Cohomology rings for quantized enveloping algebras

We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) $U_q$ associated to a finite-dimensional simple complex Lie algebra $\mathfrak{g}$. We show that the cohomology ring is generated as an exterior algebra by homogeneous elements in the same odd degrees as generate the cohomology ring for the Lie algebra $\mathfrak{g}$. Partial results are also obtained for the cohomology rings of the non-restricted quantum groups obtained from $U_q$ by specializing the parameter $q$ to a non-zero value $\epsilon \in \mathbb{C}$.

1. Introduction 1.1. Let G be a simple compact connected Lie group of dimension d. It is a famous theorem from algebraic topology that the homology and cohomology algebras for G are exterior algebras over graded subspaces concentrated in odd degrees [Sam]. By a result of Cartan, the homology and cohomology algebras for G identify with those for its Lie algebra g, so we get also the ring structure of the Lie algebra cohomology ring H • (g, C) = H • (U (g), C). Here U (g) denotes the universal enveloping algebra of g = Lie(G). In recent years there has been much interest in homological and cohomological properties for various classes of noetherian Hopf algebras [BG, BZ, Che], important examples of which being the universal enveloping algebras and quantized enveloping algebras associated to a finite-dimensional simple complex Lie algebra. A common theme to some of the recent work has been the desire to generalize Poincaré duality to these classes of noetherian Hopf algebras [BZ, KK].
Let q be an indeterminate, and set k = C(q). Let U q be the quantized enveloping algebra over k associated to the finite-dimensional simple complex Lie algebra g. Though the above cited works provide general results relating the dimensions of the homology and cohomology groups H n (U q , k) = Tor Uq n (k, k) and H n (U q , k) = Ext n Uq (k, k), namely, dim k H n (U q , k) = dim k H d−n (U q , k), there have been no explicit calculations of the dimensions of these groups, nor of the ring structure for the cohomology ring H • (U q , k). Similarly, one would like to know the dimension and ring structure of the cohomology ring H • (U ε , C) associated to the quantized enveloping algebra U ε with parameter q specialized to a value ε ∈ C × := C − {0}.
In this paper we show that the cohomology ring H • (U q , k) is an exterior algebra generated by homogeneous elements in the same odd degrees as for H • (U (g), C), and thus for each n ∈ N that the cohomology group H n (U q , k) for U q is of the same dimension as the corresponding group for U (g). Our proof relies on an integral form U A for U q , which enables us to relate, via the universal coefficient theorem, cohomology for U q to that for U (g). The main steps of this argument are carried out in Sections 3.4 and 4.1. A key step in the proof is the calculation of the restriction maps in Lie algebra cohomology associated to an inclusion F ⊂ E of simple Lie algebras; see Section 2.3. Finally, assuming ε ∈ C is a root of unity of sufficiently large prime order p, we obtain the structure of the cohomology ring H • (U ε , C) for the quantized enveloping algebra U ε . This last computation exploits a connection between U ε and the characteristic p universal enveloping algebra of g.
2010 Mathematics Subject Classification. Primary 17B37, 17B56. The author was supported in part by NSF VIGRE grant DMS-0738586.
1.2. Notation. Let N = {0, 1, 2, 3, . . .} denote the set of non-negative integers. Let (a ij ) be the r × r Cartan matrix associated to the finite-dimensional simple complex Lie algebra g, and let (d 1 , . . . , d r ) ∈ N r be the unique vector such that gcd(d 1 , . . . , d r ) = 1 and the matrix (d i a ij ) is symmetric. The ordering of the rows and columns for the matrix (a ij ) corresponds to a labeling of the Dynkin diagram associated to g; we assume this is done as in Bourbaki [Bou2,.
Let C[q, q −1 ] be the Laurent polynomial ring over C in the indeterminate q, and let k = C(q) be its quotient field. Then the quantized enveloping algebra (or quantum group) associated to g is the k-algebra U q = U q (g) defined by the generators {E i , F i , K ±1 i : i ∈ [1, r]} and the relations given in [DCK,]. The algebra U q is also a Hopf algebra via the maps in [DCK,]. For be the ℓ-th cyclotomic polynomial. Now define S ⊂ Z[q, q −1 ] to be the multiplicatively closed set generated by

{1}
if g has Lie type ADE, Then the generators for S are precisely the irreducible factors of [n] then U ε is the C-algebra with the same generators and relations as U q , but with q replaced by ε. For this reason, we call U ε a specialization of U q .
