Search

In this note, we compute the Gelfand-Kirillov dimension of cosemisimple Hopf algebras that arise as deformations of a linearly reductive algebraic group. Our work lies in a purely algebraic setting and generalizes results of Goodearl-Zhang (2007), of Banica-Vergnioux (2009), and of D'Andrea-Pinzari-Rossi (2017)., Comment: 6 pages + references ; small changes following referee comments ; to appear in Proc. AMS

We present the strongest known knot invariant that can be computed effectively (in polynomial time)., Comment: Typos fixed, length reduced for publication in PAMS

We study when a finite dimensional Hopf action on a quantum formal deformation A of a commutative domain A_0 (i.e., a deformation quantization) must factor through a group algebra. In particular, we show that this occurs when the Poisson center of the fraction field of A_0 is trivial., Comment: v3. 9 pages. To appear in Proc. Amer. Math. Soc

Foulkes discovered a marvelous set of characters for the symmetric group by summing Specht modules of certain ribbon shapes according to height. These characters have many remarkable properties and have been the subject of many investigations, including a recent one [2] by Diaconis and Fulman, which established some new formulas, a conjecture of Isaacs, and a connection with Eulerian idempotents. We widen our consideration to complex reflection groups and find ourselves equipped from the start with a simple formula for (generalized) Foulkes characters which explains and extends these properties. In particular, it gives a factorization of the Foulkes character table which explains Diaconis and Fulman’s formula for the determinant, their link to Eulerian idempotents, and their formula for the inverse. We present a natural extension of a conjecture of Isaacs, and then use properties of Foulkes characters which resemble those of supercharacters to establish the result. We also discover a remarkable refinement of Diaconis and Fulman’s determinantal formula by considering Smith normal forms. Classic type A Foulkes characters have connections with adding random numbers, shuffling cards, the Veronese embedding, and combinatorial Hopf algebras [2, 7]. Our formula brings Orlik–Solomon coexponents from [12] the cohomology theory of [10] complements into the picture with the geometry of the Milnor fiber complex [8], and it gives rise to a curious classification at the end of the paper. The paper is structured as follows. Section 1 introduces Foulkes characters for Shephard and Coxeter groups. Key properties are quickly gathered, including our main formula. In Section 2, properties of type A Foulkes characters are explained and extended from the symmetric group to the infinite family of wreath products. In Section 3, Isaacs’ type A conjecture is sharpened for the Coxeter–Shephard–Koster family. Diaconis and Fulman’s type A determinantal formula is also extended here. Lastly, we determine exactly when the Foulkes characters are a basis for the space of class functions .g/ that depend only on the dimension of the fixed space of g.

We investigate when a weak Hopf algebra H H is Frobenius. We show this is not always true, but it is true if the semisimple base algebra A A has all its matrix blocks of the same dimension. However, if A A is a semisimple algebra not having this property, there is a weak Hopf algebra H H with base A A which is not Frobenius (and consequently, it is not Frobenius “over” A A either). Moreover, we give a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of a weak Hopf algebra.

Assume that H is a finite-dimensional Hopf algebra over a field k and that A is an H-module algebra satisfying a polynomial identity (PI). We prove that if H is semisimple and A is H-semiprime, then A#H is semiprime. If H is cosemisimple, we show that the prime radical of A is H-stable.

Let R be an H-module algebra, where H is a pointed Hopf algebra acting on R finitely of dimension N. Suppose that L H ≠ 0 for every nonzero H-stable left ideal of R. It is proved that if R H satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN provided at least one of the following additional conditions is fulfilled: (1) R is semiprime and R H is almost central in R, (2) R is reduced. If we also assume that R H is central, then R satisfies the standard polynomial identity of degree 2[√N], where [√N] is the greatest integer in √N.

We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of π ∈ S n \pi \in \mathbb {S}_n is infinite-dimensional, if either the order of π \pi is odd, or all cycles in the decomposition of π \pi as a product of disjoint cycles have odd order except for exactly two transpositions.

