Pointed Hopf Algebras with Triangular Decomposition -- A Characterization of Multiparameter Quantum Groups

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (2004) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl2.

1. Introduction 1.1. What are Quantum Groups? An important problem in the theory of quantum groups is to give some definition of a class of these objects that captures known series of quantum groups, such as the quantum enveloping algebras U q pgq of [Dri86], and their finite-dimensional analogues, as examples. This was for example formulated in [BG02, Problem II.10.2]: "Given a finite-dimensional Lie algebra g, find axioms for Hopf algebras to qualify as quantized enveloping algebras of this particular g." A possible hint to the structure of quantum groups is that the quantum enveloping algebras U q pgq (as well as the small quantum groups u q pgq and multiparameter versions) are pointed Hopf algebras. Such Hopf algebras were studied by several authors (see e.g. [AS02]). Classification results as in [AS10] suggest a strong resemblance of all finite-dimensional pointed Hopf algebras over abelian groups with small quantum groups. Another paper [AS04] gives a characterization of quantum groups at generic parameters using pointed Hopf algebras of finite Gelfand-Kirillov-dimension with infinitesimal braiding of positive generic type.
A further hint to the structure of quantum groups is that they can be decomposed in a triangular way (via the PBW theorem) as U q pgq " U q pn`q b kZ n b U q pn´q.
Here, the positive and negative part are perfectly paired braided Hopf algebras, and the relation with the group algebra kZ n is governed by semidirect product relations. The positive (and negative) part are so-called Nichols algebras.
A third aspect -observed already in the original paper [Dri86] -is that quantum groups are (quotients of) quantum or Drinfeld doubles. It was shown in [Maj99] that U q pgq in fact is a braided Drinfeld double (which are referred to as a double bosonization there). It was proved in [BW04] that also two-parameter quantum groups are Drinfeld doubles.
In this paper, we aim to provide an axiomatic approach to the definition of (multiparameter) quantum groups by combining the pointed Hopf algebra and the triangular decomposition approach. Under the additional assumption of what we call a triangular decomposition of weakly separable type, the only indecomposable examples are close generalizations of multiparameter quantum groups. In particular, assuming further non-degeneracy, they are examples of a more general version of braided Drinfeld doubles, which we refer to as asymmetric braided Drinfeld doubles. Further, under certain assumptions on the group and the parameters, we can recover Lie algebras from these Hopf algebras, after introducing a suitable integral form.
1.2. This Paper's Results. This paper starts by recalling the necessary technical background, including a brief overview on classification results of finite-dimensional pointed Hopf algebras, as well as structural results by [BB09] on algebras with triangular decomposition, in Section 2. Next, we give the definition of a bialgebra with a triangular decomposition over a Hopf algebra H in Section 3. This adapts the two-step approach used for algebras in [BB09] to the study of bialgebras. Namely, we first consider the free case of a bialgebra T pV q b H b T pV˚q where the positive and negative parts (T pV q, respectively T pV˚q) are tensor algebras, and then specify by what ideals (called triangular Hopf ideals) we can take the quotient.
The core of this paper is formed by a partial classification of bialgebras with triangular decomposition over a group algebra kG. We again proceed in two steps. First, we determine all pointed bialgebras with free positive and negative part over kG in Section 4.2, and then look at pairs of ideals I, I˚such that the quotient A{xI, I˚y is still a bialgebra in Section 4.3. We find that indecomposable examples are automatically pointed Hopf algebras, and can only arise over finitely-generated abelian groups. Multiparameter quantum groups share these features. Indeed, the only possible commutator relations (2.10) closely resemble those of multiparameter quantum groups: rf i , v j s " γ i,j pk j´li q P kG, @i " 1, . . . , n. (1.1) We further observe that there exists a natural generalization of the definition of a braided Drinfeld double to the setting of primitively generated braided Hopf algebras in the category of Yetter-Drinfeld modules (YD-modules) over H. For this, the base Hopf algebra H does not need to be quasitriangular. We need two braided Hopf algebras which are only required to be dually paired considered as braided Hopf algebra in the category of modules (rather than YD-modules). That is, the requirement that is weakened compared to the definition of a braided Drinfeld double (as in [Maj99] or [Lau15]) is that the comodule structures do not need to be dually paired. We refer to this generalization as the asymmetric braided Drinfeld double. It gives a natural way of producing Hopf algebras with triangular decomposition -which are not necessarily quasitriangular. We show in Theorem 4.3.2 that the Hopf algebras arising in the classification 4.2.2 are of this form (provided that the parameters γ ii are non-zero).
In Section 4.4 we show that from these asymmetric braided Drinfeld doubles of separable type we can recover Lie algebras provided that there exists a well-defined morphism of rings to Z when setting the parameters equal to 1. Hence, in the spirit of the question asked in Section 1.1, we can relate the outcome of our classification back to Lie algebras, which are always generated by Lie subalgebras isomorphic to sl 2 .
Here is an overview of the increasingly stronger assumptions on the Hopf algebras A and H used in the classification: ‚ Section 3: H any Hopf algebra over a field k, A a bialgebra with triangular decomposition ‚ Section 4: H " kG, A a bialgebra with triangular decomposition -Section 4.1-4.2: A is of weakly separable type and indecomposable after 4.1.3 -Section 4.3: A is indecomposable of separable type, the scalars γ ii are non-zero.
-Section 4.4: In addition to the assumptions of 4.3, we require that char k " 0, and that setting the parameters equal to 1 gives a well-defined homomorphism of rings to Z.
The final Section 5 contains different classes of indecomposable pointed Hopf algebras with triangular decomposition over a group kG that arise as examples in the main classification. The first class we discuss are the multiparameter quantum groups U λ,p pgl n q introduced by [FRT88] (adapting the presentation in [CM96]). They are asymmetric braided Drinfeld doubles, which is a generalization of the result of [BW04] showing that two-parameter quantum groups are Drinfeld doubles. In section 5.2 we bring results of [Ros98] on growth condition (finite Gelfand-Kirillov dimension) and classification of Nichols algebras from [AS04] into the picture. We use these results to characterize the Drinfeld-Jimbo type quantum groups at generic parameters q within the classification of this paper under the additional assumption that the triangular decomposition is what we call symmetric. Further, classes of finite-dimensional pointed Hopf algebras by Radford can naturally be included as examples in this framework (Section 5.3). The more flexible approach to defining quantum groups of this paper can be used to construct examples where a certain classical limit is not a semisimple Lie algebra. A small example is given in 5.2.8. This example does not satisfy the assumptions from Section 4.4.
To conclude this paper, we suggest in Section 5.4 that future research could focus on the search for Hopf algebras with triangular decomposition over other Hopf algebras H (replacing the group algebra kG). This might give interesting monoidal categories, or even knot invariants in other contexts. As the first -most classical -example, if we take H to be a polynomial ring krx 1 , . . . , x n s. In this case, the only examples are universal enveloping algebras of Lie algebras.
1.3. Notations and Conventions. In this paper, adapted Sweedler's notation (see e.g. [Swe69]) is used to denote coproducts and coactions omitting sums. Unless otherwise stated, we work with Hopf algebras over an arbitrary field k. A Hopf algebra always has an invertible antipode S. The category of left YD-modules over a Hopf algebra H is denoted by H H YD, while left modules are denoted by H-Mod, and right modules by Mod-H.
We denote the module spanned by generators S over a commutative ring R as RxSy, while RrSs denotes the R-algebra generated by elements S (subject to some specified relations). Groups generated by elements of a set S are denoted by xSy. A basic observation is that if dim A " 1, then A can be written as kg, for a generator g P H such that ∆pgq " g b g. Such elements are called grouplike. Indeed, if H is a Hopf algebra, then the set of all grouplike elements GpHq has a group structure. A Hopf algebra is pointed if all simple subcoalgebras are one-dimensional. This notion can be traced back to [Swe69] and classifying all finite-dimensional pointed Hopf algebras can be taken as a first step in the classification of all finite-dimensional Hopf algebras (see e.g. [And14] for a recent survey).
In the late 1980s and early 1990s, large classes of pointed Hopf algebras have been discovered with the introduction of the quantum groups (and their small analogues). Due to the vast applications of and attention to these Hopf algebras in the literature, the study of pointed Hopf algebras has become an important algebraic question.
2.2. Link-Indecomposability. In the early 1990s, Montgomery asked the question, which groups may occur as GpHq where H is an indecomposable pointed Hopf algebras. In [Mon95], an appropriate notion of indecomposability is discussed in different ways. We will briefly recall the description in terms of link-indecomposability which is equivalent to indecomposability as a coalgebra and indecomposability of the Ext-quiver of simple comodules.
Given a pointed Hopf algebra H, we define a graph Γ H with vertices being the simple subcoalgebras of H (that is, the grouplike elements). There is an edge h Ñ g if there exists a pg, hq-skew-primitive element v P H, i.e. ∆pvq " v b g`h b v, which is not contained in kGpHq. We say that H is indecomposable if Γ H is connected. As an example, group algebras kG are only indecomposable if G " 1. The quantum group U q psl 2 q is indecomposable if the coproducts are e.g. defined as ∆pEq " E b 1`K b E and ∆pF q " F b 1`K´1 b F . There are other versions of the coproduct which are not indecomposable (see [Mon95]).