For each i ∈ [1, r], let T i be the braid group operator on U q as defined in [DCK,§1.6], and let Φ be the root system associated to g. Then for each positive root β ∈ Φ + , there exist root vectors E β , F β ∈ U q , defined in terms of the T i [DCK,§1.7]. By the definition of the denominator set S, the T i restrict to automorphisms of the algebra U Z , so also E β , F β ∈ U Z ⊂ U A . (This is why we work with the coefficient rings Z and A instead of with Z[q, q −1 ] and C[q, q −1 ].) 2. Lie algebra cohomology 2.1. An isomorphism with U 1 cohomology. We begin with an observation on the relationship between the cohomology spaces for the universal enveloping algebra U (g) and for the specialization U 1 . Recall from [DCK,Proposition 1.5] that U 1 is a central extension of U (g) by the group algebra over C for the finite group (Z/2Z) r , so there exists a surjective Hopf algebra homomorphism U 1 → U (g).
Lemma 2.1. The homomorphism U 1 → U (g) induces an algebra isomorphism Proof. Set G = (Z/2Z) r , and consider the Lyndon-Hochschild-Serre (LHS) spectral sequence for the algebra U 1 and its normal Hopf-subalgebra isomorphic to CG: . The group algebra CG is a semisimple algebra, so E i,j 2 = 0 for all j > 0. Since the spectral sequence respects cup products, it follows that the edge map E •,0 2 = H • (U (g), C) → H • (U 1 , C) is an algebra isomorphism.
2.2. The structure of Lie algebra cohomology. Lemma 2.1 reduces the problem of studying the cohomology ring H • (U 1 , C) to the classical problem of studying the cohomology ring H • (U (g), C). We summarize some details on the computation of H • (E, C) = H • (U (E), C) for E an arbitrary finite-dimensional reductive Lie algebra over C. Our main reference is [GHV, Chapters V-VI].
Let Λ • (E * ) denote the exterior algebra on the dual space E * = Hom C (E, C), considered as a graded complex with E * concentrated in degree 1. The map Λ 2 (E) → E defined by x ∧ y → [x, y] induces a map d : E * → Λ 2 (E * ), which extends by derivations to a differential on Λ • (E * ), also denoted d. Then H • (E, C) is the cohomology of the complex Λ • (E * ) with respect to the differential d. The space Λ • (E * ) is also naturally an E-module, with E-action induced by the coadjoint The addition map E × E → E, (x, y) → x + y, induces on Λ • (E * ) the structure of a bialgebra, and the bialgebra structure restricts to one on Λ • (E * ) E ∼ = H • (E, C). Then H • (E, C) is generated as an algebra by its subspace of primitive elements, which we denote by P E . The subspace P E is concentrated in odd degrees, and the induced map Λ( Theorem 2.2. [GHV, Sam] Let g be a finite-dimensional simple complex Lie algebra. Then H • (U (g), C) is an exterior algebra generated by homogeneous elements in the odd degrees listed in Table 1.

Restriction maps.
Let E and F be finite-dimensional reductive Lie algebras over C with F ⊆ E. Write j : F → E for the inclusion map. Let W (E) and W (F ) be the Weyl groups associated to E and F , respectively. The cohomological restriction map H • (E, C) → H • (F, C) is completely determined by the induced map j * : P E → P F on the spaces of primitive elements.
Let H ⊂ F be a Cartan subalgebra of F , and let H ′ ⊂ E be a Cartan subalgebra of E containing H. Let S(E * ) be the ring of polynomial functions on E, but with the subspace E * concentrated in degree 2. Similarly, define S(F * ), S(H * ), and S(H ′ * ) to be the evenly graded rings of polynomial functions on F , H, and H ′ . The coadjoint action of E on E * extends to an action of E on S(E * ), and similarly for F on S(F * ). Then the restriction map S(E * ) → S(F * ) induces a map S(E * ) E → S(F * ) F . By [GHV,§11.9], the restriction maps S(E * ) → S(H ′ * ) and S(F * ) → S(H * ) induce isomorphisms S(E * ) E ∼ = S(H ′ * ) W (E) and S(F * ) F ∼ = S(H * ) W (F ) . Since W (E) and W (F ) are finite reflection groups, the rings S(H ′ * ) W (E) and S(H * ) W (F ) are generated by algebraically independent homogeneous elements.