Let k be an algebraically closed field. The main goal of this paper is to classify the finite-dimensional pointed Hopf algebras over k of finite corepresentation type. To do so, we give a necessary and sufficient condition for a basic Hopf algebra over k to be of finite representation type firstly. Explicitly, we prove that a basic Hopf algebra over k is of finite representation type if and only if it is Nakayama. By this conclusion, we classify all finite-dimensional pointed Hopf algebras over k of finite corepresentation type.

We show that the formal A-module Adams-Novikov spectral sequence of Ravenel does not naturally arise from a filtration on a map of spectra by examining the case A = Z[i]. We also prove that when A is the ring of integers in a nontrivial extension of Qp, the map (L, W) → (L A , W A ) of Hopf algebroids, classifying formal groups and formal A-modules respectively, does not arise from compatible maps of E ∞ -ring spectra (MU, MU^MU)→ (R, S).

The quantum permutation group of the set $X_n=\{1,..., n\}$ corresponds to the Hopf algebra $A_{aut}(X_n)$. This is an algebra constructed with generators and relations, known to be isomorphic to $\cc (S_n)$ for $n\leq 3$, and to be infinite dimensional for $n\geq 4$. In this paper we find an explicit representation of the algebra $A_{aut}(X_n)$, related to Clifford algebras. For $n=4$ the representation is faithful in the discrete quantum group sense., 9 pages

Let g be a finite-dimensional complex Lie algebra, and let U(g) be its universal enveloping algebra. We prove that if \hat{U}(g), the Arens-Michael envelope of U(g), is stably flat over U(g) (i.e., if the canonical homomorphism U(g)-->\hat{U}(g) is a localization in the sense of Taylor), then g is solvable. To this end, given a cocommutative Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H as an ``analytic smash product'' of their completions w.r.t. certain families of seminorms., 11 pages

We prove that Ext^*_A(k,k) is a Gerstenhaber algebra, where A is a Hopf algebra. In case A=D(H) is the Drinfeld double of a finite dimensional Hopf algebra H, our results implies the existence of a Gerstenhaber bracket on H^*_{GS}(H,H). This fact was conjectured by R. Taillefer in math.KT0207154. The method consists in identifying Ext^*_A(k,k) as a Gerstenhaber subalgebra of H^*(A,A) (the Hochschild cohomology of A)., Comment: 5 pages

The harmonic Hopf construction is an equivariant ansatz for harmonic maps between Euclidean spheres. We prove existence of solutions in the case that has been open. Moreover, we show that the harmonic Hopf construction on every bi-eigenmap with at least one large eigenvalue has a countable family of solutions (if it has one).

It is shown that the mod 3 cohomology of a homotopy associative mod 3 H-space which is rationally equivalent to the Lie group E 7 and which has integral 3-torsion is isomorphic to that of E7 as a Hopf algebra over the mod 3 Steenrod algebra.

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample., Comment: AMS-TeX; 8 Pages; no figures

Since an H-separable extension A|B is of depth two, we associate to it dual bialgebroids S:= End B A B and T:= (A ⊗B A) B over the centralizer R as in Kadison-Szlachanyi. We show that S has an antipode τ and is a Hopf algebroid. T op is also Hopf algebroid under the condition that the centralizer R is an Azumaya algebra over the center Z of A. For depth two extension A|B, we show that End A A ⊗ B A ≅ T α End B A.

Let H H be a Hopf algebra over a commutative ring R R such that H H is a finitely generated, projective module over R R , let A A be a right H H -comodule algebra, and let B B be the subalgebra of H H -coinvariant elements of A A . If A A is a Galois extension of B B and B B is a local subalgebra of the center of A A , then A A is a cleft right H H -comodule algebra or, equivalently, there is a normal basis for A A over B B .

