Hans-Jürgen Schneider

Principal homogeneous spaces for arbitrary Hopf algebras

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗BA →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

Representation theory of Hopf galois extensions

LetH be a Hopf algebra over the fieldk andB ⊂A a right faithfully flat rightH-Galois extension. The aim of this paper is to study some questions of representation theory connected with the ring extensionB ⊂A, such as induction and restriction of simple or indecomposable modules. In particular, generalizations are given of classical results of Clifford, Green and Blattner on representations of groups and Lie algebras. The stabilizer of a leftB-module is introduced as a subcoalgebra ofH. Very often the stabilizer is a Hopf subalgebra. The special case whenA is a finite dimensional cocommutative Hopf algebra over an algebraically closed field,B is a normal Hopf subalgebra andH is the quotient Hopf algebra was studied before by Voigt using the language of finite group schemes.

Frobenius extensions of subalgebras of Hopf algebras

We consider when extensions S ⊂ R of subalgebras of a Hopf algebra are β-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra H, we show that when S ⊂ R are Hopf algebras in the Yetter-Drinfeld category for H, the extension is β-Frobenius provided R is finite over S and the extension of biproducts S ? H ⊂ R ?H is cleft. More generally we give conditions for an extension to be β-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants. We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras. 0. Introduction In this paper we consider when various extensions S ⊂ R of subalgebras of a Hopf algebra are β-Frobenius; such extensions generalize the usual notion of Frobenius extensions by having the module action on one side twisted by an automorphism β of S. It was already known that any extension of finite-dimensional Hopf algebras is β-Frobenius (a result of the third author [Sch 92]) as is any finite extension U(K) ⊂ U(L) of enveloping algebras of Lie superalgebras (a result of Bell and Farnsteiner [BF]). Note that U(L) is not an ordinary Hopf algebra, but rather a Hopf algebra in the category of Z2-graded modules. These results were an important motivation for this paper, and raised the question as to when an extension of Hopf algebras in a category was β-Frobenius. One of the main results of this paper is that an extension S ⊂ R of Hopf algebras of finite index in the Yetter-Drinfeld category HYD for a given Hopf algebra H is β-Frobenius provided that the associated extension of Hopf algebras S ?H ⊂ R?H (the biproducts of S and R with H) has a normal basis (Theorem 5.6); this will happen whenever R and H are finite-dimensional, or when R ? H is pointed. As an application we are able to generalize the [BF] result to Lie coloralgebras: if U(K) ⊂ U(L) is a finite extension of enveloping algebras of Lie coloralgebras, then it is β-Frobenius. Moreover we give an explicit description of the automorphism β of U(K), and of the Frobenius homomorphism f : U(L)→ U(K) (Corollary 6.3). Along the way we prove a number of other results about β-Frobenius extensions and conditions that ensure that an extension is β-Frobenius. In Section 1 we give a short direct proof of the fact that an extension B ⊂ A of finite-dimensional Hopf algebras is always β-Frobenius; moreover we give explicit Received by the editors December 10, 1995. 1991 Mathematics Subject Classification. Primary 16W30; Secondary 17B35, 17B37. c ©1997 American Mathematical Society

Different origins of metal binding sites in binuclear copper proteins, tyrosinase and hemocyanin

The primary structures of five binuclear copper proteins (two tyrosinases, two arthropodan, and one molluscan hemocyanin) are compared. All proteins show a highly homologous region of 56 amino acids in the C-terminal parts containing three invariant histidines previously identified as ligands to Cu B in P. interruptus hemocyanin by x-ray crystallography (Gaykema et al., Nature 309, 23–29 (1984). In contrast, a very different ligand environment is observed for Cu A. It is concluded that the Cu B site in tyrosinases and hemocyanins originated during the earliest period of life, whereas Cu A must have evolved independently subsequent to the invention of Cu B. Furthermore, the two copper sites appear to be unique for these proteins because they are absent in ceruloplasmin, a multicopper protein containing one binuclear and two different mononuclear sites.

Quantum homogeneous spaces with faithfully flat module structures

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB.In the last few years many interesting examples of quantumG-spaces for concrete quantum groupsG have been constructured by Podleś, Noumi, Dijkhuizen and others (as analogs of classical compact symmetric spaces). In these examplesB consists of infinitesimal invariants of the function algebraA of the quantum group. As a consequence of a general theorem we show that in all these casesA as a left or rightB-module is faithfully flat. Moreover, the coalgebraA/AB+ is cosemisimple.

Über Untergruppen Endlicher Algebraischer Gruppen

Let k be a commutative ring, G′⊃G finite affine algebraic k-groups, and H′⊃H the dual Hopfalgebras of the affine algebras of G′ resp. G. The main results of this paper are: (I) If k is semilocal (e.g. k a field) there is an H′-linear, H∥H′-colinear, unitary, augmented isomorphism H→H∥H′◯ H′, where H∥H′ is the coalgebra belonging to G/G′. (II) If the k-submodule of the fixelements of (H∥H′)* is isomorphic to k (e.g. k principal or semilocal), then H′⊃H is a Frobeniusextension of the second kind.

Finite quantum groups over abelian groups of prime exponent

Abstract We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius–Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.

A characterization of quantum groups

Abstract We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension, which are domains, whose groups of group-like elements are finitely generated and abelian, and whose infinitesimal braidings are positive.

Some properties of factorizable Hopf algebras

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.

Prime ideals in Hopf galois extensions

For a finite-dimensional Hopf algebraH, we study the prime ideals in a faithfully flatH-Hopf-Galois extensionR ⊂A. One application is to quotients of Hopf algebras which arise in the theory of quantum groups at a root of 1. For the Krull relations betweenR andA, we obtain our best results whenH is semisolvable; these results generalize earlier known results for crossed products for a group action and for algebras graded by a finite group. We also show that ifH is semisimple and semisolvable, thenA is semiprime providedR isH-semiprime.