On the Hopf algebra structure of the Lusztig quantum divided power algebras

We study the Hopf algebra structure of Lusztig's quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build them from the plus, minus and zero parts by means of suitable actions and coactions within the formalism presented by Sommerhauser to describe triangular decompositions.


Introduction
There are two versions of quantum groups at roots of 1: the one introduced and studied by De Concini, Kac and Procesi [DK, DP] and the quantum divided power algebra of Lusztig [Lu1,Lu2,Lu3,Lu4]. The small quantum groups (aka Frobenius-Lusztig kernels) appear as quotients of the first and Hopf subalgebras of the second; in both cases they fit into suitable exact sequences of Hopf algebras.
The key actor in all these constructions is what we now call a Nichols algebra of diagonal type. Indeed all the Hopf algebras involved have triangular decompositions compatible with the mentioned exact sequences; the positive part of the small quantum group is a Nichols algebra. The celebrated classification of the finite-dimensional Nichols algebras of diagonal type was achieved in [H]. The positive parts of the small quantum groups correspond to braidings of Cartan type, but there are also braidings of super and modular types in the list, see the survey [AA].
The question of defining the versions of the quantum groups of De Concini, Kac and Procesi on one hand, and of Lusztig on the other, for every Nichols algebra in the classification arises unsurprisingly. The first was solved in [Ang2] introducing Hopf algebras also with triangular decompositions and whose positive parts are now the distinguished pre-Nichols algebras of diagonal type. These were introduced earlier in [Ang1], instrumental to the description of the defining relations of the Nichols algebras. A distinguished pre-Nichols algebra projects onto the corresponding Nichols algebra and the kernel is a normal Hopf subalgebra that is even central under a mild technical 2010 Mathematics Subject Classification. 16T05. The work of N. A., I. A. and C. V. was partially supported by CONICET and Secyt (UNC). The work of C.V. was partially supported by Foncyt PICT 2016-3927. hypothesis, see [Ang2,AAR2]. The geometry behind these new Hopf algebras is studied in the forthcoming paper [AAY] for Nichols algebras coming in families.
Towards the second goal, the graded duals of those distinguished pre-Nichols algebras were studied in [AAR1] under the name of Lusztig algebras; when the braiding is of Cartan type one recovers in this way the positive (and the negative) parts of Lusztig's quantum groups. A Lusztig algebra contains the corresponding Nichols algebra as a normal Hopf subalgebra and the cokernel is an enveloping algebra U (n) under the same mild technical hypothesis mentioned above, see [AAR2]. In [AAR2,AAR3] it was shown that n is either 0 or the positive part of a semisimple Lie algebra that was determined explicitly in each case.
In order to construct the analogues of Lusztig's quantum groups at roots of one for Nichols algebras of diagonal type, we still need to define the 0-part and the interactions with the positive and negative parts. This leads us to understand the Hopf algebra structure of a Lusztig's quantum group which is the objective of this Note.
Let V be the Z[v, v −1 ]-Hopf algebra as in [Lu3,2.3]; the quantum group is defined by specialization of V . Our first goal is to describe the specialization of the 0-part V 0 , a commutative and cocommutative Hopf subalgebra of V . We show in Theorem 3.9 that it splits as the tensor product of the group algebra of a finite group and the enveloping algebra of the Cartan subalgebra of the corresponding Lie algebra. For this we use some skew-primitive elements h i,n ∈ V 0 , cf. Definition 3.4, defined from the elements K i ;0 t and K n i of the original presentation of [Lu3] via elements p n,s ∈ Z[v, v −1 ]. These are defined recursively in Lemma 3.3.
In [S] it is explained that Hopf algebras U with a triangular decomposition U ≃ A ⊗ H ⊗ B, where H is a Hopf subalgebra, A is a Hopf algebra in the category of left Yetter-Drinfeld modules and B is the same but right, plus natural compatibilities, can be described by some specific structure that we call a TD-datum. Our second goal is to spell out the TD-datum corresponding to the quantum group, see Theorem 4.4.
The paper is organized as follows. In Section 2 we set up some notation and recall the formalism of [S]. Section 3 contains the analysis of the Hopf algebra V 0 from [Lu3]. In Section 4 we recall the definition of Lusztig's version of quantum groups at roots of 1, show that it fits into the setting of [S] and prove Theorem 4.4. For simplicity of the exposition we assume that the underlying Dynkin diagram is simply-laced; in the last Subsection we discuss how one would extend the material to the general case.
The Lusztig's quantum groups enter into a cleft short exact sequence of Hopf algebras [A, 3.4.1,3.4.4] and contain an unrolled version of the finite quantum groups [Le] but as is apparent from the description here, they are not unrolled Hopf algebras. We are not aware of other papers containing information on the matter of our interest. Other versions of triangular decompositions similar to [S] appear in [Ma, L].