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory., 7 pages, 5 eps files

If (Q, n) -* (R, m) is a surjective local homomorphism with kernel I, such that I C n2 and the conormal module I/12 has a free summand of rank n, then the degree 2 central subspace of the homotopy Lie algebra of R has dimension greater than or equal to n. This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module. Let R be a noetherian local ring with maximal ideal m and residue field k. Ring theoretic properties of R are reflected on the algebra structures carried by TorR (k, k) and EXtR (k, k) . Recall that the former has the rh-product of Cartan and Eilenberg, and the latter the Yoneda multiplication. The k-algebra structure of TorR (k, k) is rather "simple": It is free in the appropriate category, and so determined by the dimension of the k-vector space TornR (k, k), for all integers i > 0, that is to say, the Betti numbers of k over R. The multiplicative structure of EXtR (k, k) is quite another matter. The simplicity of the product on TorR (k, k) arises from the fact that it is endowed with additional structures: It is a commutative algebra, in the graded sense, with a family of divided powers, and has a diagonal map that is compatible with the divided powers algebra structure on it. In other words, the k-algebra TorR (k, k) is a commutative Hopf algebra with divided powers, and so is free as a divided powers algebra. In contrast, the multiplication on EXtR(k, k) is decidedly non-commutative, unless R happens to be a complete intersection of a special kind. The graded kdual of the product on TorR (k, k) turns EXtR (k, k) = TorR (k, k) * into a Hopf algebra. Attention has focussed on a certain subspace of primitives of this Hopf algebra, denoted ir(R), which is a Lie algebra with the bracket operation defined by the commutator in EXtR (k, k) . This object is called the homotopy Lie algebra of R. The importance of this Lie algebra is attested to by its defining property: Its universal enveloping algebra is EXtR (k, k) . In this note, we are concerned with the centre of the homotopy Lie algebra of R, denoted ((R). This subspace, besides being a measure of the non-commutativity of ir(R), and hence of ExtR (k, k), plays an important part in the change of rings Received by the editors April 7, 1999 and, in revised form, May 12, 1999. 1991 Mathematics Subject Classification. Primary 13C15, 13D03, 13D07, 18G15.

A direct proof without modular category theory is given of a recent theorem of Etingof and Gelaki (1998) on the dimensions of irreducible representations. Factorizable Hopf algebras are characterized in terms of their Drinfeld double, and their character rings and the group-like elements of their duals are described.

We calculate the Brauer group of the four dimensional Hopf algebra H 4 H_4 introduced by M. E. Sweedler. This Brauer group B M ( k , H 4 , R 0 ) {\mathrm {BM}}(k,H_4,R_0) is defined with respect to a (quasi-) triangular structure on H 4 H_4 , given by an element R 0 ∈ H 4 ⊗ H 4 R_0\in H_4\otimes H_4 . In this paper k k is a field . The additive group ( k , + ) (k,+) of k k is embedded in the Brauer group and it fits in the exact and split sequence of groups: 1 ⟶ ( k , + ) ⟶ B M ( k , H 4 , R 0 ) ⟶ B W ( k ) ⟶ 1 \begin{equation*} 1\longrightarrow (k,+)\longrightarrow {\mathrm {BM}}(k,H_4,R_0)\longrightarrow {\mathrm {BW}}(k)\longrightarrow 1 \end{equation*} where B W ( k ) {\mathrm {BW}(k)} is the well-known Brauer-Wall group of k k . The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.

In this note we describe nonsemisimple Hopf algebras of dimension pn with coradical isomorphic to kC, C abelian of order pn-l, over an algebraically closed field k of characteristic zero. If C is cyclic or C = (Cp)n-1, then we also determine the number of isomorphism classes of such Hopf algebras. 0. INTRODUCTION AND PRELIMINARIES In recent years considerable effort has been made to classify finite dimensional Hopf algebras over an algebraically closed field k of characteristic 0. In [17], Zhu proved that a Hopf algebra of prime dimension p is isomorphic to kCp. For the semisimple case, a series of results has appeared. Masuoka has classified semisimple Hopf algebras of dimensions 6,8, p27 p3 and 2p for p an odd prime (see [8], [9], [10], [11]). Larson and Radford [7] showed that a semisimple Hopf algebra of dimension 3, which have coradical kO, C abelian of order pn-7 and note that the situation is somewhat different from the dimension p2 case. The Hopf algebras which occur can be obtained by an Ore extension construction as in [2] or [3]; if C is cyclic, there are p[ 2] +PI'I + p 3 nonisomorphic such Hopf algebras; if C = (Cp)n1, then there are p-I isomorphism classes. Throughout, k is an algebraically closed field of characteristic 0. We follow the standard notation in [12]. For H a Hopf algebra, G(H) will denote the group of grouplike elements and Ho, H1, H2, ... will denote the coradical filtration of H. H is called pointed if Ho = kG(H). If g, h E G(H) then the set of (g, h)-primitive elements is Pg,h = {x E H1A(x) = x X g + h X x}. Since g-h E Pg,h we can choose a subspace P of Ph such that P =,h k(g h) @ P',h. We will need the Received by the editors October 7, 1997 and, in revised form, April 3, 1998. 1991 Mathematics Subject Classification. Primary 16W30. The first and third authors research was partially supported by NSERC. ?)1999 American Mathematical Society

