical enveloping algebra , such that the weights of the Cartan subalgebra lie in the weight lattice of .
The quantum Frobenius
An important feature of Lusztig’s integral form is that it admits a homomorphism of Hopf algebras,
uniquely defined so that for all simple roots , we have:
The “quantum Frobenius” map Fr is a surjective homomorphism of quasitriangular Hopf algebras, whose kernel is the twosided ideal generated by the augmentation ideal of the small quantum group . Thus, we have Lusztig’s resulting “short exact sequence” of Hopf algebras:
(1.1) 
Basic references for the quantum Frobenius include [ChariGuideQuantumGroups1995], [LusztigQuantumgroupsroots1990], [LusztigIntroductionQuantumGroups2010], [lentner_frobenius_2015],[lentner_factorizable_2017], [arkhipov_another_2003], [kremnizer_proof_2006], [NegronLogModular]. This setup gives rise to a remarkable adjoint pair of braided tensor functors,
the pullback via Fr, and the passage to invariants, respectively. We note that is the identity functor on ; in particular, the functors form an adjoint pair. The functor is braided monoidal, and maps into the Müger center of . That is, for any object in and any object in , the two braidings,
coincide. In fact, each is the switchoffactors map on the underlying vector space.
Frobenius Poisson orders
The notion of a Poisson order, introduced in [BrownPoissonorderssymplectic2003a], consists of a noncommutative algebra , a central subalgebra of , a Poisson bracket on , and a linear map from to . The general and elegant formalism developed in loc. cit. produces isomorphisms between fibers of any two points in the same symplectic core of , in particular on any two points of the open symplectic leaf. For instance, if is generically Azumaya over , we get that it is in fact Azumaya over the whole open symplectic leaf. Poisson orders have been applied more recently to the theory of discriminants of PI algebras [BrownYakimov], [NgyTramYak] and Sklyanin algebras [YakWalWang]. The general method to obtain Poisson orders is as follows. Suppose is a family of associative algebras parametrized by together with a central subalgebra at the special value . Then under a mild assumption (see Proposition LABEL:prop:PoissonOrderDegeneration) we get the structure of a Poisson order on the pair . For instance, this assumption is satisfied when is the whole center of (see Lemma LABEL:lm:PoissonOrderAutomatic which goes back to Hayashi [HayashiSugawara]). In our examples is a family of algebras in , so we combine the quantum Frobenius map and the notion of a Poisson order into the notion of a Frobenius Poisson order. Namely, a Frobenius Poisson order consists of an algebra , a Poisson algebra and a central embedding such that form a Poisson order. If is a flat family of associative algebras in and is a central subalgebra at the special value , then under the same mild assumption form a Frobenius Poisson order, see Proposition LABEL:prop:FrobeniusPoissonOrderDegeneration.
Frobenius quantum Hamiltonian reduction
If is a PoissonLie group and is a Poisson variety equipped with a classical moment map (we recall the relevant formalism of groupvalued moment maps in Section 2), the affine quotient carries a natural structure of a Poisson variety, see Proposition 2.8. Similarly, if is an algebra in equipped with a quantum moment map from the reflection equation algebra (see Sections LABEL:sec:REA and LABEL:sect:quantummomentmaps for the relevant definitions), then we can form the quantum Hamiltonian reduction which is still an associative algebra (see Proposition LABEL:prop:qhamreduction). We are interested in obtaining a Poisson order structure on the Hamiltonian reduction of , so we need to assume the two moment maps are compatible with each other: this leads to the notion of a Frobenius quantum moment map for a Frobenius Poisson order . Namely, we assume that the composite factors through the central subalgebra which gives rise to a classical moment map . Our first result in this direction is that given a Frobenius Poisson order equipped with such a Frobenius quantum moment map satisfying some extra compatibilities (see Definition LABEL:def:HamiltonianFrobeniusPoissonOrder for the notion of a Hamiltonian Frobenius Poisson order) the Hamiltonian reduction LABEL:prop:FrobeniusPoissonReduction). Our second result in this direction is the following mechanism for obtaining Hamiltonian Frobenius Poisson orders (see Proposition LABEL:prop:StrongHamiltonianOrder). Suppose is a flat family of algebras such that the map on invariants is surjective, is a central subalgebra and is a quantum moment map. Then becomes a Hamiltonian Frobenius Poisson order. In examples we check the surjectivity assumption using a good filtration on (see Section LABEL:sect:goodfiltrations). becomes a Poisson order (see Proposition
Distinguished fibers
In each of our examples there is a distinguished point – corresponding to the trivial local system, and the trivial quiver representation, respectively, where the Azumaya property can be checked directly: for character varieties this proceeds by reducing to the small quantum group, while for quiver varieties it requies a long computation.