By [GHV,§6.7], there exists a canonical linear map ρ E : S(E * ) E → Λ • (E * ) E , homogeneous of degree −1, and natural with respect to the inclusion F ⊆ E. By [GHV,§6.14], im ρ E = P E , and so to compute the map j * : P E → P F , and hence the map j * : In the following theorem we explicitly describe the cohomological restriction map H • (E, C) → H • (F, C) for certain simple pairs (E, F ). Specifically, let E be a simple complex Lie algebra with associated root system Φ, and let ∆ = {α 1 , . . . , α r } be a set of simple roots for Φ, ordered as in [Bou2,. Then we will assume that F is a simple subalgebra of E of rank r − 1 corresponding to removing some simple root α F from ∆. For ease in stating the theorem, we identify the Lie algebras E and F with their respective Lie types.
and H • (F, C) = Λ(y j 1 , . . . , y j r−1 ) as in Theorem 2.2, with the x i and y j homogeneous of degrees i and j, respectively. If one of E or F is of type D, write x i or y j for the generator of the last degree listed in Table 1. Then the homogeneous generators can be chosen so that the cohomological restriction map H • (E, C) → H • (F, C) admits the following description: (3) If (E, F ) = (D r , D r−1 ), r ≥ 5, and α F = α 1 , then x 3 → y 3 , x 7 → y 7 , . . . , x 4r−9 → y 4r−9 , x 4r−5 → 0, x 2r−1 → 0.

Cohomology for the integral form U A
Next we study the cohomological properties of a certain integral form U A of U q , to be defined in Section 3.3, which will enable us to relate cohomology for U q to that for the Lie algebra g. First we collect some results on the algebra U Z .
3.1. A resolution of the trivial module. We begin with the following lemma, which is wellknown for the Lusztig integral form of U q , though we could find no analogous statement in the literature for the De Concini-Kac integral form U Z as we have defined it here. We thus record the result now.
Lemma 3.1. The algebra U Z is a free Z-module. Consequently, for any Z-algebra B, the algebra ]. The identity also shows that [DCK,(1.5.4)] and [Lus,6.4(b2) and 6.7(i)(c)] to deduce that this set is linearly independent over Z, and hence forms a Z-basis for A i . Thus we conclude that U Z is free over Z. The second claim of the lemma is now immediate.
Proof sketch. The argument is due to Brown and Goodearl [BG,§2.2]. In [DCP,§10.1], De Concini and Procesi define a sequence of degenerations of the algebra U q , each of which is the associated graded ring of the previous algebra with respect to a multiplicative N-filtration. The definition of the degenerations relies on the commutation relations between the root vectors in U q . Since the root vectors in U q are elements of U Z by our choice for the denominator set S, one can define a similar sequence of degenerations is an iterated twisted polynomial ring over the torus U 0 B . The torus U 0 B is generated as a B-algebra by the finite set of commuting elements [DCK,(1.5.4)]), so is noetherian because B is noetherian. Then U B is noetherian by [MR, Theorems 1.2.9 and 1.6.9].
Corollary 3.3. Let B be a Z-algebra. There exists a resolution of the trivial U B -module B by finitely-generated free U B -modules: Proof. First consider the case B = Z. Set P −1 = Z, P 0 = U Z , and let P 0 → P −1 be the augmentation map. Now given P n with n ≥ 0, let I n be the kernel of the map P n → P n−1 . Since by induction P n is a finitely-generated U Z -module, and since U Z is noetherian by Lemma 3.2, the U Z -submodule I n of P n is also finitely-generated as a U Z -module. Then there exists a finitely-generated free U Zmodule P n+1 mapping onto I n . Take P n+1 → P n to be the composite map P n+1 ։ I n ֒→ P n . We thus inductively construct the resolution P • → Z of Z by finitely-generated free U Z -modules. Since U Z is free over Z by Lemma 3.1, P • → Z is a complex of free Z-modules, hence splits over Z. It then follows for any Z-algebra B that P • ⊗ Z B → B is a resolution of B by finitely-generated free U B -modules.
3.2. Base change and the universal coefficient theorem. The crux of our argument for computing the cohomology ring H • (U q , k) relies on the universal coefficient theorem, which we now recall.
Theorem 3.4 (Universal Coefficient Theorem for Homology). [Rot,Theorem 7.55] Let R be a ring, A a left R-module, and (K, d) a chain complex of flat right R-modules such that the subcomplex of boundaries also consists of flat R-modules. Then for each n ∈ Z, there exists a short exact sequence with respect to both K and A, such that λ n : cls(z) ⊗ a → cls(z ⊗ a).