We consider the problem of the classification of semisimple Hopf algebras A of dimension pq where p < q are two prime numbers. First we prove that the order of the group of grouplike elements of A is not q, and that if it is p, then q = 1 (mod p). We use it to prove that if A and its dual Hopf algebra A* are of Frobenius type, then A is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension 3p, and a partial classification in dimensions 5p and 7p. In this paper we consider semisimple Hopf algebras of dimension pq over an algebraically closed field k of characteristic 0, where p and q are distinct prime numbers. Masuoka has proved that a semisimple Hopf algebra of dimension 2p over k, where p is an odd prime, is trivial (i.e. is either a group algebra or a dual of a group algebra) [Mal]. Izumi and Kasaki have proved that Kac algebras (i.e. semisimple Hopf algebras over the field of complex numbers, with an additional condition on the existence of an involution), of dimension 3p over k, where p is prime, are trivial [IK]. Thus, a natural conjecture is: Conjecture 1. Any semisimple Hopf algebra of dimension pq over k, where p and q are distinct prime numbers, is trivial. A well known property of A, a finite dimensional semisimple group algebra or a dual of a group algebra, is that it is of Frobenius type; that is, the dimension of any irreducible representation of A divides the dimension of A (the definition is due to Montgomery [Mo]). A special case of Kaplansky's 6th conjecture [K] is: Conjecture 2. Any semisimple Hopf algebra of dimension pq over k, where p and q are distinct prime numbers, is of Frobenius type. In this paper we prove among the rest that Conjecture 1 is equivalent to Conjecture 2 (see Theorem 3.5). A major role in the analysis is played by G(A) (where G(A) denotes the group of grouplike elements of A). By [NZ], IG(A)I is either l,p, q or pq. We prove in Theorem 2.1 that if p < q, then IG(A)| /: q, and if [G(A)| = p, then q = 1 (modp). Consequently, we prove in Theorem 2.2 that if IG(A)| 71 and q 1 (modp), then A is a commutative group algebra. Received by the editors August 1, 1997 and, in revised form, March 17, 1998. 1991 Mathematics Subject Classification. Primary 16W30. The second author's research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. (1999 American Mathematical Society

We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebraically closed field of characteristic 0.

By a finite quantum group, we will mean in this paper a finitedimensional Hopf algebra. A left Haar measure on such a quantum group is a linear functional satisfying a certain invariance property. In the theory of Hopf algebras, this is usually called an integral. It is well-known that, for a finite quantum group, there always exists a unique left Haar measure. This result can be found in standard works on Hopf algebras. In this paper we give a direct proof of the existence and uniqueness of the left Haar measure on a finite quantum group. We introduce the notion of a faithful functional and we show that the Haar measure is faithful. We consider the special case where the underlying algebra is a *-algebra with a faithful positive linear functional. Then the left and right Haar measures coincide. Finally, we treat an example of a root of unity algebra. It is an example of a finite quantum group where the left and right Haar measures are different. This note does not contain many new results but the treatment of the finite-dimensional case is very concise and instructive.

