Showing papers on "Representation theory published in 2002"

Kac-Moody Groups, their Flag Varieties and Representation Theory

Abstract: Introduction * Kac--Moody Algebras -- Basic Theory * Representation Theory of Kac--Moody Algebras * Lie Algebra Homology and Cohomology * An Introduction to ind-Varieties and pro-Groups * Tits Systems -- Basic Theory * Kac--Moody Groups -- Basic Theory * Generalized Flag Varieties of Kac--Moody Groups * Demazure and Weyl--Kac Character Formulas * BGG and Kempf Resolutions * Defining Equations of G/P and Conjugacy Theorems * Topology of Kac-Moody Groups and Their Flag Varieties * Smoothness and Rational Smoothness of Schubert Varieties * An Introduction to Affine Kac-Moody Lie Algebras and Groups * Appendix A. Results from Algebraic Geometry * Appendix B. Local Cohomology * Appendix C. Results from Topology * Appendix D. Relative Homological Algebra * Appendix E. An Introduction to Spectral Sequences * Bibliography * Index of Notation * Index

Operads in algebra, topology, and physics

Abstract: 'Operads are powerful tools, and this is the book in which to read about them' - ""Bulletin of the London Mathematical Society"". Operads are mathematical devices that describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of 'homotopy', where they play a key role in organizing hierarchies of higher homotopies. Significant examples from algebraic topology first appeared in the sixties, although the formal definition and appropriate generality were not forged until the seventies. In the nineties, a renaissance and further development of the theory were inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, last but not least, theoretical physics, especially string field theory and deformation quantization. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature.

Lectures on Algebraic Quantum Groups

Abstract: Preface.- I. BACKGROUND AND BEGINNINGS.- I.1. Beginnings and first examples.- I.2. Further quantized coordinate rings.- I.3. The quantized enveloping algebra of sC2(k).- I.4. The finite dimensional representations of Uq(5r2(k)).- I.5. Primer on semisimple Lie algebras.- I.6. Structure and representation theory of Uq(g) with q generic.- I.7. Generic quantized coordinate rings of semisimple groups.- I.8. 0q(G) is a noetherian domain.- I.9. Bialgebras and Hopf algebras.- I.10. R-matrices.- I.11. The Diamond Lemma.- I.12. Filtered and graded rings.- I.13. Polynomial identity algebras.- I.14. Skew polynomial rings satisfying a polynomial identity.- I.15. Homological conditions.- I.16. Links and blocks.- II. GENERIC QUANTIZED COORDINATE RINGS.- II.1. The prime spectrum.- II.2. Stratification.- II.3. Proof of the Stratification Theorem.- II.4. Prime ideals in 0q (G).- II.5. H-primes in iterated skew polynomial algebras.- II.6. More on iterated skew polynomial algebras.- II.7. The primitive spectrum.- II.8. The Dixmier-Moeglin equivalence.- II.9. Catenarity.- II.10. Problems and conjectures.- III. QUANTIZED ALGEBRAS AT ROOTS OF UNITY.- III.1. Finite dimensional modules for affine PI algebras.- 1II.2. The finite dimensional representations of UE(5C2(k)).- II1.3. The finite dimensional representations of OE(SL2(k)).- III.4. Basic properties of PI Hopf triples.- III.5. Poisson structures.- 1II.6. Structure of U, (g).- III.7. Structure and representations of 0,(G).- III.8. Homological properties and the Azumaya locus.- II1.9. Muller's Theorem and blocks.- III.10. Problems and perspectives.

Lie Groups: An Introduction through Linear Groups

Abstract: Preface 1 The exponential map 2 Lie theory 3 The classical groups 4 Manifolds, homogeneous spaces, Lie groups 5 Integration 6 Representations Appendix: Analytic Functions and Inverse Function Theorem References Index

Localization of modules for a semisimple Lie algebra in prime characteristic

Abstract: We show that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of D-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra Thus the “derived” version of the Beilinson-Bernstein localization theorem holds in sufficiently large positive characteristic Next, one finds that for any smooth variety this algebra of differential operators is an Azumaya algebra on the cotangent bundle In the case of the flag variety it splits on Springer fibers, and this allows us to pass from D-modules to coherent sheaves The argument also generalizes to twisted D-modules As an application we prove Lusztig’s conjecture on the number of irreducible modules with a fixed central character We also give a formula for behavior of dimension of a module under translation functors and reprove the Kac-Weisfeiler conjecture The sequel to this paper [BMR2] treats singular infinitesimal characters