We apply the universal coefficient theorem as follows: Lemma 3.5. Let B be a Z-algebra, and Γ a B-algebra. Suppose B is a principal ideal domain. Then for each n ∈ N, there exists a short exact sequence Proof. Let P • → B be a resolution of B by finitely-generated free U B -modules as in Corollary 3.3, and set K n = Hom U B (P −n , B). Then the chain complex K • consists of finitely-generated free B-modules. Since every submodule of a free module over a PID is again free, the subcomplex of boundaries in K is also free, hence flat, over B. Also, since P n is free over U B , there exists for each n ∈ N a natural isomorphism Then applying the universal coefficient theorem with R = B, A = Γ, and K as above, one obtains the short exact sequence (3.3). Now let α ∈ H a (U B , B) and β ∈ H b (U B , B) be represented by cocycles f α ∈ K −a and f β ∈ K −b , respectively, and let ∆ : P → P ⊗ B P be a U B -module chain map lifting the isomorphism B ∼ = B ⊗ B B. Then the product αβ is represented by the cocycle ( is a chain map lifting the isomorphism Γ ∼ = Γ ⊗ Γ Γ. Then making the identification (3.4), one sees for all γ α , γ β ∈ Γ that λ(αβ ⊗ B γ α γ β ) and the product λ a (α ⊗ γ α )λ b (β ⊗ γ β ) are both represented by the cocycle and hence that λ is an algebra homomorphism.
In Lemma 3.5 we assumed that B was a principal ideal domain to conclude that the subcomplex of boundaries in K was flat. This conclusion would also hold under the weaker assumption that B is right semihereditary, or perhaps under even weaker assumptions on B, but we will not require such a generalization in this paper.
We now collect some results useful for analyzing the Tor-group in (3.3).
Lemma 3.6. Let B be a noetherian Z-algebra. Then for each n ∈ N, the cohomology group H n (U B , B) is a finitely-generated B-module.
Proof. Let K = Hom U B (P • , B) be the complex of finitely-generated free B-modules considered in the proof of Lemma 3.5. Since B is noetherian, any subquotient of a finitely-generated B-module is again finitely-generated. In particular, H n (U B , B) is a B-module subquotient of K −n , so is finitely-generated over B. Proof. This follows from [Bou1, II.3.2 Corollary 2 of Proposition 5].
In a similar vein, one has: Lemma 3.8. Let B be an integral domain, b ∈ B, and M a B-module. Then Proof. Compute the Tor-group using the resolution 0 → B ×b → B → B/bB → 0.
3.3. The integral form U A . We now define the integral form U A , and describe how we will apply the results of Section 3.2 to relate the cohomology theories for U q , U A , and U (g). To begin, set A = C[q] (q−1) , the localization of C[q] at the maximal ideal generated by q − 1. Then A is a local principal ideal domain, with quotient field k = C(q) and residue field C. As in Section 1.2, we write C 1 for the field C considered as an A-algebra via the map q → 1. The field k is A-flat by [Rot,Corollary 3.50] because it is torsion-free, so applying Lemma 3.5 with B = A and Γ = k, we get for each n ∈ N the isomorphism On the other hand, U 1 = U A ⊗ A C 1 , so applying Lemma 3.5 with B = A and Γ = C 1 , we get for each n ∈ N the short exact sequence It follows from Lemmas 3.6 and 3.7 that the map λ n is an isomorphism if and only if H n+1 (U A , A) is free as an A-module. In particular, if the algebra homomorphism λ : Our strategy for computing H • (U q , k) is now as follows. We first verify that the injective algebra homomorphism λ : H • (U A , A) ⊗ A C 1 → H • (U 1 , C) is an isomorphism, and hence that H • (U A , A) is A-free of rank dim C H • (U (g), C), by showing that the odd degree homogeneous generators for H • (U 1 , C) ∼ = H • (U (g), C) all lie in the image of λ. We verify this for g not of type D r or E 6 in Section 3.4, and for types D r and E 6 in Sections 4.2 and 4.3. Next, using the fact that H • (U A , A) is A-free and that H • (U A , A) ⊗ A C 1 ∼ = H • (U (g), C) is an exterior algebra, we deduce in Section 4.1 that H • (U A , A) is an exterior algebra generated in the same odd degrees as is H • (U (g), C). Finally, we apply (3.5) to deduce the structure of H • (U q , k).

3.4.