Preliminaries
2.1. Conventions. We adhere to the notation in [Lu3,Lu4] as much as possible. If t ∈ N 0 , θ ∈ N and t < θ, then I t,θ := {t, t + 1, . . . , θ}, I θ := I 1,θ . Let , the ring of Laurent polynomials in the indeterminate v, A ′ = Q(v), its field of fractions; later we also need The v-numbers are the polynomials If B is a commutative ring and ν ∈ B is a unit, then B is an As in [Lu3,5.1,pp. 287 ff], we fix ℓ ′ ∈ N and set We also have that Hence for all m ≥ n ∈ N 0 and j ∈ I ℓ−1 , we have Let k be a field; all algebras, coalgebras, etc. below are over k unless explicitly stated otherwise. If A is an associative unital k-algebra, then we identify k with a subalgebra of A.

2.2.
Hopf algebras with triangular decomposition. Let H be a Hopf algebra with multiplication m, comultiplication ∆ (with Sweedler notation ∆(h) = h (1) ⊗ h (2) ), counit ε and bijective antipode S; we add a subscript H when precision is desired. We denote by H H (2.5) (iv) ↼: B ⊗ A → B is a right action so that B is a right A-module, and the following identities hold for all a ∈ A, b ∈ B and h ∈ H: (2.6) both actions also satisfy for all a ∈ A and b ∈ B: Compatibility of ♯ with the structure of H:

8)
Compatibility of ♯ with the products of A and B: (2.9)

Compatibility of the actions with the multiplications via
(2.10)

Compatibility of the coactions with the comultiplications via
Hopf algebra with multiplication, comultiplication and antipode: [S, 3.5] Let U be a Hopf algebra. Let A and B be Hopf algebras in H H YD and YD H H respectively, provided with injective algebra maps Assume that Clearly these constructions are mutually inverse. In the setting of the Proposition, we say that U ≃ A ⊗ H ⊗ B is a triangular decomposition.
As observed in [S], the verification of the conditions in the definition of TD-datum is easier when H is commutative and cocommutative.
3. The algebra V 0 3.1. Basic definitions. We fix θ ∈ N. For simplicity, set I := I θ . Following [Lu3,2.3,pp. 268 ff] we consider the A-algebra V 0 presented by generators and relations for all i ∈ I, tagged as in loc. cit., the generators (3.1) commute with each other, (g6) Observe that (g9) and (g10) actually define the elements See also §4.4 for an equivalent formulation. Set actually it is a form of the Hopf algebra structure as we see next.
Lemma 3.2. The A-algebra V 0 is a Hopf algebra with comultiplication determined by (g9) and (g10), it is enough to show that for all i ∈ I and t ∈ N. We proceed by induction on t. If t = 1, then which completes the inductive step.
3.2. Some skew-primitive elements. We introduce some notation: It is easy to see that (3.10) defines an algebra map. Notice that Then Proof. Fix i ∈ I. By Proposition 3.1 (a), K n i − K −n i is a linear combination of k i,t , k i,t K i , t ∈ N 0 . Indeed, it can be shown by induction on n that K ±n i belongs to the A-submodule spanned by k i,t , k i,t K i , t ≤ n. Using the involution Φ, we see by (3.11) that there are a n,t ∈ A, t ∈ I n such that We extend scalars as in Proposition 3.1 (b) and consider the algebra maps (3.14) j ∈ N 0 . Notice that, with the convention N n v = 0 when n > N , Applying Ξ i,0 and Ξ i,1 to (3.13), we see that 0 = a n,0 , v n − v −n = a n,1 v φn(1) . Now we apply Ξ i,s , s > 1, to (3.13): v ns − v −ns = t∈Is a n,t s t v v φn(t)s ; this implies the recursive formula holds since Definition 3.4. Let n ∈ N. We set Lemma 3.5. Let n ∈ N. Then In particular, Looking at the equality (3.12), K −n i appears only in one summand, p n,n k i,n , on the right hand side. Hence and the claimed equality follows.
Given ℓ ∈ N we consider the lower triangular matrix , and the column vectors where h i,n := nh i,n K −n i .