Controllability of quantum mechanical systems by root space decomposition of su(N)

Abstract: The controllability property of the unitary propagator of an N-level quantum mechanical system subject to a single control field is described using the structure theory of semisimple Lie algebras Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases

Unitary Representations of Uq(??}(2,ℝ)),¶the Modular Double and the Multiparticle q-Deformed¶Toda Chain

Abstract: The paper deals with the analytic theory of the quantum q-deformed Toda chains; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L. Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin–Barnes type. For the periodic chain the two dual Baxter equations are derived.

Paths, crystals and fermionic formulae

Abstract: We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one-dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowsky—Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one-dimensional sums conjecturally give rise to branching functions of an integrable G 2 (1) -module related to the embedding G 2 (1) ↪ B 3 (1) ↪ D 4 1 .

Banach space representations and Iwasawa theory

Abstract: We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringoK[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗oK[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL2(ℤp).

Several Complex Variables With Connections to Algebraic Geometry and Lie Groups

Abstract: Selected problems in one complex variable Holomorphic functions of several variables Local rings and varieties The Nullstellensatz Dimension Homological algebra Sheaves and sheaf cohomology Coherent algebraic sheaves Coherent analytic sheaves Stein spaces Frechet sheaves--Cartan's theorems Projective varieties Algebraic vs. analytic--Serre's theorems Lie groups and their representations Algebraic groups The Borel-Weil-Bott theorem Bibliography Index.

Fredholm determinants, Jimbo‐Miwa‐Ueno τ‐functions, and representation theory

Abstract: The authors show that a wide class of Fredholm determinants arising in the representation theory of “big” groups, such as the infinite-dimensional unitary group, solve Painleve equations. Their methods are based on the theory of integrable operators and the theory of Riemann-Hilbert problems. © 2002 Wiley Periodicals, Inc.

A Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems

Abstract: Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product C*-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green's Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have C*-algebras with actions or coactions (or both) of a fixed locally compact group G as their objects, and equivariant equivalence classes of right-Hilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules. The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction of representations.

Gelfand-Tsetlin bases for classical Lie algebras

Abstract: This is a review paper on the Gelfand-Tsetlin type bases for representations of the classical Lie algebras. Different approaches to construct the original Gelfand-Tsetlin bases for representations of the general linear Lie algebra are discussed. Weight basis constructions for representations of the orthogonal and symplectic Lie algebras are reviewed. These rely on the representation theory of the B,C,D type twisted Yangians

Dynamics of Subgroup Actions on Homogeneous Spaces of Lie Groups and Applications to Number Theory

Abstract: Publisher Summary This chapter presents an exposition of homogeneous dynamics—that is, the dynamical and ergodic properties of actions on the homogeneous spaces of Lie groups. Many concepts of the modern theory of dynamical systems appeared in connection with the study of the geodesic flow on a compact surface of constant negative curvature. The algebraic nature of the phase space and the action itself allows obtaining much more advanced results as compared to the general theory of smooth dynamical systems. This can be seen in the example of smooth flows with polynomial divergence of trajectories. Homogeneous actions are discussed in the chapter and some basic examples include rectilinear flow on a torus, solvable flows on a three-dimensional locally Euclidean manifold, suspensions of toral automorphisms, nilflows on homogeneous spaces of the three-dimensional Heisenberg group, the geodesic and horocycle flows on a constant negative curvature surface, and geodesic flows on locally symmetric Riemannian spaces. The chapter presents the main link between homogeneous actions and number theory (Mahler's criterion and its consequences).

Essential Stability Theory

Abstract: Stability theory began in the early 1960s with the work of Michael Morley, and progressed in the 1970s through Shelah's research in model-theoretic classification theory. In the mid-1990s, stability theory both influences and is influenced by number theory, algebraic group theory, Riemann surfaces and representation theory of modules. The aim of this text is to provide students with a quick route from basic model theory to research in stability theory, and to give an introduction to classification theory with an exposition of Morley's categoricity theorem.

