230 results
Search Results
2. On the Hyperinvariant Subspace Problem. IV
- Author
-
Carl Pearcy, H. Bercovici, and C. Foias
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,0101 mathematics ,01 natural sciences ,Subspace topology ,Mathematics - Abstract
This paper is a continuation of three recent articles concerning the structure of hyperinvariant subspace lattices of operators on a (separable, infinite dimensional) Hilbert space . We show herein, in particular, that there exists a “universal” fixed block-diagonal operator B on such that if ε > 0 is given and T is an arbitrary nonalgebraic operator on , then there exists a compact operator K of norm less than ε such that (i) Hlat(T) is isomorphic as a complete lattice to Hlat(B + K) and (ii) B + K is a quasidiagonal, C00, (BCP)-operator with spectrum and left essential spectrum the unit disc. In the last four sections of the paper, we investigate the possible structures of the hyperlattice of an arbitrary algebraic operator. Contrary to existing conjectures, Hlat(T) need not be generated by the ranges and kernels of the powers of T in the nilpotent case. In fact, this lattice can be infinite.
- Published
- 2008
3. A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups
- Author
-
Arash Ghaani Farashahi
- Subjects
Pure mathematics ,Linear representation ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Convolution power ,01 natural sciences ,Algebra ,Compact group ,Homogeneous ,Homogeneous space ,Invariant measure ,0101 mathematics ,Mathematics - Abstract
This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. LetGbe a compact group andHa closed subgroup ofG. Letμbe the normalizedG-invariant measure over the compact homogeneous spaceG/Hassociated with Weil's formula and. We then present a structured class of abstract linear representations of the Banach convolution function algebrasLp(G/H,μ).
- Published
- 2018
4. On the Curves Associated to Certain Rings of Automorphic Forms
- Author
-
Kamal Khuri-Makdisi
- Subjects
Pure mathematics ,Quaternion algebra ,General Mathematics ,010102 general mathematics ,Automorphic form ,Complex multiplication ,Congruence relation ,01 natural sciences ,Algebra ,Elliptic curve ,0103 physical sciences ,Representation ring ,010307 mathematical physics ,Compactification (mathematics) ,0101 mathematics ,Hecke operator ,Mathematics - Abstract
In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.
- Published
- 2001
5. Characters of Depth-Zero, Supercuspidal Representations of the Rank-2 Symplectic Group
- Author
-
Clifton Cunningham
- Subjects
Symplectic group ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Symplectic representation ,01 natural sciences ,Algebra ,Character (mathematics) ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Symplectomorphism ,Moment map ,Springer correspondence ,Mathematics - Abstract
This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of p-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.
- Published
- 2000
6. Factorization in the Invertible Group of a C*-Algebra
- Author
-
Michael J. Leen
- Subjects
Group (mathematics) ,General Mathematics ,010102 general mathematics ,Compact operator ,01 natural sciences ,law.invention ,Combinatorics ,Algebra ,Nilpotent ,Invertible matrix ,law ,0103 physical sciences ,Homogeneous space ,Algebra representation ,010307 mathematical physics ,Compact quantum group ,Identity component ,0101 mathematics ,Mathematics - Abstract
In this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.
- Published
- 1997
7. Automorphisms of the Lie Algebras W* in Characteristic 0
- Author
-
J. Marshall Osborn
- Subjects
Automorphisms of the symmetric and alternating groups ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Lie conformal algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In a recent paper [2] we defined four classes of infinite dimensional simple Lie algebras over a field of characteristic 0 which we called W*, S*, H*, and K*. As the names suggest, these classes generalize the Lie algebras of Cartan type. A second paper [3] investigates the derivations of the algebras W* and S*, and the possible isomorphisms between these algebras and the algebras defined by Block [1]. In the present paper we investigate the automorphisms of the algebras of type W*.
- Published
- 1997
8. A Schensted Algorithm Which Models Tensor Representations of the Orthogonal Group
- Author
-
Robert A. Proctor
- Subjects
Algebra ,General Mathematics ,Tensor (intrinsic definition) ,010102 general mathematics ,0103 physical sciences ,Orthogonal group ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.