2.3.
Classification Results for Pointed Hopf Algebras. It was understood early that pointed Hopf algebras can be obtained as bosonizations A " BpV q¸kG of so-called Nichols (or Nichols-Woronowicz ) algebras BpV q associated to YD-modules over a group G (see e.g. [AS02] for definitions). In this case, the coproducts are given by ∆pvq " v p0q b v p´1q`1 b v using Sweedler's notation. That is, if v is a homogeneous element, then ∆pvq " v b g`1 b v for the degree g P GpAq of v and A is indecomposable over the group generated by g P G with V g ‰ 0. Thus, the question of finding finite-dimensional pointed Hopf algebras is linked to finding finite-dimensional Nichols algebras 2 . Although both questions remain open in general, vast progress has been made in a series of papers by Andruskiewitsch and Schneider (see [AS02,AS10]) for abelian groups G, and more recently for symmetric and alternating groups [AFGV11], or groups of Lie type [ACA13,ACA14]. See [And14] for more detailed references.
Let us briefly recall the classification results of [AS10] here in order to provide the basis for comparison to our own classification in Section 4 later. To fix notation, let D denote a finite Cartan datum. That is, a finite abelian group Γ, a Cartan matrix A " pa ij q of dimension nˆn with a choice of group elements g i , characters χ i for i " 1, . . . , n. Then define q ij :" χ j pg i q and impose the conditions that We can associate to the Cartan matrix A a root system Φ (with positive roots Φ`). The simple roots α i of Φ can be indexed by i " 1, . . . , n. Denote by χ the set of connected components of the corresponding diagram, and by Φ J the root system restricted to the component J P χ, and write i " j if i and j are in the same connected component. Denote further To state the classification of finite-dimensional pointed Hopf algebras, some technical assumptions need to be made. To construct pointed Hopf algebra from a Cartan datum D, we need two families of parameter.
(d) Let λ " pλ ij q be a nˆn-matrix of elements in k such that for all i  j, g i g j " 1 or χ i χ j ‰ ε implies λ ij " 0. (e) Further let µ " pµ α q Φ`b e elements in k such that for any α P ΦJ , for J P χ, Definition 2.3.1 ( [AS10]). Given the a Cartan datum D with families of parameters λ, µ as above, there is a Hopf algebra u " upD, λ, µq. The algebra u is generated by elements g P Γ (define u α pµq P kΓ, see [AS10, 2.14] for α P Φ`), and x i for i " 1, . . . , n, subject to the relations for all i ă j, i  j, (2.4) Here, adpxqpyq is the braided commutator xy´m˝Ψpx b yq where m denotes multiplication and Ψ is the YD-braiding. The comultiplication is given by ∆px i q " Theorem 2.3.2 ([AS02, 0.1]). Under the above assumptions (a)-(e) on a Cartan datum D with parameters λ, µ, the Hopf algebra upD, λ, µq is pointed with Gpuq " Γ and of finite dimension. Moreover, if |G| is not divisible by 2, 3, 5 or 7, then any finite-dimensional pointed Hopf algebra is of this form.
2 However, a pointed Hopf algebra is not necessary bosonizations of this form. Important tools available are the coradical filtration (see e.g. [Mon93]) and the lifting method of Andruskiewitsch and Schneider [AS02].

Algebras with Triangular Decomposition (Free Case).
A triangular decomposition of algebras means that an intrinsic PBW decomposition exists, similar to universal enveloping algebras of Lie algebras. This is a common feature of quantum groups and rational Cherednik algebras, but more generally shared by all braided Drinfeld or Heisenberg doubles (cf. [Lau15,3.4]). Here, we are using the definitions introduced in [BB09] to study such algebras with triangular decomposition (so-called braided doubles).
From a deformation theoretic point of view, triangular decomposition can be viewed as follows. Let V , V˚be dually paired finite-dimensional vector spaces and H a Hopf algebra over a field k, such that V is a left H-module, and V˚carries the dual right H-action. That is, for the evaluation map x , y : V˚b V Ñ k, we have (i.e. the bosonizations T pV q¸H and H˙T pV˚q are subalgebras), and rf, vs " 0. In [BB09], a family of deformations of A 0 pV, V˚q over Hom k pV b V˚, Hq is defined. The algebra A β pV, V˚q over a parameter β : V b V˚Ñ H is defined using the same generators in V , V˚and H with the same bosonization relations, but the commutator relation (2.8) rf, vs " βpf, vq.
In order to obtain flat deformations we restrict to maps β such that the multiplication is an isomorphism of k-vector spaces.
Definition 2.4.1. In the case where m is an isomorphism of k-vector spaces, we say that A β pV, V˚q is a free braided double.
In this case, we call pV, δq a quasi-YD-module and we have (2.10) rf, vs " βpf b vq " v r´1s xf, v r0s y.
2.5. Triangular Ideals. So far, the braided Hopf algebras T pV q and T pV˚q were assumed to be free. We can bring additional relations into the picture, defining braided double that are not necessarily free. Let I Ÿ T pV q and I˚Ÿ T pV˚q be ideals. We want to determine when the quotient map m : T pV q{I b H b T pV˚q{I˚" ÝÑ A β pV, V˚q{xI, I˚y is still an isomorphism of k-vector spaces. In [BB09] it is show that this is the case if and only if J :" xI, I˚y is a so-called triangular ideal. That is, where I Ÿ T ą0 pV q, I˚Ÿ T ą0 pV˚q such that I and I˚are H-invariant and This is equivalent to the commutator rf, Is and rI˚, vs being contained in J for all degree one elements v P V , f P V˚. For each quasi-YD-module, there exists a unique largest triangular ideal I max , and thus a unique maximal quotient referred to as a minimal braided double.
If δ is a YD-module, then the maximal quotient T pV q{I max is the Nichols algebra BpV q of V , and the braided double on BpV q b H b BpV˚q is a generalization of the Heisenberg double, a so-called braided Heisenberg double.
For the purpose of this paper, we need ideals I such that T pV q{I is a braided bialgebra, where V is a YD-module. That is, not a bialgebra object in the category of k-vector spaces but in the category of YD-modules over kG (see e.g. [AS02]). However, if I is a homogeneous ideal in T ą1 pV q which is a coideal and a YD-submodule, then T pV q{I is a braided Hopf algebra. We denote the collection of such ideals by I V . In fact I max P I V as the Nichols algebra BpV q is a braided Hopf algebra.