Cohomology for U A . Following the strategy outlined in Section 3.3, we first verify that λ is an isomorphism when g is not of type D r or E 6 . Theorem 3.9. Suppose g is not of type D r or E 6 . Then the injective algebra map is an isomorphism. In particular, H • (U A , A) is a finitely-generated free A-module.
Proof. We prove the theorem by showing that the odd-degree homogeneous generators for H • (U 1 , C) described in Theorem 2.2 all lie in the image of λ. First suppose g is of type A 1 , A 2 , B 2 , C 2 , E 7 , E 8 , F 4 , or G 2 , and let n be one of the odd degrees listed in Table 1. Using Theorem 2.2 and Table  1 one can check that H n+1 (U (g), C) = 0. Then (3.6) implies that and hence H n+1 (U A , A) = 0 by Nakayama's Lemma. Then λ n : H n (U A , A) ⊗ A C 1 → H n (U 1 , C) is an isomorphism by (3.6), so for these Lie types we conclude that the odd-degree homogeneous generators for H • (U 1 , C) all lie in the image of λ. Now suppose that g is of type X r , with X ∈ {A, B, C} and r ≥ 3. Let g ′ ⊂ g be the subalgebra of g of type X r−1 as defined in cases (1) and (2) of Theorem 2.3. Define U q (g ′ ) and U A (g ′ ) to be the subalgebras of U q and U A , respectively, generated by the set Then U q (g ′ ) is isomorphic to the quantized enveloping algebra associated to g ′ , and U A (g ′ ) is its corresponding integral form. By induction on the rank of g, we may assume for each n ∈ N that the space H n (U A (g ′ ), A) is A-free of rank dim C H n (U (g ′ ), C). Let n 1 < · · · < n r be the degrees listed in Table 1 of the homogeneous generators for H • (U (g), C) ∼ = H • (U 1 , C). As in Theorem 2.3, write H • (U (g), C) ∼ = Λ(x n 1 , . . . , x nr ), with x n i of degree n i , and set z i = x n i . Let j ∈ [1, r], and assume by induction that z 1 , . . . , z j−1 ∈ im(λ). To show that z j ∈ im(λ), it suffices to show that H n j +1 (U A , A) is A-free, since this implies by (3.6) that λ n j : By Theorem 2.2, the space H n j +1 (U 1 , C) is spanned by certain monomials in the generators z 1 , . . . , z r , but since n i = 1 for any i, no nonzero monomial can involve a generator z i with i ≥ j. Then H n j +1 (U 1 , C) is spanned by certain monomials in the generators z 1 , . . . , z j−1 ∈ im(λ), and it follows that these monomials are in the image of λ, and hence that λ n j +1 is an isomorphism. Now consider the following diagram, where the vertical arrows are the corresponding restriction maps: The commutativity of the diagram follows from the fact that the universal coefficient theorem (Theorem 3.4) is natural with respect to the complex K. The bottom map in the diagram is an isomorphism by induction on the rank of the Lie algebra. The right-hand restriction map is also an isomorphism, since by Theorem 2.3 the homogeneous generators z 1 , . . . , z j−1 for H • (U (g), C) can be chosen so that the restriction map H • (U (g), C) → H • (U (g ′ ), C) maps them onto the corresponding generators for H • (U (g ′ ), C). This implies that the left-hand restriction map is an isomorphism as well, hence that the map H n j +1 (U A (g), A) → H n j +1 (U A (g ′ ), A)/(q − 1) H n j +1 (U A (g ′ ), A) is surjective. Then the restriction map H n j +1 (U A , A) → H n j +1 (U A (g ′ ), A) is surjective by Nakayama's Lemma. By induction on the rank of the Lie algebra, the space H n j +1 (U A (g ′ ), A) is A-free of rank dim C H n j +1 (U (g ′ ), C) = dim C H n j +1 (U (g), C). Then the map H n j +1 (U A , A) → H n j +1 (U A (g ′ ), A) is a split surjection of A-modules, and H n j +1 (U A , A) has A-rank at least dim C H n j +1 (U (g), C). Now dim C H n j +1 (U (g), C) ≤ dim k H n j +1 (U A , A) ⊗ A k by the bound on the A-rank, 4. Cohomology for the quantized enveloping algebra U q 4.1. Cohomology ring structure. We now deduce the structure of H • (U q , k) in any case for which λ : C) is an isomorphism. Then the cohomology rings H • (U A , A) and H • (U q (g), k) are exterior algebras generated by homogeneous elements in the odd degrees listed in Table 1.