- Published
- 1990
9. Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces
- Author
-
Tuomas Hytönen, Dongyong Yang, Dachun Yang, and Suile Liu
- Subjects
General Mathematics ,010102 general mathematics ,Foundation (engineering) ,01 natural sciences ,Measure (mathematics) ,Algebra ,Non homogeneous ,0103 physical sciences ,Metric (mathematics) ,Natural science ,Maximal operator ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let (𝒳, d, μ) be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that μ(﹛x﹜) = 0 for all x ∈ 𝒳. In this paper, we show that the boundedness of a Calderón–Zygmund operator T on L2(μ) is equivalent to that of T on Lp(μ) for some p ∈ (1,∞), and that of T from L1(μ) to L1,∞(μ). As an application, we prove that if T is a Calderón–Zygmund operator bounded on L2(μ), then its maximal operator is bounded on Lp(μ) for all p ∈ (1,∞) and from the space of all complex-valued Borel measures on 𝒳 to L1,∞(μ). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.
- Published
- 2012
10. The Genuine Omega-regular Unitary Dual of the Metaplectic Group
- Author
-
Annegret Paul, Susana A. Salamanca-Riba, and Alessandra Pantano
- Subjects
Algebra ,Infinitesimal character ,Tensor product ,Metaplectic group ,Oscillator representation ,General Mathematics ,Representation (systemics) ,Omega ,Unitary state ,Mathematics ,Dual (category theory) - Abstract
We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.
- Published
- 2012
11. Iwahori-Hecke Algebras of SL2 over 2-Dimensional Local Fields
- Author
-
Kyu-Hwan Lee
- Subjects
Algebra ,Hecke algebra ,General Mathematics ,Subalgebra ,Division algebra ,Algebra representation ,Cellular algebra ,Universal enveloping algebra ,Iwahori subgroup ,Mathematics::Representation Theory ,Hecke operator ,Mathematics - Abstract
In this paper we construct an analogue of Iwahori–Hecke algebras of SL2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL2, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori–Matsumoto type relations.
- Published
- 2010
12. Reducibility of the Principal Series for (F) over a p-adic Field
- Author
-
Christian Zorn
- Subjects
Algebra ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Principal (computer security) ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Field (mathematics) ,010307 mathematical physics ,0101 mathematics ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,01 natural sciences ,Mathematics - Abstract
Let Gn = Spn(F) be the rank n symplectic group with entries in a nondyadic p-adic field F. We further let be the metaplectic extension of Gn by defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of . In most cases, we will use some techniques developed by Tadić that analyze the Jacquetmodules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters χ with χ2 = 1. Because such representations π are unitary, to show the irreducibility of π, it suffices to show that . We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of . We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in for the π in question.
- Published
- 2010
13. Verma Modules over Quantum Torus Lie Algebras
- Author
-
Kaiming Zhao and Rencai Lu
- Subjects
Pure mathematics ,Verma module ,General Mathematics ,010102 general mathematics ,Generalized Verma module ,Kac–Moody algebra ,01 natural sciences ,Affine Lie algebra ,Lie conformal algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,0103 physical sciences ,Maximal torus ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.
- Published
- 2010
14. On Hankel Forms of Higher Weights: The Case of Hardy Spaces
- Author
-
Marcus Sundhäll and Edgar Tchoundja
- Subjects
Class (set theory) ,Pure mathematics ,Group (mathematics) ,General Mathematics ,Hardy space ,Characterization (mathematics) ,Carleson measure ,Algebra ,symbols.namesake ,Bounded function ,symbols ,Hankel matrix ,Möbius transformation ,Mathematics - Abstract
In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundh¨all for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.
- Published
- 2010
15. Uniqueness of Shalika Models
- Author
-
Chufeng Nien
- Subjects
General Mathematics ,010102 general mathematics ,Linear model ,Field (mathematics) ,01 natural sciences ,Algebra ,Finite field ,0103 physical sciences ,Division algebra ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Quaternion ,Mathematics - Abstract
Let 𝔽q be a finite field of q elements, 𝓕 a p-adic field, and D a quaternion division algebra over 𝓕. This paper proves uniqueness of Shalika models for GL2n(𝔽q) and GL2n(D), and re-obtains uniqueness of Shalika models for GL2n(𝔽q) and GL2n(D), and re-obtains uniqueness of Shalika models for GL2n (𝓕) for any n ∈ ℕ.