Hopf Algebras with Triangular Decomposition
In this section, we let k be a field of arbitrary characteristic and H a Hopf algebra over k. We introduce a notion of a Hopf algebra with triangular decomposition.
3.1. Definitions. We refer to the grading of a braided double T pV q{I b H b T pV˚q{I˚given by as the natural grading. We want to study Hopf algebras with triangular decomposition preserving this grading.
Definition 3.1.1. A bialgebra (or Hopf algebra) A with triangular decomposition over a Hopf algebra H is a braided double H " T pV q{I b H b T pV˚q{I˚which is a bialgebra (respectively Hopf algebra) such that H is a subcoalgebra of A with respect to the original coproduct of H, Note that (3.3) implies that εpvq " εpf q " 0 for all v P V , f P V˚. We further observe that assumption (3.2) and (3.3) combined with the counit property, give that ∆pV The coalgebra axioms imply that δ l and δ r are left (respectively right) H-coactions. In particular, as the semidirect product relations in A are preserved by ∆, δ l (and δ r ) are left (respectively right) YD-compatible with the given action of H on V . Similarly, we can obtain a left and right YD-module structure over H on the dual V˚from the coproduct. These are denoted by δl and δr .
Definition 3.1.2. Given a bialgebra A with triangular decomposition over H, we define the right (respectively, left) YD-structure of A to be δ r (respectively, δ l ). We refer to δr and δl as the right and left dual YD-structures.
To fix Sweedler's notation for the different coactions, denote δ r pvq " v p0q b v p´1q and δ l pvq " v p´1q b v p0q and use similar notations for f P V˚. We will reformulate the definition of a bialgebra with triangular decomposition in terms of conditions on the YD-structures of A in (3.6)-(3.10) in the free case first.

Lemma 3.1.3. A bialgebra with triangular decomposition A is a Hopf algebra with triangular decomposition if and only if
In this case, the antipode extends uniquely to all of A.
Proof. This follows (under use of the semidirect product relations) by restating the antipode axioms for the coproduct of a Hopf algebra with triangular decomposition, which has the form ∆pvq " v p0q b v p´1q`vp´1q b v p0q . Note that εpvq " 0 as we require the counit to be a morphism of graded algebras.
3.2. The Free Case. Let A be a free braided double, i.e. A " T pV q b H b T pV˚q. We can now state necessary and sufficient conditions on the YD-structures of A to make the algebra A a bialgebra with triangular decomposition. In the following, we stick to the notation of [BB09] denoting the quasicoaction determining the commutator relation between elements of V and V˚by δpvq " v r´1s b v r0s , for v P V .
Lemma 3.2.1. A free braided double A on T pV q b H b T pV˚q is a bialgebra with triangular decomposition if and only if there exist YD-structures δ l , δ r , δl , and δr such that the following conditions hold for v P V , f P V˚: Proof. The conditions are easily checked to be equivalent -under use of the relations in A and the PBW theorem -to the requirement that (2.10) is preserved by ∆. This gives the relations (3.8)-(3.10), as well as These relations are equivalent to (3.6) and (3.7) under use of the counit of H, applying the coaction axioms.
Conversely, given δ r , δ l as well as their dual counterparts δr , δl , the bosonization relations are preserved by the coproduct defined as for v P V , f P V˚by YD-compatibility.
It will become apparent in Section 4 what constraints on the structure of A conditions (3.6)-(3.10) give working over a group, and over a polynomial ring in Section 5.4.
3.3. Triangular Hopf ideals. We are looking for triangular ideals J " I bH bT pV˚q`T pV qbH bI( cf. [BB09, Appendix A] or Section 2.5) which are also coideals, and hence A{J is a triangular bialgebra or Hopf algebra. Using the description of the coproduct ∆ in terms of the left and right YD-structures on A, the triangular ideas J that are also coideals are simply those triangular ideals for which I (and I˚) are YD-submodules for both δ l and δ r (respectively, δl and δr ).
If A is a triangular Hopf algebra with antipode given as in Lemma 3.1.3, then every triangular ideal which is also a coideals is automatically a Hopf ideal. Definition 3.3.1. We denote the collection of ideals of the form or I Ÿ T pV q and I˚Ÿ T pV˚q which are also YD-submodules for δ r , δ l (respectively for δr , δl ) by I ∆ pAq. Such ideals J are called triangular Hopf ideals.

Asymmetric Braided Drinfeld doubles.
A special class of Hopf algebras with triangular decomposition can be provided by braided Drinfeld doubles of primitively generated Hopf algebras over a quasitriangular base Hopf algebra H. This form of the Drinfeld double was introduced as the double bosonization in [Maj95,Maj99], see also [Lau15] for the presentation used here. We now give a more general definition of an asymmetric braided Drinfeld double which is suitable to capture the more general class of Hopf algebras that we find in Section 4, including multiparameter quantum groups, as examples. In this construction, the base Hopf algebra H need not be quasitriangular, and the asymmetric braided Drinfeld double is also not quasitriangular in general.
To define the braided Drinfeld double of dually paired braided Hopf algebras C and B in the category DrinpHq-Mod " H H YD we require that x , y : B b C Ñ k is a morphism of YD-modules. This implies that the actions and coactions on C and B are dual to one-another (by means of the antipode of H). A weaker requirement is that we consider the images of C and D under the forgetful functor F : H H YD ÝÑ H-Mod, and require that F pCq and F pBq are dually paired Hopf algebras in H-Mod, while C and B may not be dually paired in H H YD. Hence the coactions on C and B do not necessarily have to be related via the antipode, but the actions and resulting braidings need to be related by duality. In this case, we say that C, B are weakly dually paired braided Hopf algebras in H H YD. This weaker duality is equivalent to an analogue of condition (3.7). Assuming this axiom, we can define an analogue of the braided Drinfeld double on the k-vector space B b H b C (rather than using B b DrinpHq b C) with this weaker requirement of duality on C and B. The definition of the asymmetric braided Drinfeld double can be given using Tannakian reconstruction theory by describing their category of modules. This is similar to the approach used for the braided Drinfeld double in [Maj99,Lau15]. The condition (3.7) can be rephrased as We can visualize condition using graphical calculus (using the conventions from [Lau15]): Definition 3.4.2. Let C, B be weakly dually paired braided Hopf algebras in H H YD. We define the category B YD C asy pHq of asymmetric YD-modules over B, C as having objects V which are left Hmodules (also viewed as right modules by means of the inverse antipode), equipped with a left C-action and a right B-action (by morphisms of H-modules) which satisfy the compatibility condition asy pHq are required to commute with the actions of H, B and C. Note that on V the right action is induced from the left action via It may help to visualize the condition (3.13) using graphical notation: Proposition 3.4.3. The category B YD C asy pHq is monoidal, with a monoidal fiber functors.

Mod-BpH-Modq
Proof. This monadicity statement can for example be checked directly using graphical calculus. Note that condition (3.12) is crucial. The fiber functors simply forget the additional structure at each step.
Definition 3.4.4. The asymmetric braided Drinfeld double Drin H pB, Cq is defined as the algebra obtained by Tannakian reconstruction 3 on B b H b C applied to the functor B YD C asy pHq ÝÑ Vect k . Hence Drin H pB, Cq-Mod and B YD C asy pHq are canonically equivalent as categories. Proposition 3.4.5. An explicit presentation for the asymmetric braided Drinfeld double Drin H pB, Cq on the k-vector space B b H b C can be given as follows: the multiplication on B is opposite, and for c P C, b P B and h P H we have The coproducts are given by and the antipode is Proof. This follows under application of reconstruction (in Vect k ) applied to B YD C asy pHq. See e.g [Lau15, 2.3] for formulas on how to obtain the structures, including the antipode (Figure 2.1).
An important feature of the braided Drinfeld double is that it has braided categories of representations. For the asymmetric braided Drinfeld double to be quasitriangular, we either H to be quasitriangular. If H is not quasitriangular, this can be achieved by working with over DrinpHq instead of H as a base Hopf algebra.
From now on, we restrict to the important special case where B and C are primitively generated by finite-dimensional YD-modules. This way, we obtain examples of Hopf algebras with a triangular decomposition over H.
Lemma 3.4.6. Let V , V˚be left YD-modules over H, such that the action on V˚is dual to the action on V . Then the algebras T pV q op and T pV˚q cop are dually paired Hopf algebras 4 in the monoidal category of right modules over H. Further assume that the compatibility condition (3.12) holds.
Then the asymmetric braided Drinfeld double Drin H pT pV˚q cop , T pV q cop q is given on A " T pV q b H b T pV˚q subject to the usual bosonization relations (2.7) and the cross relation 3 See e.g. [Lau15,2.3]. 4 We choose the opposite T pV q op and coopposite T pV˚q cop in order to avoid having to take the opposite multiplication in the resulting double (cf. 3.4.5). As tensor algebras are cocommutative, this choice does not affect the formulas for the coproduct .