Nondegenerate Hamiltonian varieties
In order to fully exploit the method of Poisson orders, we require a description of the open symplectic leaf of the framed moduli space. It turns out in our examples these can be described purely in terms of the moment map, because both of our examples are nondegenerate Poisson varieties as defined in [AlekseevQuasiPoisson]. For framed character varieties, this is proved in loc. cit., while for framed quiver varieties, we use a convenient characterization of nondegeneracy due to LiBland–Severa [LiBlandSevera] to prove that they are nondegenerate by a direct computation in coordinates. We show (Theorem 2.14) that given a nondegenerate Poisson variety equipped with a moment map , the open symplectic leaf of is given by the preimage of the big cell .
Langlands duality at even roots of unity
Throughout the paper we make simplifying assumptions on and the order of the root of unity , e.g. that in the semisimple case is of adjoint type and is odd. This ensures that the Müger center of is identified with the symmetric monoidal category equipped with the obvious braiding. This no longer holds if we relax the assumptions on and . Let us now assume that is divisible by and for . Then the Müger center of coincides with , where , the adjoint group of type . Therefore, in this case the quantum character variety for forms a sheaf of algebras over the classical character variety for . Note that in these cases one still has a factorizable braided monoidal category , but it is not given by modules over a Hopf algebra. We expect that our approach nevertheless admits a minor modification which would prove that the quantum character variety is Azumaya over the classical character variety for a more general class of and .
1.5 Outline
We briefly describe the contents of this paper. In Section 2, we recall factorizable PoissonLie groups (Section 2.1), multiplicative moment maps (Section 2.2), and nondegenerate Poisson varieties (Section 2.3). The main result therein (Theorem 2.14, Section 2.4) is a description of symplectic leaves of a nondegenerate Poisson variety as the preimages under the multiplicative moment map of the orbits on (i.e. orbits under the dressing action), where is the PoissonLie dual group of . In Section LABEL:sec:frobenius, we first consider reflection equation algebras, for generic parameters, for Lusztig’s integral form, and at a root of unity (Section LABEL:sec:REA). We then set up the notions of Frobenius Poisson orders and Frobenius quantum moment maps (Section LABEL:subsec:Frobeniuspairs), and discuss degenerations of quantum groups and quantum algebras in light of these notions (Section LABEL:subsec:degenqalgebras). We also formulate the procedure of Frobenius quantum Hamiltonian reduction and prove that Azumaya algebras descend to Azumaya algebras under this procedure (Theorem LABEL:thm:frobtwistedAzumaya, Section LABEL:sec:templateforresults). In Section LABEL:sec:CharVar, we apply the techniques developed in Section LABEL:sec:frobenius to the setting of character varieties. After recalling the construction of quantum character varieties and stacks, we show that the framed character variety, together with its “quantum character sheaf”, form a Frobenius Poisson order (Section LABEL:subsec:CharVarFrobPoissonOrder). We exhibit a Frobenius quantum moment map in this setting (Section LABEL:subsec:CharVarFrobqmm), and run the procedure of Frobenius quantum Hamiltonian reduction (Section LABEL:subsec:CharVarFrobqHred) to obtain Azumaya algebras over classical affine character varieties (Theorem LABEL:ChGthmbody). In Section LABEL:sec:DqMat, we turn our attention to multiplicative quiver varieties and their quantizations at a root of unity. We then recall in Section LABEL:subsec:mqv the construction of the multiplicative quiver variety, as a Hamiltonian reduction of the framed multiplicative quiver variety. We recall in Section LABEL:subsec:quantizationofQ the quantization of the framed multiplicative quiver variety, and in Section LABEL:subsec:quantizationofQ we identify the central subalgebra, establish the existence of good filtrations, and finally construct the Frobenius Poisson order, in Theorem LABEL:thm:quivercenter. In Section LABEL:subsec:qmqvnondeg we show that the resulting Poisson variety is nondegenerate (Theorem LABEL:thm:qmqvnondeg), and in Section LABEL:subsec:qmqvazumaya we prove that the zero representation is an Azumaya point. In Section LABEL:subsec:qmqvmommap, we construct the Frobenius quantum moment map (Theorem LABEL:thm:mmgeneral). In Section LABEL:subsec:QMQVrootofunity, we define the quantized multiplicative quiver variety associated to an arbitrary GIT parameter via Frobenius quantum Hamiltonian reduction. Finally, in Section LABEL:subsec:sheaf, we bring all the elements together to establish the Azumaya property on the smooth locus of the quantum multiplicative quiver variety (Theorem 1.4).