- Published
- 2009
16. Prehomogeneity on Quasi-Split Classical Groups and Poles of Intertwining Operators
- Author
-
Xiaoxiang Yu
- Subjects
Classical group ,Algebra ,Pure mathematics ,Operator (computer programming) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Tempered representation ,Field (mathematics) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Suppose that P = MN is amaximal parabolic subgroup of a quasisplit, connected, reductive classical group G defined over a non-Archimedean field and A is the standard intertwining operator attached to a tempered representation of G induced from M . In this paper we determine all the cases in which Lie(N ) is prehomogeneous under Ad(m) when N is non-abelian, and give necessary and sufficient conditions for A to have a pole at 0.
- Published
- 2009
17. On Reducibility and Unitarizability for Classical p-Adic Groups, Some General Results
- Author
-
Marko Tadić
- Subjects
Algebra ,Infinitesimal character ,Reduction (recursion theory) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Parabolic induction ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The aim of this paper is to prove two general results on parabolic induction of classical p-adic groups (actually, one of them holds also in the archimedean case), and to obtain from them some consequences about irreducible unitarizable representations. One of these consequences is a reduction of the unitarizability problem for these groups. This reduction is similar to the reduction of the unitarizability problem to the case of real infinitesimal character for real reductive groups.
- Published
- 2009
18. Klyachko Models for General Linear Groups of Rank 5 over a p-Adic Field
- Author
-
Chufeng Nien
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Extension (predicate logic) ,Rank (differential topology) ,01 natural sciences ,Unitary state ,Algebra ,0103 physical sciences ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
This paper shows the existence and uniqueness of Klyachko models for irreducible unitary representations of GL5 (ℱ), where ℱ is a p-adic field. It is an extension of the work of Heumos and Rallis on GL4(ℱ).
- Published
- 2009
19. Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences
- Author
-
Chun-Yen Shen, Hao-Wei Huang, and Chang-Pao Chen
- Subjects
Algebra ,Pure mathematics ,Matrix (mathematics) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let A = (aj,k)j,k≥1 be a non-negative matrix. In this paper, we characterize those A for which ∥A∥E,F are determined by their actions on decreasing sequences, where E and F are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: ℓp, d(w, p), and ℓp(w). The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.
- Published
- 2008
20. On Certain Classes of Unitary Representations for Split Classical Groups
- Author
-
Goran Muić
- Subjects
Classical group ,Unitarity ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Unitary state ,split classical groups ,tempered representations ,unitarizability ,Aubert-Schneider- Stuhler involution ,Algebra ,Mathematics::Quantum Algebra ,0103 physical sciences ,Dual polyhedron ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper we prove the unitarity of duals of tempered representations supported onminimal parabolic subgroups for split classical p-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups.
- Published
- 2007
21. Decomposability of von Neumann Algebras and the Mazur Property of Higher Level
- Author
-
Zhiguo Hu and Matthias Neufang
- Subjects
Jordan algebra ,General Mathematics ,010102 general mathematics ,Group algebra ,01 natural sciences ,Filtered algebra ,Combinatorics ,Algebra ,symbols.namesake ,Von Neumann algebra ,symbols ,Division algebra ,Cellular algebra ,0101 mathematics ,Abelian von Neumann algebra ,Affiliated operator ,Mathematics - Abstract
The decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.
- Published
- 2006
22. A Steinberg Cross Section for Non–Connected Affine Kac—Moody Groups
- Author
-
Stephan Mohrdieck
- Subjects
Connected component ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Coxeter group ,01 natural sciences ,Algebra ,Mathematics::Group Theory ,0103 physical sciences ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Mathematics::Representation Theory ,Crucial point ,Coxeter element ,Quotient ,Mathematics - Abstract
In this paper we generalise the concept of a Steinberg cross section to non–connected affine Kac–Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac–Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a “twisted Coxeter cell”, a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a 𝕔*-action.
- Published
- 2006
23. Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori
- Author
-
Saeid Azam
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Killing form ,01 natural sciences ,Affine Lie algebra ,Lie conformal algebra ,Reductive Lie algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Generalized Kac–Moody algebra ,Mathematics - Abstract
We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study themin this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.