Robert Laugwitz
The coalgebra structure is given by The counit is given by εpvq " εpf q " 0 and the antipode can be computed using the conditions from equations (3.4) and (3.5) as We can also consider quotients of the form A{J for any triangular Hopf ideal J P I ∆ pAq. The quotient of A by the maximal triangular Hopf ideal in I ∆ pAq is denoted by Drin H pV, V˚q.
Lemma 3.4.7. Let A " Drin H pT pV q op , T pV˚q cop q for V , V˚as in Lemma 3.4.6. Then the maximal ideal I max pAq in I ∆ pAq is given by where I max pV q is the maximal ideal for the left coaction on V , and I max pV˚q is the maximal ideal for the left coaction on V˚. Hence is an isomorphism of k-vector spaces (PBW theorem).
Proof. This is clear as we know that T pV q op {I max pV q and T pV˚q cop {I max pV˚q are weakly dually paired braided Hopf algebras and their asymmetric braided Drinfeld double is given by the quotient Drin H pT pV q op , T pV˚q cop q{I max pAq, which must be the minimal double Drin H pV, V˚q.
A perfect pairing between the positive and negative part of Drin H pV, V˚q implies the existence of a formal power series coev satisfying the axioms of coevaluation. This can be used to give a braiding on a suitable category of modules over Drin H pV, V˚q (where BpV q acts integrally), and all modules have the structure of being YD-modules over H.
3.5. Symmetric Triangular Decompositions. The rest of this section will be devoted to the question of recovering the braided Drinfeld double over a quasitriangular base Hopf algebra H as a special case of the asymmetric braided Drinfeld double. For this, we introduce the idea of a Hopf algebra with a symmetric triangular decomposition: Definition 3.5.1. Given a bialgebra with triangular decomposition. If the associated coactions satisfy that the right coaction δr of V˚is the dual coaction to δ l , i.e.
(3.24) xf p0q b vyf p´1q " xf b v p0q yv p´1q , and the coactions δ r and δl are compatible in the same way, then we call the triangular decomposition symmetric.
In the case where H is a quasitriangular Hopf algebra, we can recover a special case of the definition of the braided Drinfeld double given in [Lau15, 3.5.6] from the more general form given in Definition 3.4.4, and the resulting triangular decomposition will be symmetric. For this, note that the universal R-matrix and its inverse give functors Given a right H-module V we can hence give V the left YD-module structure using R´1, and Vt he dual YD-module structure. Note that (3.12) is satisfied in this case. With these structures, the relation (3.21) becomes This is precisely the condition of [Lau15, 3.5.6]. Note that we use R " pS´1 b Id H qR´1. This proves the following: Note that a partial converse statement also holds: Given an asymmetric braided Drinfeld double that is symmetric, then it can be displayed as a braided Drinfeld double in the sense of [Lau15,Maj99], but unless H is quasitriangular (and the coaction induced by the R-matrix), we need to view if over the base Hopf algebra DrinpHq. If the positive and negative part are perfectly paired, then we can give a formal power series describing the R-matrix and an appropriate subcategory (corresponding to the Drinfeld center) is braided.
Particularly interesting examples of such braided Drinfeld doubles include the quantum groups U q pgq for generic q, and the small quantum groups u q pgq (see [Maj99]). Their construction uses the concept of a weak quasitriangular structure for which a similar statement to 3.5.2 can be made. We will see in Section 5 that multiparameter quantum groups can be viewed as examples of asymmetric braided Drinfeld doubles that are not symmetric. Further, all the pointed Hopf algebras classified in the main result of this paper (Theorem 4.2.2), under the additional assumption that the braiding is of separable type and some commutators do not vanish, are asymmetric braided Drinfeld doubles.

Classification over a Group
In this section, we denote by A " T pV q b kG b T pV˚q a bialgebra with triangular decomposition over a group algebra kG. Note that we do not assume G to be finite.

Preliminary Observations.
Hopf algebras that are generated by grouplike and skew primitive elements are always pointed. We show that assuming a Hopf algebra has triangular decomposition over a group and is of what we call weakly separable type, it is generated by skew-primitive elements and hence pointed.
where v 1 1 , . . . , v 1 n is another basis of V , and f 1 1 , . . . , f 1 n of V˚. Proof. Let v 1 , . . . , v n be a homogeneous basis for the YD-compatible grading δ r and v 1 1 , . . . , v 1 n a homogeneous basis for δ l . The form (4.1) of the coproducts is obtained by letting M be the base change matrix from tv i u to tv 1 i u. The same argument works for the dual V˚, denoting the base change matrix from tf i u to tf 1 i u by N . Lemma 4.1.2. A bialgebra A with a triangular decomposition over kG as above is a Hopf algebra, with antipode S given on generators of the form v i , f i as in (4.1) by Proof. The antipode axioms require that S is of the form stated, using that kG is a Hopf subalgebra, cf. (3.4)-(3.5). As T pV q and T pV˚q are free, defining S on the generators extends uniquely to an antialgebra and coalgebra map on all of A.
Definition 4.1.3. A Hopf algebra with triangular decomposition A is called of weakly separable type if the degrees right g i , . . . , g n of V are pairwise distinct group elements, and the same holds for the left degrees h 1 , . . . , h n of V as well as the dual degrees.
We observe that being of weakly separable type over a group implies that V and V˚have 1dimensional homogeneous components. This gives that for a homogeneous basis element v i of degree a i , g Ź v i ‰ 0 is homogeneous of degree ga i g´1 which hence has to be a scalar multiple of a basis element v gpiq where gpiq is an index 1, . . . , n. Hence we obtain an action of G on t1, . . . , nu. To fix notation, we write We will see that for A of weakly separable type, the bases change matrices M , N are diagonal matrices and can be chosen to be the identity matrix by rescaling of the diagonal bases. This implies that A is generated by primitive and group-like elements and hence pointed. It is a conjecture in [AS02] that all finite-dimensional pointed Hopf algebras over a field of characteristic zero are in fact generated by skew-primitive and group-like elements.
Proposition 4.1.4. If A is of weakly separable type, then there exists a basis tv i u of V and tf i u of V˚consisting of pg i , h i q-skew primitive elements, i.e. (4.4) and the antipode on a these skew-primitive elements is given by Spv Proof. Consider the right and left coactions δ r and δ l from Section 3.1. Choosing a basis v 1 , . . . , v n homogeneous for δ l and v 1 1 , . . . , v 1 n homogeneous for δ r , (4.1) gives where M " pM ij q is the base change matrix. By coassociativity, we find that By weak separability of δ r and δ l we now have for each j " 1, . . . , n: Note that M ij ‰ 0 for at least some i. This implies that pM´1q jk " 0 unless k " i as the g i are all distinct. Further, if M ij ‰ 0, then v i and v 1 j are proportional. This can only be true for at most one i for given index j by weak separability. Hence by reordering the basis v 1 1 , . . . , v 1 n we find that M is a diagonal matrix and can rescale the basis tv 1 i u such that M is the identity matrix. Hence we have ∆pv i q " v i b g i`hi b v i . The antipode conditions for A give (using Lemma 3.1.3) that S is of the form claimed.
Remark 4.1.5. The bases tv i u and tf i u do not necessarily need to be orthogonal with respect to the pairing x , y. We will see in Theorem 4.2.2 that if the characters λ i are all distinct, then the bases can be chosen to be dual bases.
Notation 4.1.6. In the following, we fix a basis v 1 , . . . , v n for V and f 1 , . . . , f n for V˚such that A direct observation from Proposition 4.1.4 is that the algebra A is generated by primitive and grouplike elements (which are precisely the group G) and hence pointed. Even in the general case (not assuming that A is of weakly separable type), we have the following restrictions on the group structure.
Proposition 4.1.7. In the group G, the relations rg i , a j s " rh i , a j s " 1 and rh i , b j s " rg i , b j s " 1 hold for all i, j " 1, . . . , n. In particular, if A has a symmetric triangular decomposition, then the subgroup of G generated by all degrees is abelian.
Further, the following identities for the characters of the group action hold: Proof. The commutator relations follow by applying (3.8) and (3.9) to a pair of homogeneous basis elements of V and V˚with respect to δ l , δr (or δ r , δl ). Then, even without weak separability, it follows from (3.6) and (3.7) that h i pjq " j, a j piq " i, g i pjq " j and b j piq " i by the PBW theorem. This implies the relations (4.8). In the symmetric case, a i " g´1 i and b i " h´1 i which forces the subgroup generated by all degrees to be abelian.