Baby Verma modules for rational Cherednik algebras

Abstract: Symplectic reflection algebras arise in many different mathematical disciplines: integrable systems, Lie theory, representation theory, differential operators, symplectic geometry. In this paper, we introduce baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter t=0 (the so--called rational Cherednik algebras at parameter t=0) and present their most basic properties. By analogy with the representation theory of reductive Lie algebras in positive characteristic, we believe these modules are fundamental to the understanding of the representation theory and associated geometry of the rational Cherednik algebras at parameter t=0. As an example, we use baby Verma modules to answer several problems posed by Etingof and Ginzburg, and give an elementary proof of a theorem of Finkelberg and Ginzburg.

Construction and classification of complex simple Lie algebras via projective geometry

Abstract: We construct the complex simple Lie algebras using elementary algebraic geometry. We use our construction to obtain a new proof of the classification of complex simple Lie algebras that does not appeal to the classification of root systems.

Point Groups and Space Groups in Geometric Algebra

Abstract: Geometric algebra provides the essential foundation for a new approach to symmetry groups. Each of the 32 lattice point groups and 230 space groups in three dimensions is generated from a set of three symmetry vectors. This greatly facilitates representation, analysis and application of the groups to molecular modeling and crystallography.

Algebras whose groups of units are Lie groups

Abstract: Let A be a locally convex, unital topological algebra whose group of units A is open and such that inversion : A ! A is continuous. Then inversion is analytic, and thus A is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then A has a locally dieomorphic ex- ponential function and multiplication is given locally by the Baker{Campbell{Hausdor series. In contrast, for suitable non-Mackey complete A, the unit group A is an analytic Lie group without a globally dened exponential function. We also discuss generalizations in the setting of \convenient dieren tial calculus", and describe various examples.

Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group

Abstract: Let F denote a complete nonarchimedean local field with perfect residue field. Let G be a connected reductive group defined over F with Lie algebra g. This paper exploits the formalism of Moy and Prasad to sharpen and extend familiar harmonic analysis results for g = g(F ). We show that the G-orbits of the Moy-Prasad filtration lattices are asymptotic to the set of nilpotent elements in g. In g we define G-domains in terms of the filtration lattices and explore their properties. We then show that the domain where the local expansion for G-invariant distributions on g is valid behaves well with respect to parabolic induction.

Dual canonical bases, quantum shuffles and q-characters

Abstract: Rosso and Green have shown how to embed the positive part $U_q(n)$ of a quantum enveloping algebra $U_q(g)$ in a quantum shuffle algebra. In this paper we study some properties of the image of the dual canonical basis $B^*$ of $U_q(n)$ under this embedding $\Phi$. This is motivated by the fact that when $g$ is of type $A_r$, the elements of $\Phi(B^*)$ are $q$-analogues of irreducible characters of the affine Iwahori-Hecke algebras attached to the groups $GL(m)$ over a $p$-adic field.

Classification of graded Hecke algebras for complex reflection groups

Abstract: The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups.

Some Comments on the Representation Theory of the Algebra Underlying Loop Quantum Gravity

Abstract: Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance. Recently, in the context of the quest for semiclassical states, states of the theory in which the quantum gravitational field is close to some classical geometry, it was realized that it might also be worthwhile to study different representations of the algebra A of observables. The content of the present note is the observation that under some mild assumptions, the mathematical structure of representations of A can be analyzed rather effortlessly, to a certain extent: Each representation can be labeled by sets of functions and measures on the space of (generalized) connections that fulfill certain conditions. These considerations are however mostly of mathematical nature. Their physical content remains to be clarified, and physically interesting examples are yet to be constructed.

Representation theory of symmetric groups and their double covers

Abstract: In this article we will give an overview of the new Lie theoretic approach to the p-modular representation theory of the symmetric groups and their double covers that has emerged in the last few years. There are in fact two parallel theories here: one for the symmetric groups Sn involving the affine Kac-Moody algebra of type A p−1, and one for their double covers Ŝn involving the twisted algebra of type A p−1. In the case of Sn itself, the theory has been developed especially by Kleshchev [19], Lascoux-Leclerc-Thibon [21], Ariki [1] and Grojnowski [9], while the double covers are treated for the first time in [4] along the lines of [9], after the important progress made over C by Sergeev [35, 36] and Nazarov [30, 31]. One of the most striking results at the heart of both of the theories is the explicit description of the modular branching graphs in terms of Kashiwara’s crystal graph for the basic module of the corresponding affine Lie algebra. Note that the results described are just a part of a larger picture: there are analogous results for the cyclotomic and affine Hecke algebras, and their twisted analogues, the cyclotomic and affine Hecke-Clifford superalgebras. However we will try here to bring out only those parts of the theory that apply to the symmetric group, since that is the most applicable to finite group theory.