- Published
- 2006
24. The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras
- Author
-
Salem Ben Said
- Subjects
Algebra ,Pure mathematics ,Simple (abstract algebra) ,Non-Euclidean geometry ,General Mathematics ,Euclidean geometry ,Functional equation ,Type (model theory) ,Bernstein polynomial ,Mathematics - Abstract
This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.
- Published
- 2006
25. Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
- Author
-
Michail Zhitomirskii
- Subjects
Generalization ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Zero (complex analysis) ,Submanifold ,Space (mathematics) ,01 natural sciences ,Manifold ,Algebra ,Simple (abstract algebra) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset N of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to N of the contact structure. This is a generalization of the Darboux-Givental’ theoremfor smooth submanifolds of a contactmanifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold N in terms of the algebraic restriction of the contact structure to N. In particular, the algebraic restriction is zero if and only if N is contained in a smooth Legendrian submanifold of M.
- Published
- 2005
26. Reducibility of Generalized Principal Series
- Author
-
Goran Muić
- Subjects
Classical group ,classical groups ,p-adic fields ,irreducible square integrable representations ,induced representations ,generalized principal series ,cuspidal representations ,Pure mathematics ,Series (mathematics) ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Principal (computer security) ,01 natural sciences ,Algebra ,Discrete series ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper we describe reducibility of non-unitary generalized principal series for classical p-adic groups in terms of the classification of discrete series due to Moeglin and Tadić.
- Published
- 2005
27. On Local L-Functions and Normalized Intertwining Operators
- Author
-
Henry H. Kim
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Holomorphic function ,Residual ,Constant term ,01 natural sciences ,Algebra ,symbols.namesake ,0103 physical sciences ,Eisenstein series ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we make explicit all L-functions in the Langlands–Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local L-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for Re(s) ≥ 1/2 in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrumand determining poles of automorphic L-functions.
- Published
- 2005
28. Duality and Supports of Induced Representations for Orthogonal Groups
- Author
-
Chris Jantzen
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Duality (optimization) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we construct a duality for p-adic orthogonal groups.
- Published
- 2005
29. Equivariant Formality for Actions of Torus Groups
- Author
-
Laura Scull
- Subjects
General Mathematics ,010102 general mathematics ,Torus ,Minimal models ,Formality ,Rationalization (economics) ,01 natural sciences ,Circle group ,Algebra ,Minimal model ,0103 physical sciences ,Equivariant map ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use S1-equivariant minimal models to give a number of examples of S1-spaces illustrating the properties of the various definitions.
- Published
- 2004
30. Meromorphic Functions Sharing the Same Zeros and Poles
- Author
-
Günter Frank, Rémi Vaillancourt, and Xinhou Hua
- Subjects
010101 applied mathematics ,Algebra ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Argument principle ,0101 mathematics ,01 natural sciences ,Meromorphic function ,Mathematics - Abstract
In this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function f is determined by its zeros and poles and the zeros of f(j) for j = 1, 2, 3, 4.
- Published
- 2004
31. Similarity Classification of Cowen-Douglas Operators
- Author
-
Chunlan Jiang
- Subjects
General Mathematics ,010102 general mathematics ,01 natural sciences ,Centralizer and normalizer ,law.invention ,Combinatorics ,Algebra ,Operator (computer programming) ,Invertible matrix ,Integer ,Similarity (network science) ,law ,Bounded function ,0103 physical sciences ,Idempotence ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
Let ℋ be a complex separable Hilbert space and ℒ(ℋ) denote the collection of bounded linear operators on ℋ. An operator A in ℒ(ℋ) is said to be strongly irreducible, if , the commutant of A, has no non-trivial idempotent. An operator A in ℒ(ℋ) is said to be a Cowen-Douglas operator, if there exists Ω, a connected open subset of C, and n, a positive integer, such that(a)Ω ⊂ σ(A) = ﹛z ∈ C | A – z not invertible﹜;(b)ran(A – z) = ℋ, for z in Ω;(c)Vz∈Ω ker(A – z) = ℋ and(d)dim ker(A – z) = n for z in Ω.In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the K0-group of the commutant algebra as an invariant.