Classification in the Free Case of Weakly Separable Type.
We are now in the position, that we can classify all Hopf algebras A with triangular decomposition of weakly separable type (cf. Definition 4.1.3). This will enable us view the Hopf algebras arising from this classification as analogues of multiparameter quantum groups in Section 5. We start by considering the case A " T pV q b kG b T pV˚q which is referred to as the free case.
Proposition 4.2.1. For the Hopf algebra A with triangular decomposition of weakly separable type to be indecomposable as a coalgebra it is necessary that G is generated by elements k 1 , . . . , k n , l 1 , . . . , l n such that there exist generators v i of V and f i of V˚which are skew-primitive of the form with rk i , l j s " 1 for all i, j. For the characters of the actions on the homogeneous components of V and V˚we require that (4.10) µ j pk i q " λ i pl j q´1.
Proof. To determine when pointed Hopf algebras are indecomposable as coalgebras, consider the graph Γ A described in 2.2. Assume that A has generators given as in 4.1.6. We claim that the connected components of Γ A are in bijection with the double cosets of the subgroup Z :" xg´1 1 h 1 , . . . , g´1 n h n , a´1 1 b 1 , . . . , a´1 n b n y in G which partition G. Indeed, using that the elements gv i and gf i are skew-primitive of type pgg i , gh i q and pga i , gb i q, we find that the connected component of g contains, for i " 1, . . . , n, the strands for i " 1, . . . , n and the same strand with a´1 i b i instead of g´1 i h i (and with g multiplied on the right). Moreover, as the elements gv i , gf i , v i g, f i g (and possibly linear combinations of products of them, which would again be of type given by elements in Z) are the only skew-primitive elements in A, and thus give the only arrows in Γ A , two elements g and h are in the same connected component if and only if z 1 gz 2 " z 3 hz 4 , for some z i P Z. Thus, A is indecomposable if and only if G equals the connected component of 1 in the graph Γ A , hence if G " Z which is the finitely-generated group generated by the elements k i :" h´1 i g i , l i :" a´1 i b i for i " 1, . . . , n. Hence, in order to obtain indecomposability, the coproducts are of the form as stated in (4.9). This is achieved by replacing the generators v i by v i h´1 i and f i by a´1 i f i . The rest of the statements follow directly from Proposition 4.1.7.
Theorem 4.2.2. For an indecomposable pointed Hopf algebra A as in Theorem 4.2.1 of weakly separable type, the commutator relation (2.10) is of the form rf i , v j s " γ ij pk j´li q @1 ď i, j ď n, (4.11) where γ ij are scalars in k such that γ ij " 0 whenever λ i ‰ λ j in which case also xf i , v j y " 0. Conversely, any choice of such scalars gives a pointed Hopf algebra of this form.
Proof. With the work done in Proposition 4.1.3, it remains to verify that the form of the commutator relation (2.10) is as stated. Recall that in [BB09], the commutator relation is given by means of a quasi-coaction. That is a morphism δ : V Ñ kG b V satisfying (2.9) and (2.10). Such a morphism has the general form on the basis elements. Then (3.10), which is required for A to be a bialgebra, rewrites as ÿ For each i, there exists k such that xf i , v k y ‰ 0. For given i, we denote the set of indices such that xf i , v k y ‰ 0 by I i . For such k P I i , we find that α j k,g " 0 for g ‰ k j , l i , and α j k,kj "´α j k,li . Thus, we obtain that δ is of the form where γ ij " ř kPIi α j k,kj 1{xf i , v k y |I i | and tv 1 i u is the dual basis of V to tf i u. Conversely, given arbitrary scalars γ ij for i, j " 1, . . . , n, we can define a quasi-coaction by the same formula (4.13). Then δ is YDcompatible with the given action of G on V if and only if (cf. to condition (A) in [BB09, Theorem A]) As A is indecomposable of weakly separable type, G is abelian and hence this condition is equivalent to λ j " µ i whenever γ ij ‰ 0. But by duality of the action, if xf i , v j y ‰ 0 then λ i " µ j .
As for given i " 1, . . . , n, xf i , v j y ‰ 0 for some j we have that λ i " µ j for at least some j, and vice versa. Hence, the set of characters and dual characters are in bijection. We can change the numbering and assume without loss of generality (recall that we are in the weakly separable case) to obtain (4.14) From now on, we will hence only use the notation λ i .
The situation, where tv i u and tf i u are orthogonal bases deserves particular attention. In this case, the scalars γ ij " 0 for i ‰ j. The following concept of separability ensure this.
Such a change of generators causes the commutators ad " r , s to become braided commutators ad " Id V b2´Ψ. The scalars λ ij satisfy the condition (d) in 2.3, where for the characters χ i χ j ‰ ε implies λ ij " 0. This is the analogue of our condition λ i ‰ λ j implying γ ij " 0.
This linking relation also appears in the quantum group characterization of [AS04, Theorem 4.3]. Hence we can conclude that the classification in this section gives Hopf algebras with similar relations as appearing in the work of Andruskiewitsch and Schneider.
Example 4.2.5. The most degenerate case, where γ ij " 0, gives the Hopf algebra pT pV q ' T pV˚qq¸kG where the tensor algebras are again computed in the category of YD-modules over kG.
Assuming the non-degeneracy that γ ii ‰ 0, we can adapt the terminology of [BB09] that the braided doubles in this case come from mixed YD-structures. A mixed YD-structure is a quasi-coaction δ that is a weighted sum ř t i δ i , where δ i are YD-modules compatible with the same action, and t i are generic scalars. The quasi YD-module in the theorem is the sum δ " δ r´p δl q˚, where pδl q˚is the YD-module given by v j Þ Ñ l j b v j , which is dual to δl . We will see that in this case all the Hopf algebras arising are certain asymmetric braided Drinfeld doubles (as defined in 3.4).
In the symmetric case, these algebras are in fact braided Drinfeld doubles. In particular, their adequately defined module categories (resembling the category O, see [Lau15, 3.9]) are braided.