Proof. Since λ is an isomorphism, we have for each n ∈ N that H n (U A , A) is a free A-module of rank dim C H n (U (g), C) by the discussion in Section 3.3. Choose homogeneous elements z 1 , . . . , z r ∈ H • (U A , A) such that their images under λ in H • (U 1 , C) ∼ = H • (U (g), C) are the homogeneous generators described in Theorem 2.2. Since U A is a Hopf algebra over the commutative ring A, the cohomology ring H • (U A , A) is graded-commutative [ML,Corollary VIII.4.3]. The elements z 1 , . . . , z r ∈ H • (U A , A) are each homogeneous of odd degree, so z 2 i = 0 for each i ∈ [1, r], and there exists a well-defined map ϕ : Λ(z 1 , . . . , z r ) → H • (U A , A) of graded A-algebras. The induced map ϕ ⊗ A C 1 : Λ(z 1 , . . . , z r ) ⊗ A C 1 → H • (U A , A) ⊗ A C 1 is surjective by the choice of the z i , so we conclude by Nakayama's Lemma that ϕ is surjective, hence a graded algebra isomorphism because Λ(z 1 , . . . , z r ) and H • (U A , A) are each A-free of the same finite rank. Extending scalars to k, we obtain via (3.5) the graded algebra isomorphism ϕ ⊗ A k : Λ(z 1 , . . . , z r ) ⊗ A k ∼ → H • (U q (g), k).

Type D.
To extend Theorem 3.9 to the case when g is of type D r , we consider cohomological restrictions maps corresponding not only to a Lie subalgebra g ′ of g of type D r−1 , but also to a Lie subalgebra g ′′ of g of type A r−1 . In the latter case, we also require the explicit understanding of the ring structure for H • (U q (g ′′ ), k) that comes from Theorem 4.1.
Theorem 4.2. The conclusion of Theorem 3.9 holds if g is of type D r .
Proof. Suppose g is of type D r with r ≥ 4. The overall strategy is similar to that in the proof of Theorem 3.9 for types A, B, and C, though some subtleties arise because the right-hand column of (3.7) need not be an isomorphism when g is of type D. As in the proof of Theorem 3.9, we consider a subalgebra g ′ ⊂ g of type D r−1 , as defined in case (3) of Theorem 2.3, and also a subalgebra g ′′ ⊂ g of type A r−1 , as defined in case (4) of Theorem 2.3. (If r = 4, then g ′ is of type A 3 , and cases (3) and (4) of Theorem 2.3 coincide.) For j ∈ [1, r − 1] set n j = 4j − 1, and set n r = 2r − 1, so that n 1 , . . . , n r are the degrees listed in Table 1 for type D r .
Our first step is to show for all n ∈ [1, 2r] that H n (U A , A)⊗ A C 1 ∼ = H n (U 1 , C). Since H • (U 1 , C) is an exterior algebra generated in the odd degrees n 1 , . . . , n r , this is equivalent to showing H n j (U A , A)⊗ A C 1 ∼ = H n j (U 1 , C) whenever n j ≤ 2r − 1. First let j ∈ [1, r] with n j ≤ 2r − 3. It follows from Theorem 2.3 that the restriction map H n j +1 (U (g), C) → H n j +1 (U (g ′ ), C) is an isomorphism; cf. the analysis of (3.7). Also, by induction on the rank of g, we may assume for all n ∈ [1, 2(r − 1)] that H n (U A (g ′ ), A) ⊗ A C 1 ∼ = H n (U (g ′ ), C), and hence that H n (U A (g ′ ), A) is A-free of rank dim C H n (U (g ′ ), C); cf. Section 3.3. Now one can imitate the proof of Theorem 3.9, arguing by induction on the rank and the degree, to show for all n j ≤ 2r−3 that H n j (U A , A)⊗ A C 1 ∼ = H n j (U 1 , C). Then to complete the first step, we must now show that H 2r−1 (U A , A) ⊗ A C 1 ∼ = H 2r−1 (U 1 , C).