We prove that if H is a semisimple Hopf algebra, then the action of the Drinfeld double D(H) on H and the action of the character algebra on H form a commuting pair. This result and a result of G. I. Kats imply that the dimension of every simple D(H)-submodule of H is a divisor of dim (H). Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, D(H) be the Drinfeld double of H, and C(H) be the character algebra of H. C(H) is spanned by the characters of H-modules and is an associative subalgebra of H*. It is known that D(H) acts on H and that C(H) acts on H by the restriction of the action "-" of H* on H (these actions will be recalled below). The purpose of this note is to prove that these two actions form a commuting pair. Using this result, we prove that the dimension of every simple D(H)-submodule of H is a divisor of dim(H). It would be interesting if there exists an analog of this commuting pair in the context of Poisson Lie groups. We first recall the construction of the Drinfeld double (cf. [D], [M]) and fix necessary notations. Let H be a finite dimensional Hopf algebra over a field k (here we do not need any additional assumptions on H and k). The Drinfeld double of H, denoted by D(H), as a vector space, is the tensor space H* 0 H. The comultiplication of D(A) is given by A(f 0 a)= S (f(2) 0 a(,)) 0 (.f(i) 0 a(2)) E D(H) 0 D(H), where Af = f(l) (0 f(2), Aa =a(l) ( a(2) are comultiplictions in H and H* respectively. The multiplication in D(H) is defined as follows: for f 0 a and g 0 b in D(H), (1) (f oa)(gob) f (a(,) D 9(2)) 0 (a(2) g is the action of H on H* given by a > g = a(l)' 9 -Sla(2) and a

We study coactions of Hopf algebras coming from compact quantum groups on the Cuntz algebra. These coactions are the natural generalization to the coalgebra setting of the canonical representation of the unitary matrix group U ( d ) U(d) as automorphisms of the Cuntz algebra O d O_d . In particular we study the fixed point subalgebra under the coaction of the quantum compact groups U q ( d ) U_q(d) on the Cuntz algebra O d O_d by extending to any dimension d > ∞ d>\infty a result of Konishi (1992). Furthermore we give a description of the fixed point subalgebra under the coaction of S U q ( d ) SU_q(d) on O d O_d in terms of generators.

We describe the monoidal structure of the category of Hopf bimodules of a finite group and we derive a surjective ring map from the Grothendieck ring of the category of Hopf bimodules to the center of the integral group ring. We consider analogous results for the multiplicative structure of the Hochschild cohomology.

In this paper we prove some properties of the set of group-like elements of A, G(A), for a pointed minimal quasitriangular Hopf algebra A over a field k of characteristic 0, and for a pointed quasitriangular Hopf algebra which is indecomposable as a coalgebra. We first show that over a field of characteristic 0, for any pointed minimal quasitriangular Hopf algebra A, G(A) is abelian. We show further that if A is a quasitriangular Hopf algebra which is indecomposable as a coalgebra, then G(A) is contained in AR, the minimal quasitriangular Hopf algebra contained in A. As a result, one gets that over a field of characteristic 0, a pointed indecomposable quasitriangular Hopf algebra has a finite abelian group of group-like elements.

We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C. This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element.

We construct a derivation D in the Hopf algebra TcV, the cofree coassociative coalgebra on a vector space V. We then define the subspace of TcV consisting of dual Lie elements, which is analogous to the subspace of the Hopf algebra TV, the free associative algebra on V, consisting of Lie elements. Thereafter, we formulate a dual Dynkin-Specht-Wever theorem. Using our map D, we then give very short proofs of both the dual Dynkin-Specht-Wever and dual Friedrichs’ theorems, each of which characterizes the space of dual Lie elements in TcV at characteristic 0.

Let B = A # σ H B = A{\# _\sigma }H denote a crossed product of the associative algebra A with the Hopf algebra H. We investigate the weak dimension and the global dimension of B and show that wdim B ≤ wdim H + wdim A {\text {wdim}}\;B \leq {\text {wdim}}\;H + {\text {wdim}}\;A and l.gldim B ≤ r.gldim H + l.gldim A {\text {l.gldim}} \; B \leq {\text {r.gldim}} \; H + {\text {l.gldim}} \; A .