Asymptotic combinatorics with application to mathematical physics

Abstract: Preface. Program. List of participants. Part One: Matrix Models and Graph Enumeration. Matrix Quantum Mechanics V. Kazakov. Introduction to matrix models E. Brezin. A Class of the Multi-Interval Eigenvalue Distributions of Matrix Models and Related Structures V. Buslaev, L. Pastur. Combinatorics and Probability of Maps V.A. Malyshev. The Combinatorics of Alternating Tangles: from theory to computerized enumeration J.L. Jacobsen, P. Zinn-Justin. Invariance Principles for Non-uniform Random Mappings and Trees D. Aldous, J. Pitman. Part Two: Integrable Models (of Statistical Physics and Quantum Field Theory). Renormalization group solution of fermionic Dyson model M.D. Missarov. Statistical Mechanics and Number Theory H.E. Boos, V.E. Korepin. Quantization of Thermodynamics and the Bardeen-Cooper-Schriffer-Bogolyubov Equation V.P. Maslov. Approximate Distribution of Hitting Probabilities for a Regular Surface with Compact Support in 2D D.S. Grebenkov. Part Three: Representation Theory. Notes on homogeneous vector bundles over complex flag manifolds S. Igonin. Representation Theory and Doubles of Yangians of Classical Lie Superalgebras V. Stukopin. Idempotent (asymptotic) Mathematics and the Representation theory G.L. Litvinov, et al. A new approach to Berezin kernels and canonical representations G. van Dijk. Theta Hypergeometric Series V.P. Spiridonov.

Representation Theory and Quantum Inverse Scattering Method: Open Toda Chain and Hyperbolic Sutherland Model

Abstract: Using the representation theory of $\frak{gl}(N,\RR)$ we explain the expression through integrals for $GL(N,\RR)$ Toda chain wave function obtained recently by Quantum Inverse Scattering Method. The main tool is the generalization of the Gelfand-Tsetlin method to the case of the infinite-dimensional representations of $\frak{gl}(N,\RR)$. The interpretation of this generalized construction in terms of the coadjoint orbits is given and the connection with Yangian $Y(\frak{gl}(N))$ is discussed. We also provide the representation by integrals for the hyperbolic Sutherlend model eigenfunctions in Gelfand-Tsetlin representation. Using the example of the open Toda chain we discuss the connection between the Quantum Inverse Scattering Method and Representation Theory.

Sobolev norms of automorphic functionals

Abstract: 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.

Tilting modules for Lie superalgebras

Abstract: This is a companion article to my papers on Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebras gl(m|n) (much revised!) and q(n). The goal is to develop the general theory of tilting modules for Lie superalgebras, working in a general graded setting very similar to work of Soergel (Character formulae for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432-438) in the Lie algebra case. Examples are given involving the Lie superalgebras gl(m|n) and q(n), but maybe this will also be useful for the other classical and affine Lie superalgebras.

The Adjoint Representation and the Adjoint Action

Abstract: The purpose of this article is to study in detail the actions of a semisimple Lie or algebraic group on its Lie algebra by the adjoint representation and on itself by the adjoint action. We will focus primarily on orbits through nilpotent elements in the Lie algebra; these are called nilpotent orbits for short. Many deep results about such orbits have been obtained in the last thirty-five years; we will collect some of the most significant of these that have found wide application to representation theory. We will primarily work in the setting of a semisimple Lie algebra and its adjoint group over an algebraically closed field of characteristic zero, but we will extend much of what we do to semisimple Lie algebras over the reals or an algebraically closed field of prime characteristic, and to conjugacy classes in semisimple algebraic groups. We will give detailed proofs of many results, including some which are difficult to ferret out of the literature. Other results will be summarized with reasonably complete references. The treatment is a more comprehensive version of that in [CM93]; there is also some overlap with Humphreys’s book [Hu95]. In the last chapter we summarize some of the most recent work being done in this topic and indicate some directions of current research.