Given y ∈ H • (U A , A), set y = λ(y ⊗ A 1) ∈ H • (U 1 , C). By the previous paragraph, we can choose y 1 , . . . , y s ∈ H • (U A , A) such that y 1 , . . . , y s ∈ H • (U 1 , C) are representatives for the homogeneous generators for H • (U 1 , C) of degrees less than or equal to 2r − 3. Then H 2r (U 1 , C) is spanned over C by certain monomials in the vectors y 1 , . . . , y s . Let m 1 , . . . , m t ∈ H 2r (U A , A) be monomials in the y i such that m 1 , . . . , m t form a basis for H 2r (U 1 , C). We want to show that dim k H 2r (U q , k) ≥ t, for this implies by (3.5) and [APW,Lemma 1.21 . By Theorems 3.9 and 4.1, H • (U q (g ′′ ), k) is an exterior algebra generated by homogeneous elements of certain odd degrees. Moreover, it follows from Theorem 2.3 and the proof of Theorem 4.1 that we can take certain of the generators for H • (U q (g ′′ ), k) to be the vectors ρ( y 1 ), . . . , ρ( y s ). This implies that the vectors ρ( m 1 ), . . . , ρ( m t ) ∈ H 2r (U q (g ′′ ), k) are linearly-independent, and hence m 1 , . . . , m t ∈ H 2r (U q , k) are as well. We then conclude that dim k H 2r (U q , k) ≥ t, which completes the first step of the proof.
We have shown for all a ∈ N that if g is of type D a , then H n (U A , A) ⊗ A C 1 ∼ = H n (U 1 , C) for n ∈ [1, 2a]. Write H • (U 1 , C) ∼ = H • (U (g), C) ∼ = Λ(x 3 , . . . , x 4r−5 , x 2r−1 ) as in Theorem 2.3. Suppose n i = deg(x i ) > 2r − 1; we must show that x i ∈ im(λ). Set m = 2r − 2, and let g m be the finite-dimensional simple complex Lie algebra of type D m . The inclusion of Dynkin diagrams D r ֒→ D m induces an inclusion of algebras U A ֒→ U A (g m ); cf. Section 3.4. We thus have the following commutative diagram, where the vertical arrows are the corresponding restriction maps: Since n i ≤ 4r − 5 < 2m, the top row of (4.1) is an isomorphism by the first step of the proof. Also, since n i ≤ 4r −5 ≤ 4m−9, it follows from Theorem 2.3 that x i is in the image of the restriction map H • (U (g m ), C) → H • (U (g), C). Then from the commutativity of (4.1) we conclude that x i ∈ im(λ). This completes the proof. 4.3. Type E 6 . To extend Theorem 3.9 to the case when g is of type E 6 , we consider restriction maps like those in cases (5) and (6) of Theorem 2.3.
Proof sketch. The strategy is similar to the proofs of Theorems 3.9 and 4.2. The generators for H • (U 1 , C) are in degrees 3, 9, 11, 15, 17, and 23. One can check using Theorem 2.2 that H n (U 1 , C) = 0 for n ∈ {4, 10, 16}, so H n (U A , A) ⊗ A C 1 ∼ = H n (U 1 , C) if n ∈ {3, 9, 15}. The description of the restriction map from E 7 to E 6 in case (7) of Theorem 2.3 implies that the generators of degrees 11 and 23 are also in im(λ); cf. the analysis of (4.1). Then it remains to show that H 17 (U A , A) ⊗ A C 1 ∼ = H 17 (U 1 , C), or equivalently that H 18 (U A , A) is A-free. We can choose y 3 ∈ H 3 (U A , A) and y 15 ∈ H 15 (U A , A) such that the product y 3 y 15 spans H 18 (U 1 , C). Let g ′ ⊂ g be the subalgebra of type D 5 as defined in case (5) of Theorem 2.3, and let ρ : be the corresponding restriction map. Then the argument in the third paragraph of the proof of Theorem 3.9 shows that ρ is surjective in degrees 3 and 15. This implies by the proof of Theorem 4.1 that H 18 (U A (g ′ ), A) ⊗ A k ∼ = H 18 (U q (g ′ ), k) ∼ = k is spanned by ρ(y 3 y 15 ). Then the product y 3 y 15 ∈ H 18 (U A , A) must span a one-dimensional subspace of H 18 (U q , k). Now so H 18 (U A , A) must be A-free or rank 1 by [APW,Lemma 1.21].
Here is the main result of our computations: Theorem 4.5. The cohomology ring H • (U q , k) is an exterior algebra over a graded subspace with odd gradation. Explicitly, H • (U q , k) is generated as an exterior algebra by homogeneous elements in the same odd degrees as for H • (U (g), C).

4.4.
