65 1.32. The Andruskiewitsch-Schneider conjecture. It is easy to see that any Hopf algebra generated by grouplike and skew-primitive elements is automatically pointed. On the other hand, there exist pointed Hopf algebras which are not generated by grouplike and skew-primitive elements. Perhaps the sim-plest example of such a Hopf algebra is the algebra of regular functions on the Heisenberg group (i.e. the group of upper triangular 3 by 3 matrices with ones on the diagonal). It is easy to see that the commu-tative Hopf algebra H is the polynomial algebra in generators x, y, z (entries of the matrix), so that x, y are primitive, and Δ(z) = z ⊗ 1 + 1 ⊗ z + x ⊗ y. Since the only grouplike element in H is 1, and the only skew-primitive elements are x, y, H is not generated by grouplike and skew-primitive elements. However, one has the following conjecture, due to Andruskiewitsch and Schneider. Conjecture 1.32.1. Any finite dimensional pointed Hopf algebra over a field of characteristic zero is generated in degree 1 of its coradical filtration, i.e., by grouplike and skew-primitive elements. It is easy to see that it is enough to prove this conjecture for corad-ically graded Hopf algebras; this has been done in many special cases (see [AS]). The reason we discuss this conjecture here is that it is essentially a categorical statement. Let us make the following definition. Definition 1.32.2. We say that a tensor category C is tensor-generated by a collection of objects Xα if every object of C is a subquotient of a finite direct sum of tensor products of Xα. Proposition 1.32.3. A pointed Hopf algebra H is generated by grou-plike and skew-primitive elements if and only if the tensor category H − comod is tensor-generated by objects of length 2. Proof. This follows from the fact that matrix elements of the tensor product of comodules V, W for H are products of matrix elements of V, W . � Thus, one may generalize Conjecture 1.32.1 to the following conjec-ture about tensor categories. Conjecture 1.32.4. Any finite pointed tensor category over a field of characteristic zero is tensor generated by objects of length 2.66 As we have seen, this property fails for infinite categories, e.g., for the category of rational representations of the Heisenberg group. In fact, this is very easy to see categorically: the center of the Heisenberg group acts trivially on 2-dimensional representations, but it is not true for a general rational representation. 1.33. The Cartier-Kostant theorem. Theorem 1.33.1. Any cocommutative Hopf algebra H over an alge-braically closed field of characteristic zero is of the form k[G] � U(g), where g is a Lie algebra, and G is a group acting on g. Proof. Let G be the group of grouplike elements of H. Since H is cocommutative, it is pointed, and Ext1(g, h) = 0 if g, h ∈ G, g �= h. Hence the category C = H−comod splits into a direct sum of blocks C = ⊕g∈GCg, where Cg is the category of objects of C which have a filtration with successive quotients isomorphic to g. So H = ⊕g∈GHg, where Cg = Hg −comod, and Hg = gH1. Moreover, A = H1 is a Hopf algebra, and we have an action of G on A by Hopf algebra automorphisms. Now let g = Prim(A) = Prim(H). This is a Lie algebra, and the group G acts on it (by conjugation) by Lie algebra automorphisms. So we need just to show that the natural homomorphism ψ : U(g) A is→actually an isomorphism. It is clear that any morphism of coalgebras preserves the coradical filtration, so we can pass to the associated graded morphism ψ0 : Sg →A0, where A0 = gr(A). It is enough to check that ψ0 is an isomorphism. The morphism ψ0 is an isomorphism in degrees 0 and 1, and by Corollary 1.29.7, it is injective. So we only need to show surjectivity. We prove the surjectivity in each degree n by induction. To simplify notation, let us identify Sg with its image under ψ0. Suppose that the surjectivity is known in all degrees below n. Let z be a homogeneous element in A0 of degree n. Then it is easy to see from the counit axiom that (1.33.1) Δ(z) − z ⊗ 1 − 1 ⊗ z = u where u ∈ Sg ⊗ Sg is a symmetric element (as Δ is cocommutative). Equation 1.33.1 implies that the element u satisfies the equation (1.33.2) (Δ ⊗ Id)(u) + u ⊗ 1 = (Id ⊗ Δ)(u) + 1 ⊗ u. Lemma 1.33.2. Let V be a vector space over a field k of characteristic zero. Let u ∈ SV ⊗ SV be a symmetric element satisfying equation (1.33.2). Then u = Δ(w) − w ⊗ 1 − 1 ⊗ w for some w ∈ SV .�� � � 67 Proof. Clearly, we may assume that V is finite dimensional. Regard u as a polynomial function on V ∗ × V ∗; our job is to show that u(x, y) = w(x + y) − w(x) − w(y) for some polynomial w. If we regard u as a polynomial, equation (1.33.2) takes the form of the 2-cocycle condition u(x + y, t) + u(x, y) = u(x, y + t) + u(y, t). Thus u defines a group law on U := V ∗ ⊕ k, given by (x, a) + (y, b) = (x + y, a + b + u(x, y)). Clearly, we may assume that u is homogeneous, of some degree d = 1. Since u is symmetric, the group U is abelian. So in U we have ((x, 0) + (x, 0)) + ((y, 0) + (y, 0)) = ((x, 0) + (y, 0)) + ((x, 0) + (y, 0)) Computing the second component of both sides, we get u(x, x) + u(y, y) + 2d u(x, y) = 2u(x, y) + u(x + y, x + y). So one can take w(x) = (2d − 2)−1u(x, x), as desired. � Now, applying Lemma 1.33.2, we get that there exists w ∈ A0 such that z − w is a primitive element, which implies that z − w ∈ A0, so z ∈ A0. � Remark 1.33.3. The Cartier-Kostant theorem implies that any co-commutative Hopf algebra over an algebraically closed field of char-acteristic zero in which the only grouplike element is 1 is of the form U(g), where g is a Lie algebra (a version of the Milnor-Moore theorem), in particular is generated by primitive elements. The latter statement is false in positive charactersitic. Namely, consider the commutative Hopf algebra Q[x, z] where x, z are primitive, and set y = z + xp/p, where p is a prime. Then p−11 � p � (1.33.3) Δ(y) = y ⊗ 1 + 1 ⊗ y + p ix i ⊗ xp−i . i=1 Since the numbers 1 p pi are integers, this formula (together with Δ(x) = x ⊗ 1 + 1 ⊗ x, S(x) = −x, S(y) = −y) defines a Hopf algebra structure on H = k[x, y] for any field k, in particular, one of characteristic p.