4.3.
Interpretation as Asymmetric Braided Drinfeld Doubles. So far, we have only classified free braided doubles over kG. That is, as a k-vector space A -T pV q b kG b T pV˚q via the multiplication map.
To capture examples such as quantum groups, it is necessary to consider quotients of A by ideals J " xI, I˚y such that A{J -T pV q{I b kG b T pV˚q{I˚is still a Hopf algebra (and thus pointed). Here I Ÿ T pV q and I˚Ÿ T pV˚q are ideals and also coideals, and J P I ∆ pAq. We will now refine our considerations from Section 3.3 to find for what ideals I and I˚this is the case. We will use the notation q ij :" λ j pk i q. Then, by (4.10), we have that λ j pl i q " q´1 ji , and the matrix q " pq ij q describes the braiding on V fully, i.e. it is of diagonal type.
The collection of triangular Hopf ideals I ∆ pAq introduced in Section 2.5 can be described more concretely for A satisfying the following restrictions: we assume that the parameters γ ii ‰ 0 for all i and that V (and hence V˚) are of separable type, and that k i ‰ l i . Recall that in this situation, the algebras of the classification 4.2.2 are displayed as what is referred to in [BB09] as arising from mixed YD-structures. More specifically, the quasi-coaction δ " δ r´p δl q˚, where δl denotes the coaction on V that is obtained by dualizing the left coaction δl on V˚(this is possible as G is abelian).
By Lemma 3.4.7, the ideals in I ∆ pAq are of the form J " I b kG b T pV˚q`T pV q b kG b I˚where I is an ideal in the collection I pV,δrq for V with the right coaction given by δ r , and I˚is in T pV˚,δl q for the left dual coaction δl on V˚.
Note that by (4.10) we the braiding Ψ r coming from δ r and Ψ l from pδl q˚on V are given by Hence Ψ l " Ψ´1 r , the inverse braiding. We hence drop the subscripts l, r.
Example 4.3.1. In the quantum groups A " U q pgq, the braiding satisfies the symmetry q ij " q i¨j " q j¨i " q ji as the Cartan datum is symmetric. This implies that the relations in I are symmetric under reversing the order of tensors This can be verified explicitly by observing that in U q pgq the ideal I is generated by q-Serre relations, which carry such a symmetry.
Theorem 4.3.2. All quotients by triangular ideals J P I ∆ pAq of algebras A occurring in the classification 4.2.2, where A is of separable type with γ ii ‰ 0 for all i, are asymmetric braided Drinfeld doubles. If J is maximal of this form, then A{J -Drin kG pV, V˚q.
Proof. We have seen that the commutator relations are of the form rf i , v j s " δ ij γ ii pk i´lj q. This is precisely the form of the asymmetric braided Drinfeld double of V with right YD-module structure given by the right grading, and V˚with left YD-module structure given by the left dual grading. The pairing is given by xf i , v j y " δ ij γ ii here. We have to check that the braided Hopf algebras T pV q and T pV˚q of YD-modules over G are dually paired when viewed in the category of left kG-modules. This however follows from condition (4.10). Taking the maximal quotient by a triangular ideal (or the left and right radical of the pairing) gives the asymmetric braided Drinfeld double Drin kG pV, V˚q.
If some of the parameters γ ii are zero, then the pointed Hopf algebras obtained are not an asymmetric braided Drinfeld double any more (in the sense of Definition 3.4.4).

4.4.
Recovering a Lie Algebra. We assume that char k " 0 in this section and study Hopf algebras with triangular decomposition of separable type which are of the form Drin kG pV, V˚q (see Theorem 4.3.2). The aim is to set the characters λ i and the group elements k i , l i equal to 1. This way, we want to recover a Lie algebra g for any of the indecomposable pointed Hopf algebras of the form Drin kG pV, V˚q, relating back to the question asked in the introduction of finding quantum groups for a given Lie algebra. The tool available for this is the Milnor-Moore theorem from [MM65] (see also [Mon93,Theorem 5.6.5]) which shows that any cocommutative connected Hopf algebras is of the form U pgq for a (possibly infinite-dimensional) Lie algebra g.
There are technical problems with this naive approach. To set the elements q ij -which will be replaced by formal parameters -equal to one, we need to give an appropriate integral form to avoid that the modules collapse to zero. This rules out examples like e.g. krxs{px n q (and, more generally, the small quantum groups) which are braided Hopf algebras in the category of YD-modules over kZ, as here a generator of the group acts by a primitive nth root of unity q on x, and Zrqs X k is a cyclotomic ring.
As a first step, we introduce appropriate integral forms of Drin kG pV, V˚q, for which we need square roots of q ij . We consider the subring Z :" Zrq˘1 {2 ij s i,j Ă k adjoining all square roots of the numbers q ij and their inverses. This will now be treated as formal parameters with certain relations between them, coming from the relations we have among them in k.
Assumption 4.4.1. In this section, we assume that the ideal xq˘1 {2 ij´1 | i, j " 1, . . . , ny in Z is a proper ideal, and hence p : Z Ñ Z, q˘1 This assumption is crucial in the formal limiting process. It, for example, prevents examples in which q n`qn´1`. . .`q`1 " 0 as in cyclotomic rings.
To produce an integral form, we replace a given YD-module V over kG of separable type as in the previous sections, by a YD-module over ZG. For this, we can choose a G-homogeneous basis v 1 , . . . , v n and a homogeoneous dual basis f i , . . . , f n such that (possibly after rescaling) An important observation is that the Woronowicz symmetrizers, which are used to compute the Nichols ideal I max pV q, have coefficients in Z. Hence their kernels will be Z-modules. That is, for V int defined as Zxv 1 , . . . , v n y, which is a YD-module over the group ring ZG, the Woronowicz symmetrizer Wor n int Ψ is a Z-linear map V int bn Ñ V int bn . Hence I max pV int q :" ker Wor int Ψ is an ideal in T pV int q, the tensor algebra over Z.
In order to provide an integral form of Drin kG pV, V˚q, we will change the presentation by introducing new commuting generators, namely rf i , v i s ": t i . One verifies that the following commutator relations hold over k, as we are given the relation t i " {2 ii pk i´li q when working over the field: Definition 4.4.2. The integral form Drin ZG pV int , V˚i nt q of Drin kG pV, V˚q is defined as the graded Hopf algebra over the ring Z generated by v 1 , . . . , v n , of degree 1, f 1 , . . . , f n of degree´1, and the group elements k 1 , . . . , k n , l 1 , . . . , l n P G, and additional elements t 1 , . . . , t n of degree 0, subject to the relations of I max pV int q and Im ax pV int q, bosonization relations as well as the relations (4.18), (4.19) and The coproducts are given as before on the generators f i , v i , k i , l i and ∆pt i q " t i b k i`li b t i .
Note that as A " Drin ZG pV int , V int˚q is a Hopf algebra over the commutative ring Z, the coproduct is a map A Ñ A b Z A. For the quantum groups U q pgq at generic parameter, the integral form in this case is so-called non-restricted integral form (see e.g. [CP95,9.2]) which goes back to De Concini-Kac [DCK90]. To set the parameters equal to one, and to consider extensions of Hopf algebras to fields, we use the following Lemma: Lemma 4.4.3. Let φ : R Ñ S be a morphism of commutative algebras. We denote the category of Hopf algebra over R by Hopf R . Then base change along φ induces a functor Proof. Given a Hopf algebra A which is an R-algebra, i.e. there is a morphism R Ñ A, we induce the multiplication and comultiplication on S Þ Ñ A b R S using the isomorphism It is easy to check that the Hopf algebra axioms are preserved under base change.
Proposition 4.4.4. There is an isomorphism of graded Hopf algebras ÝÑ Drin kG pV, V˚q.
Proof. Recall that Z ď k by construction. Extending to k, we are able to divide by q ii´1 in (4.22), and recover the original commutator and bosonization relations in Drin kG pV, V˚q. It remains to verify that I max pV int q b Z k " ker Wor int Ψ b Z k " ker Wor Ψ " I max pV q.
This follows by noting that k is flat as a Z-module (since the function field KpZq is flat over Z as a localization, and k is free over KpZq), and V int b Z k -V as k-vector spaces.
Definition 4.4.5. We define the classical limit of Drin kG pV, V˚q as the algebra using the morphism p : Z Ñ Z mapping all q˘1 {2 ij to 1, and the two sided ideal xker ε G y generated by the kernel of the augmentation map ε G : kG Ñ k mapping all group elements to 1. Note that this ideal is a Hopf ideal.
That is, to obtain the classical form we first set the parameters q˘1 {2 ij equal to 1 in the integral form and then extend the resulting Z-module to a k-vector space, and finally set the group elements equal to 1 along the counit ε G : kG Ñ k. We obtain a primitively generated Hopf algebra, and hence a Lie algebra, this way: Proposition 4.4.6. The classical limit Drin cl k pV, V˚q is a connected Hopf algebra, generated by primitive elements. Hence, for the Lie algebra p V of primitive elements, U pp V q " Drin cl k pV, V˚q. This algebra is generated by triples f i , v i , t i which form a subalgebra isomorphic to U psl 2 q.
Proof. Lemma 4.4.3 ensures that Drin cl l pV, V˚q is a Hopf algebra over k, and freeness of V int over Z ensures that the positive and negative part do not collapse to the zero space. In particular, the k-vector space V int ' V int˚e mbeds into the Lie algebra p V of primitive elements. In the classical limit, we obtain the relations Hence every triple f i , v i , t i generates a Lie subalgebra of p V isomorphic to sl 2 . Note that Drin cl k pV, V˚q is generated by primitive elements: We also compute Hence, t i is skew primitive in Drin int ZG pV, V˚q and primitive in the classical limit. Thus, Drin cl k pV q is a pointed Hopf algebra over the trivial group. That is, a connected pointed Hopf algebra. It is further cocommutative and Theorem 5.6.5 in [Mon93] implies that such a Hopf algebra is of the form U pgq where g is the Lie algebra of primitive elements as char k " 0.
Note that Drin cl k pV, V˚q is a braided double over the polynomial ring SpT q, where T " kxt 1 , . . . , t n y (which is not necessarily n-dimensional). The action is given by t j Ź v i " 2δ i,j v i The quasi-coaction is given by δpv i q " t i b v i which is not a coaction, hence Drin ZG int pV, V˚q is not a braided Heisenberg double. It is also not an asymmetric braided Drinfeld double.
Example 4.4.7. For U q pgq, g a semisimple Lie algebra, viewed as a braided Drinfeld double, the classical limit is U pgq.
We can also compute examples that do not give finite-dimensional semisimple Lie algebras. As a general rule, the relations between the parameters q ij determine the relations in the Lie algebra. It is easy to construct free examples, for which there are no relations between the v 1 , . . . , v n by choosing algebraically independent parameters q ij . The work of [Ros98] and [AS04] give restrictions on examples satisfying the growth condition of finite Gelfand-Kirillov dimension. We will view their results in the setting of this paper in Section 5.2.