1.6 Acknowledgments
We would like to thank David BenZvi and Kobi Kremnitzer for their guidance and encouragement throughout our work, Nicholas Cooney for many helpful conversations, Michael Gröchenig for discussions about the Langlands dual case, Pavel Etingof for remarks about the small quantum group, and Florian Naef for helpful remarks about nondegenerate quasiPoisson structures. The work of I.G. was supported by the Advanced Grant “Arithmetic and Physics of Higgs moduli spaces” No. 320593 of the European Research Council. The work of D.J. was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [grant agreement no. 637618]. The work of P.S. was supported by the NCCR SwissMAP grant of the Swiss National Science Foundation.
2 Multiplicative Hamiltonian reduction
The goal of this section is to collect some useful results on moment maps for factorizable PoissonLie groups. Throughout this section, we assume that is an arbitrary connected reductive group. For we denote by the left and rightinvariant vector fields with value at the unit. We assume the reader is familiar with the theory of quasiPoisson groups and quasiPoisson spaces, see e.g. [AlekseevManinPairs] and [SafronovQuantumMoment, Section 4.1]. Let us just recall that a Poisson variety is a variety equipped with a Poisson structure such that the action map is Poisson. All varieties we consider in this section are affine.
2.1 Factorizable PoissonLie groups
Fix a nondegenerate element . Let . With respect to a basis of we may write,
We have the following quasiPoisson structures:

Denote by the quasiPoisson group equipped with the zero bivector and trivector .

Consider as a variety under conjugation and equip it with the bivector
By [AlekseevQuasiPoisson, Proposition 3.1] we get a quasiPoisson variety that we denote by .
Definition 2.1.
A classical twist is an element satisfying the equation
Equivalently, we may say that the classical matrix satisfies the classical Yang–Baxter equation
Example 2.2.
In the case we use the classical matrix
where is the elementary matrix with one in the th row and th column.
Using we may perform the following twists:

Twist the quasiPoisson structure into a PoissonLie structure
on that we denote by . We call PoissonLie structures obtained in this manner factorizable.

Twist the quasiPoisson variety to a Poisson variety with the Poisson bivector
where is the map given by differentiating the conjugation action of on itself. The Poisson structure was introduced in [SemenovTianShansky].
The classical twist gives the structure of a Lie bialgebra and provides an embedding of Lie algebras . In particular, the action of on by left and right translations induces a action on which is known as the dressing action. Since the diagonal subalgebra and are transverse Lagrangians, the dressing action on is free at the unit . In particular, there is an open orbit of . The following is shown in [AlekseevMalkin, Section 4].
Proposition 2.3.
The symplectic leaves of are the intersections of the conjugacy classes of with the dressing orbits.
We will assume that the Lie algebra integrates to a group and that there is a action on such that the map is equivariant.
Example 2.4.
Suppose is a connected reductive group equipped with a choice of a maximal torus and a Borel subgroup containing it. Let be a nondegenerate element. Let be the set of positive roots and pick an orthonormal basis of . Then is a basis of , where for is a standard basis of . Then we have the standard matrix
Let be the opposite Borel subgroup. Denote by the abelianization maps. Then the dual PoissonLie group is given by
The open subscheme , the image of , is given by the subscheme which is isomorphic to the big Bruhat cell. Note that in the case we have .
2.2 Multiplicative moment maps
Fix a factorizable PoissonLie structure on . Pick a basis of and let be the dual basis of . Denote by the dressing action on .
Definition 2.5.
Let be a Poisson variety. A equivariant map is a moment map if for every and we have
(2.1) 
In this case we say that is a Hamiltonian Poisson variety.
Proposition 2.6.
Suppose is a Poisson variety and is a moment map. Then is a Poisson morphism.
Proof.
By [SafronovQuantumMoment, Proposition 4.18], our definition of moment maps agrees with the definition of Alekseev and KosmannSchwarzbach introduced in [AlekseevManinPairs]. Now, let be the quasiPoisson variety obtained by twisting using . Then by [AlekseevQuasiPoisson, Proposition 3.3] the map is quasiPoisson and hence after twisting back we get that is Poisson. ∎
Using a moment map we may construct a Hamiltonian reduction of .
Definition 2.7.
Let be an element in the center and a Poisson variety equipped with a moment map . The Hamiltonian reduction is
Hamiltonian reduction carries a natural Poisson structure constructed in the following way. Let be the ideal defining . Since is reductive, the natural morphism