The third cohomology group. A famous theorem of Chevalley and Eilenberg states that H 3 (U (g), C) = 0 [CE,Theorem 21.1]. They prove the non-vanishing of H 3 (U (g), C) by showing that the Killing form on g gives rise to a nonvanishing invariant 3-cochain in g. Our analysis gives us: Corollary 4.6. Let g be a finite-dimensional simple complex Lie algebra. Then It is an interesting question whether the non-vanishing of H 3 (U q (g), k) could also be established in a manner similar to that of Chevalley and Eilenberg, perhaps by using the non-degenerate inner product on U q (g) constructed by Rosso [Ros]. 5. Cohomology for the specializations U ε 5.1. Generic behavior. Recall the set S and the ring A = S −1 C[q, q −1 ] defined in Section 1.2. We call ε ∈ C a bad root of unity if ε ∈ {±1} or if ε is the root of some polynomial in S. Define the set C g ⊂ C by C g = ε ∈ C × : ε is not a bad root of unity . Then for all ε ∈ C g , the field C is an A-algebra via the map q → ε, and we can apply the results of Section 3.2 with B = A and Γ = C ε .
Proposition 5.1. The ring H • (U A , A) is a finitely-generated A-module.
Proof. For each n ∈ N, the space H n (U A , A) is a finitely-generated A-module by Lemma 3.6. Set d = dim C g. Then for all ε ∈ C g , the ring H • (U ε , C) satisfies the Poincaré duality H n (U ε , C) ∼ = H d−n (U ε , C) by [Che,Corollary 3.2.2]. In particular, H n (U ε , C) = 0 for all n > d. This implies by Lemma 3.5 with B = A and Γ = C ε that H n (U A , A) ⊗ A C ε = 0 for all n > d. Since ε ∈ C g was arbitrary, it follows for n > d (e.g., using the fundamental theorem for finitely-generated modules over a principal ideal domain) that H n (U A , A) = 0. Then H • (U A , A) = n=d n=1 H n (U A , A), so H • (U A , A) is a finitely-generated A-module.
Corollary 5.2. For all but finitely many ε ∈ C g , H • (U A , A) ⊗ A C ε ∼ = H • (U ε , C), and for all such ε ∈ C g , H • (U ε , C) is generated as an exterior algebra by homogeneous elements in the same odd degrees as for H • (U (g), C).
Proof. It follows from Proposition 5.1 and the fundamental theorem for finitely-generated modules over a principal ideal domain that H • (U A , A) is (q − ε)-torsion free for all but finitely many ε ∈ C g . Let S ′ ⊂ A be the multiplicatively closed set generated by Set W = H • (U B , B)/ im(ϕ). Since Λ(x 1 , . . . , x r ) and H • (U B , B) are each free of the same finite rank over the principal ideal domain B, W is a finitely-generated torsion B-module. Let ε ∈ C g such that (q − ε) / ∈ S ′ and W has no (q − e)-torsion. Then Tor B 1 (W, C ε ) = Tor B 1 (W, B/(q − ε)B) = 0 by Lemma 3.8, so it follows from the long exact sequence for Tor B 1 (−, C ε ) applied to the short exact sequence 0 → Λ(z 1 , . . . , z r ) ϕ → H • (U B , B) → W → 0 that the algebra map ϕ ⊗ B C ε : Λ(x 1 , . . . , x r ) ⊗ B C ε → H • (U B , B) ⊗ B C ε is injective. Then by dimension comparison ϕ ⊗ B C ε must also be surjective, hence an algebra isomorphism. Thus, the conclusion of the corollary holds for all ε ∈ C g such that H • (U A , A) and W are (q − ε)-torsion free, and fails for only finitely-many ε ∈ C g . While Corollary 5.2 states for almost all values ε ∈ C g that H • (U ε , C) is an exterior algebra over an r-dimensional graded subspace, it unfortunately does not give any indication of the values for which this condition fails. We can at least say that the only values for which H • (U ε , C) might not be an exterior algebra are those ε that are algebraic over Q. Indeed, let B = S −1 Q[q, q −1 ], with S as defined in Section 1.2. Then for each n ∈ N, the space H n (U B , B) is a finitely-generated B-module by Lemma 3.6, and H • (U B , B) ⊗ B A ∼ = H • (U A , A). This shows that H • (U A , A) has (q − ε)-torsion if and only if there exists an irreducible polynomial f ∈ Q[q] such that (q − ε) divides f in C[q] and H • (U B , B) has f -torsion. We summarize this discussion in the following proposition: