Frobenius Reciprocity Research Articles

Frobenius Reciprocity in Tensor Categories

Since the appearance of the celebrated work by V. Jones on index of subfactors, combinatorial structures behind the theory have been exploited extensively. In the original approach of V. Jones, these are described in terms of higher relative commutants of towers of algebras, which turn out to be complete invariants for amenable subfactors by a recent result of S. Popa. Parallel to this approach, A. Ocneanu presented the invariant in a combinatorially more satisfactory way (so called paragroups) based on which he reached his classification result of subfactors of index smaller than 4. In our previous paper we have described the basics in bimodules to pursue the Ocneanu’s approach, where categorical structures underlies behind the whole theory. This point of view has been particularly useful in algebraic studies of Jones index theory ([9], [14], [15], [27]). This categorical structure turns out to be more fundamental and it deserves separate studies which is the main concern in the present paper. Since we aim at applications in operator algebras or unitary representation theory, the category considered below are mostly assumed to have ∗-structure in the sense that morphisms behave like bounded linear operators between Hilbert spaces. Clearly the category of bimodules accomodates such structures and more importantly it admits monoidal structure by taking relative tensor products. Other than such more or less explicit structures, the category of bimodules (of finite Jones index) bears some metrical information which is directly related to minimal expectations in subfactor theory and has an intimate relation to the notion of connections in paragroups. By translating some of basic properties of minimal expectations, this distinguished structure, referred to as Frobenius duality in this paper, is organized into a set of axioms, which turns out to be basic ingredients in Frobenius reciprocity. Starting with a Frobenius duality in a monoidal category, we then

Sobolev norms of automorphic functionals

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the representation theory (eventually, of adelic groups, see [7, 8, 9]). Later, Ol’shanski gave a more transparent proof of it (see [14]). However, in the subsequent rapid development of the theory of automorphic functions, Frobenius reciprocity was barely noticeable. We believe that this is due to the incompleteness of the above-mentioned results. In this paper, we prove a general theorem (see Theorem 1.1), which we view as a quantitative version of Frobenius reciprocity. We then illustrate it by looking into the example of SL(2,R). We think that these methods will play a more prominent role in the theory of automorphic functions.

Jones index theory by Hilbert C*-bimodules and K-theory

In this paper we introduce the notion of Hilbert C ∗ {\mathrm {C}}^{*} -bimodules, replacing the associativity condition of two-sided inner products in Rieffel’s imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur’s Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert C ∗ {\mathrm {C}}^{*} -bimodules and show that tensor products of minimal bimodules are also minimal. For an A A - A A bimodule which is compatible with a trace on a unital C ∗ {\mathrm {C}}^{*} -algebra A A , its dimension (square root of Jones index) depends only on its K K KK -class. Finally, we show that the dimension map transforms the Kasparov products in K K ( A , A ) KK(A,A) to the product of positive real numbers, and determine the subring of K K ( A , A ) KK(A,A) generated by the Hilbert C ∗ {\mathrm {C}}^{*} -bimodules for a C ∗ {\mathrm {C}}^{*} -algebra generated by Jones projections.

Orthogonal Frobenius reciprocity

A quantitative version of Frobenius reciprocity for isometrics is proved. It is applied to deduce recursion formulas for the rational class of some irreducible orthogonal modules of the symmetric groups.

Geometry and representations of the quantum supergroup OSPq(1|2n)

The quantum supergroup OSPq(1|2n) is studied systematically. A Haar functional is constructed, and an algebraic version of the Peter–Weyl theory is extended to this quantum supergroup. Quantum homogeneous superspaces and quantum homogeneous supervector bundles are defined following the strategy of Connes’ theory. Parabolic induction is developed by employing the quantum homogeneous supervector bundles. Quantum Frobenius reciprocity and a generalized Borel–Weil theorem are established for the induced representations.

Structure and Representations of the Quantum General Linear Supergroup

The structure and representations of the quantum general linear supergroup GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq of its representative functions. Gq is factorized into \(G_q^\pi G_q^{\bar \pi }\), and a Peter–Weyl basis is constructed for each factor. Parabolic induction for the quantum supergroup is developed. The underlying geometry of induced representations is discussed, and an analog of Frobenius reciprocity is obtained. A quantum Borel–Weil theorem is proven for the irreducible covariant and contravariant tensorial representations, and explicit realizations are given for classes of irredducible tensorial representatins in terms of sections of quantum super vector bundles over quantum projective superspaces.

Sobolev norms of automorphic functionals and Fourier coefficients of cusp forms

We propose a new approach to the study of eigenfunctions of the Laplace-Beltrami operator on a Riemann surface of curvature 1. It is based on Frobenius reciprocity from the theory of automorphic functions. We determine Sobolev class of arising automorphic functionals and discuss some applications.

Spectra of the Laplacian on the Cayley projective plane

Let M = G/K be a compact homogeneous space of a compact semi-simple Lie group G. Let V be a complex homogeneous vector bundle on M. The group G acts naturally on the space of sections T{V) of V. By a theorem of Peter and Weyl, T(V) is a unitary direct sum of finitedimensional representations of G. It is an important problem to decompose T{V) into irreducible (j-modules. By the Frobenius reciprocity theorem, the problem is divided into two parts:

Primitive Ideal Spaces, Characters, and Kirillov Theory for Discrete Nilpotent Groups

This paper is concerned with several topics in the harmonic analysis of discrete nilpotent groups. We study two topologies on the primitive ideal space and the space of characters, the separation of points in the former with the hull-kernel topology, a weak Frobenius reciprocity property and a Bochner theorem for traces on complete nilpotent groups.

Mackey Theorem and Two-Electron Wave Function of a Multi-Centre System

The two-electron wave function in a system of many equivalent atoms is investigated group-theoretically. It is shown that the classification of different types of two-electron (two-hole) localizations can be made by the double-coset decomposition of the symmetry group with respect to the local subgroup, and that the group appearing in the Mackey theorem can be used for the additional classification of states. The Mackey theorem on symmetrized squares and the generalized Frobenius reciprocity theorem are applied to the construction of two-electron states in octahedral symmetry.

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV′ which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV′-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV′ being theg-invariants in simpleV orV′ beingg-rational.

FROM SUBFACTORS TO 3-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND BACK: A detailed account of Ocneanu’s theory

A full proof of Ocneanu’s theorem is given that one can produce a rational unitary polyhedral 3-dimensional topological quantum field theory of Turaev-Viro type from a subfactor with finite index and finite depth, and vice versa. The key argument is an equivalence between flatness of a connection in paragroup theory and invariance of a state sum under one of the three local moves of tetrahedra. This was announced by A. Ocneanu and he gave a proof of Frobenius reciprocity and the pentagon relation, which produces a 3-dimensional TQFT via the Turaev-Viro machinery, but he has not published a proof of the converse direction of the equivalence. Details are given here along the lines suggested by him.

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.

Modular Theory for Bimodules

Modular theory is developed for bimodules and its basic properties are established. Among them Frobenius reciprocity is obtained in a fairly general setting.

Modified symmetry generators and the geometric phase

Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately Sd (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.

A NOTE ON OCNEANU'S APPROACH TO JONES' INDEX THEORY

Some of the basic facts in the Ocneanu's approach to Jones' index theory are proved. Among other things the biunitarity of connections in paragroups is proved generally by making use of Frobenius reciprocity.

How to obtain easily the induced representations of point groups: the icosahedral point groups

With tables of subduced representations as a starting point and use of the Frobenius reciprocity theorem, a simple method to obtain induced representations is given. Tables are given of the 22 induced representations of 532 (I) and of the 84 induced representations of {\overline 5}{\overline 3}2/m (Ih).

Induction and restriction as adjoint functors on representations of locally compact groups

In this paper the Frobenius Reciprocity Theorem for locally compact groups is looked at from a category theoretic point of view.

The multiplicity function on exponential and completely solvable homogeneous spaces

It has been shown recently that the same multiplicity function appears in direct integral decompositions of both induced and restricted representations on exponential solvable groups — that is, a generalized form of Frobenius Reciprocity is valid. Qualitative results on that multiplicity function in the completely solvable case are proven in this paper — namely, necessary and sufficient conditions for finiteness and boundedness are obtained. The proof uses techniques from ‘pseudo-algebraic geometry’.

History and variation on the theme of the frobenius reciprocity theorem

The German mathematician Georg Ferdinand Frobenius (1849-1917) was born in Berlin on 26 October 1849, the son of Christian Ferdinand Frobenius. After at tending the Joachimthal Gymnasium, Frobenius began his mathematical studies at Gottingen in 1867. Under the direction of the inimitable Karl Weierstrass he earned a doctorate three years later in Berlin. In ensuing years, Frobenius made fundamental discoveries in such diverse areas as matrix theory mod p and 0-functions in several variables. His particularly productive work during the 1870-75 period is chronicled in Crelle's Journal. The major achievements of Frobenius are undoubtedly in the theory of group representat ions--a theory he initiated in 1896 at the age of 47. Along with its three historic roots--Galois theory, Lie theory, and number theory-i t stands as a cornerstone of modern mathematics. group, so we begin with the latter notion. It appears, implicitly, as early as 1801 in the work of Gauss--viz . , in his Dzsquisitiones Arithmetzcae--and m later years, beginning in 1878, Richard Dedekind also considered the notion (again implicitly). An extensive correspondence be tween Dedekind and Frobenius, spanning periods from 1882-83 and from 1895-98, provides valuable insight into the development of Frobenius's creation. A fascinating and learned account of this cor-