Classes of Quantum groups
In this section, we relate the classification from Section 4 to various classes of examples which are often regarded as quantum groups. This includes the multiparameter quantum groups studied by [Res90,FRT88,AST91,Sud90] and others in Section 5.1, a characterization of Drinfeld-Jimbo quantum groups in Section 5.2, and classes of examples of pointed Hopf algebras from the work of Radford in Section 5.3. The classification in Theorem 4.2.2 points out natural generalizations of these classes of examples. We finally sketch how one can define analogues of quantum groups using triangular decompositions over other Hopf algebras than kG.

Multiparameter Quantum Groups.
Let k be a field of characteristic zero. For the purpose of this section, let λ P k be generic, and p ij P k for 1 ď i ă j ď n. Assume that p ii " 1 and p ji " p´1 ij . Following [AST91,CM96] and to fix notation, we set We will provide a variation of the presentation of [AST91,CM96] in order to display multiparameter quantum groups as a Hopf algebra with triangular decomposition.
Example 5.1.1 (Multiparameter quantum groups). Let F " kxf 1 , . . . , f n´1 y be the YD-module over a group algebra G with commuting generators k 1 , . . . , k n´1 , l 1 , . . . , l n´1 . Denote the dual by E " kxe 1 , . . . , e n´1 y, where the pairing is given by xe i , f j y " p1´λqδ ij . The YD-structure is of separable type, given by assigning the right degree k i to f i , and the left degree l i to e i , and actions for i, j " 1, . . . n´1. We define the multiparameter quantum group U λ,p pgl n q to be the asymmetric braided Drinfeld double Drin kG pF, Eq.
Note that the definition of Drin kG pF, Eq is possible as (4.10) holds, i.e.
The commutator relation in Drin kG pF, Eq is given by Our definition of the multiparameter quantum group is justified by the following isomorphism to an indecomposable subalgebra of the multiparameter quantum group considered in the literature: Proposition 5.1.2. There is an isomorphism of Hopf algebras U λ,p pgl n q " Drin kG pF, Eq -U 1 where U 1 is a subalgebra of the multiparameter quantum group U (as defined in the literature).
Proof. We prove the theorem by first considering the morphism Such a morphism will descent to an injective morphism φ : Drin kG pF, Eq Ñ U by the following Lemma 5.1.3. We further note that the image Im φ ": U 1 is a Hopf subalgebra isomorphic to Drin kG pF, Eq. Denote the generators of U by E i , F i for i " 1, . . . , n´1 and group elements K i , L i for i " 1, . . . , n (see [CM96,4.8]). The map φ is defined by φpe i q " λE i K´1 i`1 K i , φpf i q :" F i , φpk i q " L i`1 L´1 i , and φpl i q :" K´1 i`1 K i . One checks directly that the relations in the free braided double T pEq b kG b T pF q are preserved under this map, using the presentation in [CM96, 4.8] for U .
Lemma 5.1.3. The quantum Serre relations in the positive part of A " U λ,p pgl n q are given by the largest ideal in I ∆ pAq, making the positive part a Nichols algebra. This ideal is generated by the braided commutators adpE i q 1´aij pE j q " adpF i q 1´aij pF j q " 0, (5.4) Proof. It follows from Lemma 3.4.7 that the maximal ideal J in I ∆ pAq is given by J " xI, I˚y where I is the Nichols ideal of the YD-module F .
In U , the explicit description of the ideal the quotient of the positive (respectively negative) part is generated by quantum Serre relations. This follows from Lemma 4.5 in [CM96]. For this, it is crucial that λ is not a root of unity. The proof uses the observation in [Res90], or [AST91] for the deformed function algebra, that multiparameter quantum groups, using quantum coordinate rings, can be obtained via a 2-cocycle from a one-parameter quantum groups. The fact that the quantum Serre relations generate the ideal J follows from Theorem 4.4 in [CM96] where it is shown that these relations generate the radical of the pairing of T pF q with T pEq extending the pairing of E and F .
The result that the multiparameter quantum group U λ,p pgl n q is the asymmetric braided Drinfeld double Drin kG pF, Eq can be seen as a generalization of the result in [BW04] where the two-parameter quantum groups were shown to be Drinfeld doubles.

