Lars Kadison

New Examples of Frobenius Extensions

Introduction to Frobenius extensions The endomorphism ring theorem The Jones polynomial Frobenius algebras Azumaya algebras Hopf algebras over commutative rings Hopf subalgebras Historical notes Bibliography Index.

Bialgebroid actions on depth two extensions and duality

Abstract A general notion of depth two for ring homomorphism N → M is introduced. The step two centralizers A =End N M N and B=(M⊗ N M) N in the Jones tower above N → M are shown in a natural way via H-equivalence to be dual bimodules for Morita equivalent endomorphism rings, the step one and three centralizers, R = C M ( N ) and C =End N – M ( M ⊗ N M ). We show A and B to possess dual left and right R -bialgebroid structures which generalize Lus fundamental bialgebroids over an algebra. There are actions of A and B on M and E ′= End N M with Galois properties. If M | N is depth two and Frobenius with R a separable algebra, we show that A and B are dual weak Hopf algebras fitting into a duality-for-actions tower extending previous results in this area for subfactors and Frobenius extensions.

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.

Are biseparable extensions Frobenius

In Secion 1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension. A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent. In Section 2 the problem in the title is formulated in terms of separable bimodules. In Section 3 we specialize the problem to ring extensions, noting that a biseparable extension is a two-sided finitely generated projective, split, separable extension. Some reductions of the problem are discussed and solutions in special cases are provided. In Sec- tion 4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weak- ening the hypotheses of the question in the title. We show in Section 5 that characterizations of the separable extensions among the Frobenius extensions in (HS, K, K99) are special cases of a result for adjoint functors.

Hopf algebroids and H-separable extensions

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.

Separability and the Twisted Frobenius Bimodule

This paper begins with an introduction to β-Frobenius structure on a finite-dimensional Hopf subalgebra pair. In Section 2 a study is made of a generalization of Frobenius bimodules and β-Frobenius extensions. Also a special type of twisted Frobenius bimodule which gives an endomorphism ring theorem and converse is studied. Section 3 brings together material on separable bimodules, the dual definitions of split and separable extension, and a theorem of Sugano on endomorphism rings of separable bimodules. In Section 4, separable twisted Frobenius bimodules are characterized in terms of data that generalizes a Frobenius homomorphism and a dual base. In the style of duality, two corollaries characterizing split β-Frobenius and separable β-Frobenius extensions are proven. Suganos theorem is extended to β-Frobenius extensions and their endomorphism rings. In Section 5, the problem of when separable extensions are Frobenius extensions is discussed. A Hopf algebra example and a matrix example are given of finite rank free separable β-Frobenius extensions which are not Frobenius in the ordinary sense.

Odd H-depth and H-separable extensions

Let Cn(A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if Cn+1(A,B) is isomorphic to a direct summand of a multiple of Cn(A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B. In this paper the H-depth of semisimple subalgebra pairs is derived from the transpose inclusion matrix, and for QF extensions it is derived from the odd depth of the endomorphism ring extension. For general extensions characterizations of H-depth are possible using the H-equivalence generalization of Morita theory.

Pseudo-Galois Extensions and Hopf Algebroids

A pseudo-Galois extension is shown to be a depth two extension. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups, or more generally involutive Hopf algebras, and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observe that this Hopf algebroid is nontrivially isomorphic to a Connes-Moscovici Hopf algebroid.

Subring Depth, Frobenius Extensions, and Towers

The minimum depth 𝑑(𝐵,𝐴) of a subring 𝐵⊆𝐴 introduced in the work of Boltje, Danz and Kulshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show that 𝑑(𝐵,𝐴) < ∞ if 𝐴 is a finite-dimensional algebra and 𝐵𝑒 has finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. If 𝐴⊇𝐵 is a QF extension, minimum left and right even subring depths are shown to coincide. If 𝐴⊇𝐵 is a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depth 𝑛 extensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.

CENTRALIZERS AND INDUCTION

Given a ring homomorphism B → A, consider its centralizer R = AB, bimodule endomorphism ring S = EndBAB and sub-tensor-square ring T = (A ⊗ BA)B. Nonassociative tensoring by the cyclic modules RT or SR leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A | B is separable, H-separable, split or left depth two (D2). If RT or SR are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RT a progenerator, which takes the place of the key module AAe for an Azumaya algebra A. In addition, we characterize left D2 extensions in terms of the module TR, and show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel as well as pre-braided commutative. For example, the notion of normality yields a version for Hopf subalgebras of the fact that normal subgroups have normal centralizers, and yields a special case of a conjecture that D2 Hopf subalgebras are normal.