Normal Hopf Subalgebras Research Articles

Depth Two, Normality, and a Trace Ideal Condition for Frobenius Extensions

A ring extension A ‖ B is depth two if its tensor-square satisfies a projectivity condition w.r.t. the bimodules A A B and B A A . In this case the structures (A ⊗ B A) B and End B A B are bialgebroids over the centralizer C A (B) and there is a certain Galois theory associated to the extension and its endomorphism ring. We specialize the notion of depth two to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse, we observe that normal Hopf subalgebras over a field are depth two extensions. A generalized Miyashita–Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable extensions, which yields new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius.

Krull relations in Hopf Galois extensions: Lifting and twisting

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂ A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t -Lying Over (t -LO), and Going Up (GU), analogous to the classical Krull relations for prime ideals; we also defined three new “dual” Krull relations. We say that H itself is said to have one of the Krull relations if the relation holds for all faithfully flat H -Galois extensions. We showed in [MS] that H has one of the three “dual” Krull relations if and only if the dual Hopf algebra H ∗ of H has the original relation (hence the name). An important example of Hopf Galois extensions is given by Hopf crossed products A = Rσ # H . Moreover, Galois extensions can be useful in studying crossed products, since they satisfy a “transitivity” property which crossed products lack. That is, if K is a normal Hopf subalgebra of H with Hopf quotient H , then, in general, one cannot write A = Rσ # H = (Rσ K)#τ H , by an example of [S2]. Another basic example of a Hopf Galois extension is given by a Hopf algebra A with a normal Hopf subalgebra R of finite index such that A is faithfully flat over R: for then R ⊂ A is faithfully flat H -Galois.

Galois theory for bialgebroids, depth two and normal Hopf subalgebras

We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a leftT-Galois extension for some right finite projective left bialgebroid over some algebraR if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( $$\mathfrak{g}$$ ) containing u( $$\mathfrak{g}$$ ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G r ) u( $$\mathfrak{g}$$ ) have also been constructed, and the results are stated in this setting.

On some examples of simple quantum groups

The aim of this note is to construct noncocommutative finite dimensional simple Hopf algebras by twisting the group algebras of finite simple groups. In [N1] Nikshych described those constructions and proved that if the twist is possible then the resulting Hopf algebra will remain simple and the Grothendieck rings of the two Hopf algebras will be the isomorphic as ordered rings with involution. We show that the twisting is always possible and describe all the cases in which his construction can be done. Recently in [N2] these methods have been used to describe all simple Hopf algebras of dimension at most 60. To fix the notation let k be an algebraically closed field of characteristic zero. We will only be interested in finite dimensional Hopf algebras over k. The easiest example is provided by the group algebra kG for G a finite group and with comultiplication ∆(g) = g⊗g, counit e(g) = 1 and antipode S(g) = g−1. We will call such an example a trivial Hopf algebra. A Hopf algebra H is simple if it has no Hopf quotients. In particular in the case of a group algebra kG the normal Hopf subalgebras are in 11 correspondence with the normal subgroups of G and in particular kG is simple if and only if G is simple. Also A is cocommutative (respectively commutative) iff A is a trivial (respectively the dual of a trivial algebra). We will prove the following :

Galois Objects over Generalized Drinfeld Doubles, with an Application to uq([formula omitted][formula omitted]2)

Ralf Günther has determined all the cleft extensions over the finite quotient Hopf algebra uq(sl2) of the quantized universal enveloping algebra of sl2 at a root of unity [R. Günther, Ph.D. thesis, Universität München, 1999]. His techniques (applications of the diamond lemma) are similar to those used by A. Masuoka [Comm. Algebra22 (1994), 4537–4559] for the two-generator Taft algebras. In the present paper we give another proof of a special case of Günther's classification, namely, the case of (cleft) Galois extensions of the base field. The idea is that uq(sl2) is the quotient of the Drinfeld double of a Taft algebra by a normal Hopf subalgebra. We use techniques that allow us to calculate all Galois objects of such a composed Hopf algebra.

K 0-rings and twisting of finite dimensional semisimple hopf algebras

We show that the K 0-group of a finite dimensional semisimple Hopf algebra has a natural structure of a ring with involution and prove that this ring is a twisting invariant of the Hopf algebra.We also study relations between algebraic structure of a Hopf algebra and the one of its K 0ring and prove that the twisting of a finite simple group is a simple Hopf algebra (i.e., it does not have proper normal Hopf subalgebras).In this case the dual Hopf algebra does not have proper Hopf subalgebras.Further, we construct several series of non trivial Hopf algebras by the twisting of the classical series of finite groups and give examples of Hopf algebras which can not be described as twisting of any group.

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.