3.3.
Specializations of V 0 . Recall that ℓ ′ ∈ N is defined in Section 2.1 and that B is the field of fractions of A/ φ ℓ ′ . We study now V 0 B := V 0 ⊗ A B. We also assume that ν 2 = 1. Thus the map A → B factorizes through A ′′ .
Let Γ = (Z/2ℓ) I , with g i ∈ Γ being generators of the corresponding copies of the cyclic group Z/2ℓ. Let h be the abelian Lie algebra with basis (t i ) i∈I , so that U (h) ≃ B[t i : i ∈ I].
4. The algebra V , simply-laced diagram 4.1. Definitions and first properties. As in [Lu3], we fix a finite Cartan matrix A = (a ij ) i,j∈I whose Dynkin diagram is connected and simply-laced, that is, of type A, D or E.
Following [Lu3,2.3,pp. 268 ff] we consider the A-algebra V presented by generators (3.1), E together with the following, tagged again as in loc. cit., The following formula is analogous to (h2), cf. [Lu5,Corollary 3.19]: By [Lu3,Proposition 4.8,p. 287], we know that V has a unique Hopf algebra structure determined by (3.8) and 4.2. Specializations of V . We define next By [Lu2, Proposition 3.2 (b)], V + B is generated by E i and E (ℓ) i , i ∈ I; V − B is generated by F i and F (ℓ) i , i ∈ I. From now on, we abbreviate Lemma 4.1. Let i, j ∈ I. We have in V B : Proof. We consider first the case j = i. We take t = ℓ, c = 0 and N = 1 in (h6) and use (g9) to obtain (4.5): For (4.11), we take t = ℓ, c = 0 and N = 1 in (h6) and use (g9): hence we obtain (4.7) when N = 1, and (4.11) when N = ℓ.
(2) j = i, a ij = −1. Let b t ∈ B such that Using the algebra maps such that K i → ν s and K i → −ν s , 0 ≤ s < ℓ, we conclude from the previous equality that Hence b 0 = 0. From (4.9) and (h4): The proof is analogous to the previous case, using (4.4).
Hence c ij = 0 in all the cases, so (4.12) and (4.14) follow. 4.3. The Hopf algebra structure of V B . Recall that by Theorem 3.9, Remark 4.3. The counit on the elements K ±1 i ; c t takes the following values. (4.16) In fact, we first note that ε (4.1). The formula for c = 0 holds by (3.5). Then, for c > 0, we use (g10): While for c < 0, we use (g9): Theorem 4.4. The Hopf algebra V B has a triangular decomposition given by a TD-datum Proof. For the first claim, we just need to verify that the conditions of Proposition 2.1 (b) hold.
admit Hopf algebra sections π + and π − respectively and that Also, by [Lu3,Theorem 4.5 (a)], the multiplication induces a linear isomorphism Thus we may apply Proposition 2.1 (b).
The verification of (4.17) is direct using the formulas in the proof of [S,Theorem 3.5] and the natural projections ). If i = j, this zero by (h1). Otherwise, we use (4.2): We can verify the formulas for ↼ and ♯ in a similar way. We next show by induction that (4.17) completely determines ⇀, ↼ and ♯. We will use that the comultiplication of V ± B in the respective Yetter-Drinfeld category is given by This follows from (4.3). are determined by (4.17) for all r, s ≤ n and prove the same claim for s = r = n + 1. By (2.10), we have that (1) Hence, (4.18) is determined by (4.17) because of the inductive hypothesis. The same holds for F  (4.17) for all r, s ≥ 0. This is true for F ⇀ E and F ↼ E because ⇀ and ↼ are actions. For the others we proceed again by induction on r (or on s) using (2.9); notice that the initial inductive step r = 1 was proved above.
Remark 4.5. Here are some particular instances of the first line in (4.17): Analogously, the structure of V − B as an object in YD is as follows: the (right) action of V 0 B on V − B is given by (h4), (4.14) and (4.15); meanwhile the coaction ρ : 4.4. The multiply-laced diagrams. The arguments above can be extended to the diagrams of types B, C, F, G. We just discuss the torus part here. Let d = (d i ) i∈I ∈ N I . Following [Lu4,6.4] we consider the A-algebra V 0 that is a (multiply-laced!) variation of the V 0 studied so far. For the agility of the exposition we do not stress d in the notation. This V 0 is presented by the generators analogous to those (3.1) of V 0 : i ∈ I, c ∈ Z, t ∈ N 0 (4.19)