Let (H, R) be a triangular Hopf algebra and let V be a finite-dimensional representation of H. Following Manin we imitate the standard algebraic constructions in order to define the relativized notions of R-universal enveloping algebras of R-Lie algebras and the R-Lie algebra gl R ( V ) {\text {gl}_R}(V) . Using Majid’s "bosonization" theorem and the above we prove an R-analogue of Schur’s double centralizer theorem.

In this paper, we study the representations of Hopf C ∗ {C^ \ast } -algebras; the main result is that every irreducible left unitary representation of a Hopf C ∗ {C^\ast } -algebra with a Haar measure is finite dimensional. To prove this result, we first study the comodule structure of the space of Hilbert-Schmidt operators; then we use this comodule structure to show that every irreducible left unitary representation of a Hopf C ∗ {C^\ast } -algebra with a Haar measure is finite dimensional.

We provide a short proof of an analog of Nagata’s theorem for finite-dimensional Hopf algebras. The result, proved Hopf-algebraically by Sweedler and using group schemes by Demazure and Gabriel, says that a finite-dimensional cocommutative semisimple irreducible Hopf algebra is commutative. With mild base field assumptions such a Hopf algebra is just the dual of a p p -group algebra. We give en route an easy proof of a version of Hochschild’s theorem on semisimple restricted enveloping algebras. Let R # t H R{\# _t}H denote a crossed product with an invertible cocycle t t , where H H is a semisimple cocommutative Hopf algebra H H over a perfect field. The result above is applied to show that R # t H R{\# _t}H is semiprime if and only if R R is H H -semiprime. The approach relies on results on ideals of the crossed product that are stable under the action of the dual of H H and the Fisher-Montgomery theorem for crossed products of finite groups.

If H is a co-Frobenius Hopf algebra over a field, having a Galois H-object A which is separable over AcoH , its ring of coinvariants, then H is finite dimensional. Let H be a Hopf algebra over a field k and (A, ρ) a right H-comodule algebra such that A is H-Galois over its ring of coinvariants A . If H is a group algebra kG so that then A is strongly graded, it is well known that if A is separable over Ae, its ring of coinvariants, then the group G is finite. We show in this note that this result holds for all co-Frobenius Hopf algebras H . Recall that H is co-Frobenius if H∗rat, the rational submodule of H∗ as a (left or right) H∗-module, is nonzero. If H∗rat is nonzero, it is dense in H∗. A right Hcomodule algebraA is rightH-Galois if the canonical map can: A⊗AcoHA→ A⊗H , a⊗ b 7−→ ∑ ab0 ⊗ b1, is a bijection. Any right H-comodule M is a left H∗-module via p · m = ∑ m0〈p,m1〉 for p ∈ H∗, m ∈ M . Also H∗ and H∗rat are right Hmodules via (p ↼ h)(f) = 〈p, hf〉 for p ∈ H∗ or H∗rat and h, f ∈ H . Details about H-Galois objects with H co-Frobenius can be found in [1]. Lemma 1. Suppose H is a Hopf algebra such that H = ⊕ λ∈I Hλ where Hλ is a left H-subcomodule. Then if A is a right H-Galois object, A = ⊕ λ∈I Aλ where Aλ = {a|a ∈ A, ρ(a) ∈ A⊗Hλ} and Aλ 6= 0 for all λ ∈ I. Proof. Let pλ ∈ H∗ be the projection defined by pλ(h) = 0 if h ∈ Hμ, μ 6= λ, and pλ(h) = (h) for h ∈ Hλ. Then pλ ·H = { ∑ h1pλ(h2)|h ∈ H} = Hλ since if h ∈ Hμ, μ 6= λ, pλ · h = 0, but if h ∈ Hλ, pλ · h = ∑ h1 (h2) = h. Now, pλ · A = { ∑ a0pλ(a1)|a ∈ A} and so for a ∈ A, ρ(pλ · a) = ∑ a0 ⊗ a1pλ(a2) = ∑ a0 ⊗ pλ · a1 ∈ A⊗Hλ so that pλ ·A ⊆ Aλ. Clearly if a ∈ Aλ, pλ ·a = a so Aλ = pλ ·A, and A = ⊕ λ∈I Aλ. Since A is Galois, the canonical map can from A ⊗AcoH A to A ⊗H , a⊗ b 7−→ ∑ ab0 ⊗ b1 is a bijection and thus Aλ 6= 0 for all λ ∈ I. Received by the editors August 12, 1998 and, in revised form, January 15, 1999. 1991 Mathematics Subject Classification. Primary 16W30. The first author’s research was partially supported by NSERC. The last two authors thank Mount Allison University for their kind hospitality. c ©2000 American Mathematical Society