- Published
- 2004
32. An Elementary Proof of a Weak Exceptional Zero Conjecture
- Author
-
Louisa Orton
- Subjects
Cusp (singularity) ,Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,01 natural sciences ,Dirichlet distribution ,Algebra ,symbols.namesake ,0103 physical sciences ,Elementary proof ,symbols ,010307 mathematical physics ,0101 mathematics ,Twist ,Mathematics - Abstract
In this paper we extend Darmon's theory of “integration on ℋp × ℋ” to cusp forms f of higher even weight. This enables us to prove a “weak exceptional zero conjecture”: that when the p-adic L-function of f has an exceptional zero at the central point, the ℒ-invariant arising is independent of a twist by certain Dirichlet characters.
- Published
- 2004
33. Ternary Diophantine Equations via Galois Representations and Modular Forms
- Author
-
Michael A. Bennett and Christopher Skinner
- Subjects
Pure mathematics ,General Mathematics ,Diophantine equation ,010102 general mathematics ,Modular form ,Type (model theory) ,Galois module ,01 natural sciences ,Differential Galois theory ,Algebra ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Ternary operation ,Mathematics - Abstract
In this paper, we develop techniques for solving ternary Diophantine equations of the shape Axn + Byn = Cz2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B andC. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.
- Published
- 2004
34. Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations
- Author
-
Yucai Su
- Subjects
General Mathematics ,010102 general mathematics ,Non-associative algebra ,010103 numerical & computational mathematics ,Killing form ,Kac–Moody algebra ,01 natural sciences ,Affine Lie algebra ,Lie conformal algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,0101 mathematics ,Generalized Kac–Moody algebra ,Mathematics - Abstract
A class of nongraded Hamiltonian Lie algebras was earlier introduced by Xu. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a ``sandwich'' method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined., Comment: 36 pages, latex; to appear in Canadian J. Math
- Published
- 2003
35. Two Algorithms for a Moving Frame Construction
- Author
-
Irina A. Kogan
- Subjects
Algebra ,Differential form ,Moving frame ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,01 natural sciences ,Algorithm ,Mathematics - Abstract
The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.
- Published
- 2003
36. Generalized Factorization in Hardy Spaces and the Commutant of Toeplitz Operators
- Author
-
Kehe Zhu and Michael Stessin
- Subjects
Mathematics::Functional Analysis ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Context (language use) ,Hardy space ,01 natural sciences ,Unit disk ,Toeplitz matrix ,Algebra ,symbols.namesake ,Factorization ,Bergman space ,Factorization of polynomials ,0103 physical sciences ,symbols ,010307 mathematical physics ,Dixon's factorization method ,0101 mathematics ,Mathematics - Abstract
Every classical inner function φ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when φ(z) = z. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.
- Published
- 2003
37. The Operator Biprojectivity of the Fourier Algebra
- Author
-
Peter J. Wood
- Subjects
Fourier algebra ,General Mathematics ,010102 general mathematics ,Group algebra ,Finite-rank operator ,Reflexive operator algebra ,Compact operator ,Shift operator ,01 natural sciences ,Compact operator on Hilbert space ,Strictly singular operator ,Algebra ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group G is operator biprojective if and only if G is discrete.
- Published
- 2002
38. On Russell-Type Modular Equations
- Author
-
Wen-Chin Liaw and Heng Huat Chan
- Subjects
Pure mathematics ,Mathematics::General Mathematics ,General Mathematics ,Mathematics::History and Overview ,010102 general mathematics ,Modular invariance ,Modular form ,01 natural sciences ,Modular curve ,Algebra ,symbols.namesake ,Modular elliptic curve ,Simultaneous equations ,0103 physical sciences ,Eisenstein series ,symbols ,Astrophysics::Solar and Stellar Astrophysics ,Dedekind eta function ,010307 mathematical physics ,0101 mathematics ,Hecke operator ,Mathematics - Abstract
In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russell’s main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell’s theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature 3.
