Induced representation and Frobenius reciprocity for compact quantum groups

Unitary representations of compact quantum groups have been described as isometric comodules. The notion of an induced representation for compact quantum groups has been introduced and an analogue of the Frobenius reciprocity theorem is established.     

Gorenstein projective modules and Frobenius extensions

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x~2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.     

Abstract Plancherel theorems and a Frobenius reciprocity theorem

A method for obtaining Plancherel theorems for unitary representations of Lie groups via C^∞ vector techniques is studied. The results are used to prove the nonunimodular Plancherel theorem of Moore and to study its convergence. A C^∞ Frobenius reciprocity theorem which generalizes Gelfand's duality theorem is also proven.     

Frobenius extensions of subalgebras of Hopf algebras

We consider when extensions of subalgebras of a Hopf algebra are -Frobenius, that is Frobenius of the second kind. Given a Hopf algebra , we show that when are Hopf algebras in the Yetter-Drinfeld category for , the extension is -Frobenius provided is finite over and the extension of biproducts is cleft.

More generally we give conditions for an extension to be -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.

     

Gorenstein homological invariant properties under Frobenius extensions

Science China Mathematics - We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein...     

A rational reciprocity law over function fields

In the classical case, reciprocity laws for power residue symbols are called rational, which means that the power residue symbols only assume the values \({\pm 1}\) and have entries in \({\mathbb{Z}}\). We establish a rational reciprocity law over function fields.     

Non-abelian local reciprocity law

Following our work on the generalized Fesenko reciprocity map, we construct the non-abelian local reciprocity map ${\pmb{\Phi}_K^{(\varphi)}}$ of a local field K as a certain isomorphism from the absolute Galois group G _K of K onto a topological group ${\nabla_{K,Y}^{(\varphi)}}$ whose definition involves Fontaine–Wintenberger theory of field of norms, and build the non-abelian local class field theory over K in the sense of Fesenko and Koch.     

On polynomial reciprocity law

The well-known law of quadratic reciprocity has over 150 proofs in print. We establish a relation between polynomial Jacobi symbols and resultants of polynomials over finite fields. Using this relation, we prove the polynomial reciprocity law and obtain a polynomial analogue of classical Burde's quartic reciprocity law. Under the use of our polynomial Poisson summation formula and the evaluation of polynomial exponential map, we get a reciprocity for the generalized polynomial quadratic Gauss sums.     

Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

A geometric approach to the quadratic reciprocity law

A geometric viewpoint of much broader potential yields a unified proof of the quadratic reciprocity law based on counting points on quadratic surfaces over finite prime fields.     

Some problems concerning the Frobenius number for extensions of an arithmetic progression

The Ramanujan Journal - We show existence, uniqueness and disjointness of Klyachko periods for certain induced representations associated by Zelevinsky to every irreducible representation of a...     

Nonexcellence of certain field extensions

Towers of fields F₁ ⊂ F₂ ⊂ F₃ are considered, where F₃/F₂ is a quadratic extension and F₂/F₁ is an extension, which is either quadratic or of odd degree or purely transcendental of degree 1. Numerous examples of the above types such that the extension F₃/F₁ is not 4-excellent are constructed. Also it is shown that if k is a field, char k ≠ 2, and l/k is an arbitrary field extension of fourth degree, then there exists a field extension F/k such that the fourth degree extension lF/F is not 4-excellent. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 213–226.     

A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.