Let H H be a commutative, cocommutative, and faithfully projective Hopf algebra over a commutative ring R R . A twisted version of inner action of a Hopf algebra, called H H -inner action is introduced, and it is shown that H H acts H H -innerly on an H H -Azumaya algebra, if Pic ⁡ ( H ∗ ) \operatorname {Pic} ({H^ * }) is trivial.

In this paper we compute the additive structure of the Morava Ktheories of SO(2n + 1) . We also obtain partial information about the bialgebra structures, and indicate how one may compute the effect of the stable BPoperations and the Milnor primitives. 1. STATEMENT OF RESULTS In this paper we bring the program started in [Ra 1] to a partial close by determining the additive structure of the Morava K-theories of SO(2n + 1). The theorem we prove contains incomplete information concerning the bialgebra structure and the effect of the stable BP-operations and the Milnor primitives. This paper is a sequel to [Ra2], to which the reader is referred for a proper introduction. Throughout this paper BP will refer to the 2-local theory. For background information on BP and related topics, see [Wi]. It is well known that BP. = Z(2)[VI I V2 I .. .] where the degree of v1 is 2(2'1) (see, for example, [Qu]). Let I > 0. Using the Sullivan-Baas technique ([Su], [Bs]), we can kill 2, vl,..., v1I. The resulting spectrum is P(l), a BP-module spectrum with P(l)* = Z/2[v , v1+I, ... ] . (This and the next few statements are due to Jack Morava. See [JW] for a source in print.) It is consistent to define P(0) = BP and P(oo) = HZ/2. Killing {vi I i $ /} gives k(l), the connective Morava K-theory. Inverting v, gives B(l) = v71 P(l) and the (periodic) Morava K-theory K(/) v lk(l). If 0 < I < oo, there are two products on P(/) which make it into a BPalgebra theory [Wu]. Select one and give B(l), k(l) and K(l) the compatible products. For any space X, B(l)*(X) is free as a B(l)*-module and K(l)*(X) K(l) *B(l)*(X) [JW]. Note that if X is an H-space, then B(l)*(X) is a bialgebra. But it need not be a Hopf algebra, as B(/) is not commutative. Our calculations are based on the bar spectral sequence [RS]. In fact, we will determine the E??-term of the bar spectral sequence (Bss) converging to Received by the editors October 19, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57T10; Secondary 55N22.

We give an algebraic version of a result of G. I. Kac, showing that a semisimple Hopf algebra A of dimension pn, where p is a prime and n > 0, over an algebraically closed field of characteristic 0 contains a non-trivial central group-like. As an application we prove that, if n = 2, A is isomorphic to a group algebra.

Let H be a finite-dimensional Hopf algebra. We prove that if M is a faithful H-module and if H 1 ≠ H 2 {H_1} \ne {H_2} are sub-Hopf algebras of H, then M H 1 ≠ M H 2 {M^{{H_1}}} \ne {M^{{H_2}}} , where M H 1 {M^{{H_1}}} and M H 2 {M^{{H_2}}} are the invariants in M under the respective actions of H 1 {H_1} and H 2 {H_2} . We also show that if H 1 ≠ H 2 {H_1} \ne {H_2} , then H 1 {H_1} and H 2 {H_2} have different left integrals. Both of these results rely heavily on the freeness theorem of Nichols-Zoeller.