- Published
- 2000
39. Subregular Nilpotent Elements and Bases in K-Theory
- Author
-
George Lusztig
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,K-theory ,01 natural sciences ,Algebra ,Nilpotent ,Simple (abstract algebra) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Lie algebra ,Standard basis ,Equivariant map ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Mathematics - Abstract
In this paper we describe a canonical basis for the equivariant K-theory (with respect to a C*-action) of the variety of Borel subalgebras containing a subregular nilpotent element of a simple complex Lie algebra of type D or E.
- Published
- 1999
40. Reflection Subquotients of Unitary Reflection Groups
- Author
-
Tonny A. Springer and Gustav I. Lehrer
- Subjects
Pure mathematics ,Finite group ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Algebra ,0103 physical sciences ,Irreducibility ,Coset ,Vector field ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Reflection group ,Subquotient ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let G be a finite group generated by (pseudo-) reflections in a complex vector space and let g be any linear transformation which normalises G. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset gG, a subquotient of G which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in G of certain elements of the coset. A criterion is also given in terms of the invariant degrees of G for an integer to be regular for G. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.
- Published
- 1999
41. Lusternik-Schnirelmann Category and Algebraic R-Local Homotopy Theory
- Author
-
H. Scheerer and D. Tanré
- Subjects
Algebra ,Higher category theory ,n-connected ,Homotopy category ,Model category ,General Mathematics ,Homotopy hypothesis ,Simplicial set ,A¹ homotopy theory ,Cofibration ,Mathematics - Abstract
In this paper, we define the notion of R*-LS category associated to an increasing system of subrings of ℚ and we relate it to the usual LS-category. We also relate it to the invariant introduced by Félix and Lemaire in tame homotopy theory, in which case we give a description in terms of Lie algebras and of cocommutative coalgebras, extending results of Lemaire-Sigrist and Félix-Halperin.
- Published
- 1998
42. Derivations and Invariant Forms of Lie Algebras Graded by Finite Root Systems
- Author
-
Georgia Benkart
- Subjects
General Mathematics ,010102 general mathematics ,Non-associative algebra ,Killing form ,Kac–Moody algebra ,01 natural sciences ,Affine Lie algebra ,Graded Lie algebra ,Lie conformal algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Lie algebras graded by finite reduced root systems have been classified up to isomorphism. In this paper we describe the derivation algebras of these Lie algebras and determine when they possess invariant bilinear forms. The results which we develop to do this are much more general and apply to Lie algebras that are completely reducible with respect to the adjoint action of a finite-dimensional subalgebra.
- Published
- 1998
43. Isomorphisms Between Generalized Cartan Type W Lie Algebras in Characteristic 0
- Author
-
Kaiming Zhao
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Cartan decomposition ,Cartan subalgebra ,Real form ,Killing form ,Kac–Moody algebra ,01 natural sciences ,Affine Lie algebra ,Algebra ,0103 physical sciences ,Cartan matrix ,010307 mathematical physics ,0101 mathematics ,Generalized Kac–Moody algebra ,Mathematics - Abstract
In this paper, we determine when two simple generalized Cartan type W Lie algebras Wd(A, T, φ) are isomorphic, and discuss the relationship between the Jacobian conjecture and the generalized Cartan type W Lie algebras.
- Published
- 1998
44. Tensor Products of Analytic Continuations of Holomorphic Discrete Series
- Author
-
Genkai Zhang and Bent Ørsted
- Subjects
Tensor contraction ,Hermitian symmetric space ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Tensor product of Hilbert spaces ,Holomorphic function ,Identity theorem ,01 natural sciences ,Plancherel theorem ,Algebra ,Tensor product ,0103 physical sciences ,Ricci decomposition ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
We give the irreducible decomposition of the tensor product of an an- alytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate. 0. Introduction. The work of Segal (IES) and Mautner (M) established the abstract Plancherel theorem for type I groups. This meant that for an arbitrary unitary represen- tation, one could find its spectral decomposition into irreducibles and a corresponding spectral measure. To make this program explicit on L 2 -spaces on homogeneous spaces is one of the main subjects of harmonic analysis. Another interesting case is that of decom- posing a tensor product of irreducible representations; our aim in this paper is to consider this for certain holomorphic representations. The problem of finding the irreducible decomposition of tensor products of holomor- phic discrete series of the group SL(2, ) has been studied by Repka (Re1). The results there were used by Howe (How) to give the decomposition of the metaplectic represen- tation for certain dual pairs. See also (OZ). For a general semisimple Lie group G of Hermitian type a similar problem is studied in (Re2). It is shown that the tensor prod- uct of a scalar holomorphic discrete series with its conjugate is unitarily equivalent to the L 2 -space on the corresponding Hermitian symmetric space, L 2 (G K). Therefore we know its decomposition from the known theory of Harish-Chandra; namely L 2 (G K) W H ( ) C( ) 2 d
- Published
- 1997
45. Central Quotients and Coverings of Steinberg Unitary Lie Algebras
- Author
-
Yun Gao and Bruce Allison
- Subjects
Algebra ,Pure mathematics ,Representation of a Lie group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Unitary state ,Quotient ,Mathematics - Abstract
In this paper, we calculate the center and the universal covering algebra of the Steinberg unitary Lie algebra stun, where is a unital nonassociative algebra with involution and n ≥ 3.
