Frobenius Property Research Articles

Ruelle operator with nonexpansive IFS

The Ruelle operator and the associated Perron–Frobenius property have been extensively studied in dynamical systems. Recently the theory has been adapted to iterated function systems (IFS) $(X, \{ w_j\} _{j=1}^m, \{ p_j\} _{j=1}^m)$, where the $w_j

Frobenius properties and Maschke-type theorems for entwined modules

Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems known for Doi-Hopf modules are extended to the case of entwined modules.

Weak Hopf Algebras: I. Integral Theory and C-Structure

We give an introduction to the theory of weak Hopf algebras proposed as a coassociative alternative of weak quasi-Hopf algebras. We follow an axiomatic approach keeping as close as possible to the “classical” theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras AL and AR in any weak Hopf algebra A that play the role of non-commutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic properties of A, such as the Frobenius property, or semisimplicity, or innerness of the square of the antipode, are related to the existence of non-degenerate, normalized, or Haar integrals. In case of C*-weak Hopf algebras we prove the existence of a unique Haar measure h∈A and of a canonical grouplike element g∈A implementing the square of the antipode and factorizing into left and right elements g=gLg−1R, gL∈AL, gR∈AR. Further discussion of the C*-case will be presented in Part II.

Frobenius extensions of subalgebras of Hopf algebras

We consider when extensions S ⊂ R S\subset R of subalgebras of a Hopf algebra are β \beta -Frobenius, that is Frobenius of the second kind. Given a Hopf algebra H H , we show that when S ⊂ R S\subset R are Hopf algebras in the Yetter-Drinfeld category for H H , the extension is β \beta -Frobenius provided R R is finite over S S and the extension of biproducts S ⋆ H ⊂ R ⋆ H S\star H\subset R\star H is cleft. More generally we give conditions for an extension to be β \beta -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.

Topological Frobenius properties for nilpotent groups.

Topological Frobenius properties for nilpotent groups.

The failure of the topological frobenius property for nilpotent Lie groups. II

The failure of the topological frobenius property for nilpotent Lie groups. II

The failure of the topological Frobenius property for nilpotent Lie groups.

The failure of the topological Frobenius property for nilpotent Lie groups.

�ber Integralzerlegungen von Darstellungen nilpotenter Liegruppen

For simply connected nilpotent Lie groups we determine explicitly the central decomposition measure of the restriction of a representation onto a subgroup as well as of the induction from a subgroup to the group. For this the Kirillov method is used. Knowing this measure we can show that in the non-abelian case there are non-trivial subgroups such that all representations of these subgroups induce a representation quasi-equivalent to the regular representation. Thus Fell's “Frobenius Property” (FP) never holds in the non-abelian case-contrary to Fell's conjecture. Finally we prove a weaker Frobenius, Property defined by decomposition measures.

A Remark on Frobenius Extensions and Endomorphism Rings

In his paper [1] F. Kasch developed a theory of Frobenius extensions as a generalization of the theory of Frobenius algebras. In it he established a very interesting relationship between the Frobenius property of an extension and that of its endomorphism ring [1, Satz 5], from which he further derived the Frobenius extension property of Galois extensions of simple rings [1, Satz 6]; with these results he kindly responded to what had been vaguely “conjectured” (as he wrote) by one of the writers on the connection between Galois theory and the theory of Frobenius algebras [2].