5.2.
Characterizations of Quantum Groups. Let char k " 0 in this section. In Section 4 we observed that for an algebra A with triangular decomposition to be an indecomposable pointed Hopf algebra, GpAq needs to be abelian acting on V by scalars. That means, in the terminology of [AS02] that the YD-braiding Ψpv b wq " v p´1q Ź w b v p0q is of diagonal type, i.e. there exist non-zero scalars q ij such that Ψpv i b v j q " q ij v j b v i for a basis tv 1 , . . . , v n u.
We assume that the braidings arise from YD-module structures over an abelian group G in this section. That is, q ij " λ j pk i q for the characters λ i by which G acts on kv i and group elements k i such that δpv i q " v i b k i . It is a basic observation that the braided Hopf algebras T pV q{I for I P I V , including the Nichols algebras for V , only depend on the braiding on V (rather than the concrete choice of λ i , k i ). However, different diagonal braidings pV, Ψq and pV, Ψ 1 q give isomorphic braided Hopf algebras T pV q{I. Such isomorphisms can be obtained using the notion of twist equivalence for diagonal braidings (which is a special case of the more general concept of twisting an algebra by a 2-cocycle).
Definition 5.2.1. Two braided k-vector spaces of diagonal type pV, Ψq, pV 1 , Ψ 1 q (given by scalars q ij , q 1 ij ) twist equivalent if V -V 1 , q ii " q 1 ii , and q ij q ji " q 1 ij q 1 ji . Lemma 5.2.2. If pV, Ψq, pV 1 , Ψ 1 q are twist equivalent of diagonal type, then T pV q -T pV 1 q as braided Hopf algebras in the category of braided k-vector spaces, preserving the natural grading.
Proof. For a proof see e.g. [AS02, 3.9-3.10]. We can find generators v i of V and v 1 i of V 1 such that the isomorphism φ is determined by v i Þ Ñ v 1 i . Defining a 2-cocycle σ by σpv i b v j q " q 1 ij q´1 ij for i ă j and 1 otherwise, we find that the product v i v j maps to the product twisted by σ. Note that the isomorphism is not an isomorphism in the category of YD-modules over kG unless pV 1 , Ψ 1 q " pV, Ψq.
For an ideal I P I V , denote the corresponding ideal under the isomorphism T pV q -T pV 1 q from Lemma 5.2.2 by I 1 . Then we conclude that T pV q{I -T pV 1 q{I 1 is also an isomorphism of braided Hopf algebras. In particular, BpV q -BpV 1 q for the corresponding Nichols algebras.
Lemma 5.2.3. If pV, Ψq and pV 1 , Ψ 1 q are twist equivalent, such that G " xk 1 , . . . , k n y -xk 1 1 , . . . , k 1 n y " G 1 via k i Þ Ñ k 1 i , then Drin kG pV, V˚q -Drin kG 1 pV 1 , V 1˚q as Hopf algebras. Proof. By Lemma 5.2.2, T pV q{I -T pV 1 q{I 1 and T pV˚q{I˚-T pV 1˚q {I 1˚. By the assumptions on the group generators, k i Þ Ñ k 1 i extends to an isomorphism kG -kG 1 . Thus we can define a morphism Drin kG pV, V˚q Ñ Drin kG pV 1 , V 1˚q which is an isomorphism of k-vector spaces. Further, preservation of the bosonization condition can be checked on generators using the isomorphism φ from Lemma 5.2.2. Finally, the commutator relation (4.11) is preserved using the isomorphism on kG.
Diagonal braidings are a very general class of braidings. Quantized enveloping algebras at generic parameters however are based on braidings of specific type, called Drinfeld-Jimbo type. Following [AS04], there are different classes of braidings which we distinguish: Definition 5.2.4 ( [AS04]). Let pq ij q be the nˆn-matrix of a braiding of diagonal type.
(a) The braiding given by pq ij q is generic if q ii is not a root of unity for any i " 1, . . . , n.
(b) In the case k " C we say the braiding pq ij q is positive if it is generic and all diagonal elements q ii are positive real numbers.
(c) The braiding pq ij q is of Cartan type if q ii ‰ 1 for all i and there exists a Z-valued nˆn-matrix pa ij q with values q ii " 2 on the diagonal and 0 ď´a ij ă ord q ii for i ‰ j, such that (5.5) q ij q ji " q aij ii for all i, j.
That implies that pa ij q is a generalized Cartan matrix which may have several connected components. We denote the collection of these by χ. (d) The braiding pq ij q is of Drinfeld-Jimbo type (DJ-type) if q ij are generic (no roots of unity) and there exist positive integers d 1 , . . . , d n such that for all i, j, d i a ij " d j a ji (hence the matrix pa ij q is symmetrizable, and for any J P χ, there exists a scalar q J ‰ 0 in k such that q ij " q diaij J for any i P I, and j " 1, . . . , n. Some observations can be made about the Nichols algebras associated to braided vector spaces of DJ-type. First, observe that for a braiding of Cartan type with connected components I 1 , . . . , I n P χ, we have that BpV q is the braided tensor product BpV I1 q b . . . b BpV In q ([AS00, Lemma 4.2]). Further, for V with braiding pq ij q of DJ-type, the Nichols algebra can be computed explicitly as the quantum Serre relations ([Ros98, Theorem 15]): BpV q " kxx 1 , . . . , x n | adpx i q 1´aij px j q " 0, @i ‰ jy.
We now bring the growth condition of finite Gelfand-Kirillov dimension (GK dimension) into the picture, using characterization results of [Ros98] of Nichols algebras with this property.
Lemma 5.2.5 ( [Ros98]). Let k " C. Let pq ij q be the matrix of a braiding of diagonal type which is generic such that the Nichols algebra BpV q has finite Gelfand-Kirillov dimension. Then pq ij q is of Cartan type.
Moreover, if the braiding is positive then the braiding is twist equivalent to a braiding of DJ-type, and this condition is equivalent to finite GK dimension.
Corollary 5.2.6. Let A " Drin CG pV, V˚q, for V or separable type, with generic positive braiding pq ij q. Then the following are equivalent (i) A -U q pgq for g a semisimple Lie algebra.
(ii) The braided C-vector space V with braiding pq ij q is twist equivalent to a braiding of DJ-type with finite type Cartan matrix. (iii) BpV q has finite Gelfand-Kirillov dimension.
Proof. The equivalence of (ii) and (iii) is the statement of Lemma 5.2.5 due to [Ros98]. Using Lemma 5.2.3 we find that (ii) implies (i), while it is clear that (i) implies (ii). In fact, the GK dimension of BpV q for V of DJ-type equals the number of positive roots [AS04, 2.10(ii)].
Corollary 5.2.7. The only indecomposable bialgebras with a symmetric triangular decomposition on BpV q b kZ n b BpV˚q of separable type, such that V " Cxv 1 , . . . , v n y is of positive diagonal type, and that no v i commutes with all of V˚are isomorphic to U q pgq for some semisimple Lie algebra g.
Proof. This follows from the classification 4.2.2, combined with the result of Rosso. The Lie algebra g is determined by the Cartan matrix one obtains under twist equivalence in Lemma 5.2.5. The technical condition that no v i commutes with all of V˚ensures that rf i , v i s ‰ 0 for a dual basis f 1 , . . . , f n of V˚, resembling the so-called non-degeneracy condition that the scalars γ ii ‰ 0 in Theorem 4.3.2. This is a characterization for quantum groups at generic parameters. The work surveyed in [AS02, AS10] on pointed Hopf algebras over finite-dimensional Hopf algebras can be viewed as a characterization of small quantum groups. The triangular decomposition can be view as the case where the graph Γ described in 2.3 has two connected components, such that the corresponding generators for the two components give dually paired braided Hopf algebras.
The characterization suggests that if we are looking for examples outside of DJ-type, we can consider braidings of generic Cartan type which are not positive. In fact, [AS04,2.6] gives an example that is generic of Cartan type, but not of DJ-type. We compute the associated quantum group here: 5.3. Classes of Pointed Hopf Algebras by Radford. In [Rad94], a class of pointed Hopf algebras U pN,ν,ωq was introduced (see also [Gel98] for generalizations). These Hopf algebras are associated to the datum of a positive integer N and 1 ď ν ă N such that N does not divide ν 2 , and ω P k is a primitive N th root of unity in a field k. Denote q :" ω ν and r " |q ν | " ω ν 2 . We let C N denote a cyclic group of order N generated by an element a.
The algebra U pN,ν,ωq is the braided Drinfeld double of the YD-module Hopf algebra U`:" krxs{px r q over C p , with grading given by x Þ Ñ a ν bx and action aŹx " q´1x. Note that U`is the Nichols algebra of the one-dimensional YD-module kx. The coalgebra structure is given by ∆pxq " x b a ν`1 b x, and ∆pyq " y b 1`a´ν b y for the dual generator y. Note further that the other Hopf algebra H pN,ν,ωq introduced by Radford is simply the bosonization U`¸kC N in this set-up. The algebras U pN,ν,ωq and H pN,ν,ωq are not indecomposable unless ν " 1. To obtain indecomposable pointed Hopf algebras, we can consider the subalgebras generated by x, y and a ν (respectively, x and a ν ). Since these only depend on the choices of r and q we denote these Hopf algebras by U pr,qq (respectively, H pr,qq ). Note that U pr,1,qq " U pr,qq . 5.4. Quantum Group Analogues in Other Contexts. To conclude this paper, we would like to adapt the point of view that quantum groups can also be studied over other Hopf algebras H than the group algebra. For this, one can, motivated by the results of this paper, look for Hopf algebras A with triangular decomposition over H. The property over a group that A is of separable type can be generalized by requiring that the YD-modules V with respect to the left and right coactions δ r and δ l are a direct sum of distinct one-dimensional simples.
As a first example, we can consider the case where H itself is primitively generated, i.e. H " krx 1 , . . . , x n s over a field of characteristic zero. If A is a bialgebra with triangular decomposition over H, then for v P V , ∆pvq P V b H`H b V implies that ∆pvq in fact equals v b 1`1 b v using the counitary condition. This gives that A is generated by primitive elements and hence is a pointed Hopf algebra that is connected (i.e. the group like elements are the trivial group). Now A is in particular cocommutative, so Theorem 5.6.5 in [Mon93] implies (for char k " 0) that A " U pgq where g is the Lie algebra of primitive elements in A. From this point of view, all quantum groups over H " krx 1 , . . . , x n s are simply the classical universal enveloping algebras. Investigating Hopf algebras with triangular decomposition over other Hopf algebras H can be the subject of future research.