- Published
- 1996
46. Closed Ideals in a Convolution Algebra of Holomorphic Functions
- Author
-
Jürgen Müller and Rainer Brück
- Subjects
General Mathematics ,010102 general mathematics ,Holomorphic functional calculus ,Holomorphic function ,Convolution power ,Identity theorem ,01 natural sciences ,Convolution ,Algebra ,Filtered algebra ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
We consider the usual topological vector space H(G) of all functions holomorphic in a region G ⊂ ℂ. If G satisfies certain conditions, it is possible to introduce the Hadamard product as multiplication in H(G), and then H(G) turns out to be a commutative topological algebra. In [5] we characterized the invertible elements in H(G), and the aim of this paper is to study the closed ideals and some further questions.
- Published
- 1995
47. Unification in Varieties of Groups: Nilpotent Varieties
- Author
-
Michael H. Albert and John Lawrence
- Subjects
Algebra ,Nilpotent ,Unification ,Generalization ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Nilpotent group ,System of linear equations ,01 natural sciences ,Unitary state ,Mathematics - Abstract
In this paper we show that any system of equations over a free nilpotent group of class c is either unitary or miliary. In fact, such a system either has a most general solution (akin to the most general solution of a system of linear dipohantine equations), or every solution has a proper generalization. In principle we provide an algorithm for determining whether or not a most general solution exists, and exhibiting it if it does.
- Published
- 1994
48. Topologizing Different Classes of Real Functions
- Author
-
Krzysztof Ciesielski
- Subjects
Algebra ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The purpose of this paper is to examine which classesof functions fromcan be topologized in a sense that there exist topologies τ1and τ2onandrespectively, such thatis equal to the class C(τ1, τ2) of all continuous functions. We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functionsfor which the sets {x : f(x) = g(x)} are small in some sense for all f, g ∈f ≠ g. The topologies will be Hausdorff and connected. It will be also shown that in some model of set theory ZFC with GCH these topologies could be completely regular and Baire. One of the corollaries of this theorem is that GCH implies the existence of a connected Hausdorff topology T onsuch that the class L of all linear functions g(x) = ax + b coincides with. This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the classof all polynomials, the classof all analytic functions and the class of all harmonic functions.We will also prove that several other classes of real functions cannot be topologized. This includes the classes of C∞functions, differentiable functions, Darboux functions and derivatives.
- Published
- 1994
49. Exact Inequalities for the Norms of Factors of Polynomials
- Author
-
Peter Borwein
- Subjects
Algebra ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,media_common ,Mathematics - Abstract
This paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.
- Published
- 1994
50. Products of Decomposable Positive Operators
- Author
-
Terrance Quinn
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In recent years there has been a growing interest in problems of factorization for bounded linear operators. We first show that many of these problems properly belong to the category of C*-algebras. With this interpretation, it becomes evident that the problem is fundamental both to the structure of operator algebras and the elements therein. In this paper we consider the direct integral algebra with separable and infinite dimensional. We generalize a theorem of Wu (1988) and characterize those decomposable operators which are products of non-negative decomposable operators. We do this by first showing that various results on operator ranges may be generalized to “measurable fields of operator ranges”.
- Published
- 1994
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.