Showing papers on "Representation theory published in 1993"

Analysis and Geometry on Groups

Abstract: The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical but have little to do with what is described these days as real analysis. Most of the results described in this book have a dual formulation; they have a 'discrete version' related to a finitely generated discrete group, and a continuous version related to a Lie group. The authors chose to centre this book around Lie groups but could quite easily have pushed it in several other directions as it interacts with opetators, and probability theory, as well as with group theory. This book will serve as an excellent basis for graduate courses in Lie groups, Markov chains or potential theory.

Nilpotent Orbits In Semisimple Lie Algebra: An Introduction

Abstract: Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques used are elementary and in the toolkit of any graduate student interested in the harmonic analysis of representation theory of Lie groups. The book develops the Dynkin-Konstant and Bala-Carter classifications of complex nilpotent orbits, derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discusses basic topological questions, and classifies real nilpotent orbits. The classical algebras are emphasized throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. The authors conclude with a survey of advanced topics related to the above circle of ideas. This book is the product of a two-quarter course taught at the University of Washington.

Polynomial invariants of finite groups

Abstract: This is the first book to deal with invariant theory and the representations of finite groups. By restricting attention to finite groups Dr Benson is able to avoid recourse to the technical machinery of algebraic groups, and he develops the necessary results from commutative algebra as he proceeds. Thus the book should be accessible to graduate students. In detail, the book contains an account of invariant theory for the action of a finite group on the ring of polynomial functions on a linear representation, both in characteristic zero and characteristic p. Special attention is paid to the role played by pseudoreflections, which arise because they correspond to the divisors in the polynomial ring which ramify over the invariants. Also included is a new proof by Crawley-Boevey and the author of the Carlisle-Kropholler conjecture. This volume will appeal to all algebraists, but especially those working in representation theory, group theory, and commutative or homological algebra.

Representations of Solvable Groups

Abstract: Representation theory plays an important role in algebra, and in this book Manz and Wolf concentrate on that part of the theory which relates to solvable groups. The authors begin by studying modules over finite fields, which arise naturally as chief factors of solvable groups. The information obtained can then be applied to infinite modules, and in particular to character theory (ordinary and Brauer) of solvable groups. The authors include proofs of Brauer's height zero conjecture and the Alperin-McKay conjecture for solvable groups. Gluck's permutation lemma and Huppert's classification of solvable two-transive permutation groups, which are essentially results about finite modules of finite groups, play important roles in the applications and a new proof is given of the latter. Researchers into group theory, representation theory, or both, will find that this book has much to offer.

Diagonalization of the $XXZ$ Hamiltonian by vertex operators

Abstract: We diagonalize the anti-ferroelectricXXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}(2))$$ . Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit—thesu(2)-invariant Thirring model.

Representation Theory of Analytic Holonomy C* Algebras

Abstract: Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections. The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum theory.

On Poisson homogeneous spaces of Poisson-Lie groups

Abstract: Poisson homogeneous spaces of a Poisson-Lie group G are described in terms of Lagrangian subalgebras of D(g), where D(g) is the double of the Lie bialgebra g corresponding to G.

Lie-groups as Spin groups.

Abstract: It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.

Representation Theory of Analytic Holonomy C* Algebras

Abstract: Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections. The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum theory.

Representations and characters of groups

Abstract: This book provides a modern introduction to the representation theory of finite groups. Now in its second edition, the authors have revised the text and added much new material. The theory is developed in terms of modules, since this is appropriate for more advanced work, but considerable emphasis is placed upon constructing characters. Included here are the character tables of all groups of order less than 32, and all simple groups of order less than 1000. Applications covered include Burnside's paqb theorem, the use of character theory in studying subgroup structure and permutation groups, and how to use representation theory to investigate molecular vibration. Each chapter features a variety of exercises, with full solutions provided at the end of the book. This will be ideal as a course text in representation theory, and in view of the applications, will be of interest to chemists and physicists as well as mathematicians.

Quantum groups, quantum categories, and quantum field theory

Abstract: and survey of results.- Local quantum theory with braid group statistics.- Superselection sectors and the structure of fusion rule algebras.- Hopf algebras and quantum groups at roots of unity.- Representation theory of U q red (s? 2).- Path representations of the braid groups for quantum groups at roots of unity.- Duality theory for local quantum theories, dimensions and balancing in quantum categories.- The quantum categories with a generator of dimension less than two.

Foundations of Lie theory ; Lie transformation groups

Abstract: IFoundations of Lie Theory- 1 Basic Notions- 1 Lie Groups, Subgroups and Homomorphisms- 11 Definition of a Lie Group- 12 Lie Subgroups- 13 Homomorphisms of Lie Groups- 14 Linear Representations of Lie Groups- 15 Local Lie Groups- 2 Actions of Lie Groups- 21 Definition of an Action- 22 Orbits and Stabilizers- 23 Images and Kernels of Homomorphisms- 24 Orbits of Compact Lie Groups- 3 Coset Manifolds and Quotients of Lie Groups- 31 Coset Manifolds- 32 Lie Quotient Groups- 33 The Transitive Action Theorem and the Epimorphism Theorem- 34 The Pre-image of a Lie Group Under a Homomorphism- 35 Semidirect Products of Lie Groups- 4 Connectedness and Simply-connectedness of Lie Groups- 41 Connected Components of a Lie Group- 42 Investigation of Connectedness of the Classical Lie Groups- 43 Covering Homomorphisms- 44 The Universal Covering Lie Group- 45 Investigation of Simply-connectedness of the Classical Lie Groups- 2 The Relation Between Lie Groups and Lie Algebras- 1 The Lie Functor- 11 The Tangent Algebra of a Lie Group- 12 Vector Fields on a Lie Group- 13 The Differential of a Homomorphism of Lie Groups- 14 The Differential of an Action of a Lie Group- 15 The Tangent Algebra of a Stabilizer- 16 The Adjoint Representation- 2 Integration of Homomorphisms of Lie Algebras- 21 The Differential Equation of a Path in a Lie Group- 22 The Uniqueness Theorem- 23 Virtual Lie Subgroups- 24 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra- 25 Deformations of Paths in Lie Groups- 26 The Existence Theorem- 27 Abelian Lie Groups- 3 The Exponential Map- 31 One-Parameter Subgroups- 32 Definition and Basic Properties of the Exponential Map- 33 The Differential of the Exponential Map- 34 The Exponential Map in the Full Linear Group- 35 Cartan's Theorem- 36 The Subgroup of Fixed Points of an Automorphism of a Lie Group- 4 Automorphisms and Derivations- 41 The Group of Automorphisms- 42 The Algebra of Derivations- 43 The Tangent Algebra of a Semi-Direct Product of Lie Groups- 5 The Commutator Subgroup and the Radical- 51 The Commutator Subgroup- 52 The Maltsev Closure- 53 The Structure of Virtual Lie Subgroups- 54 Mutual Commutator Subgroups- 55 Solvable Lie Groups- 56 The Radical- 57 Nilpotent Lie Groups- 3 The Universal Enveloping Algebra- 1 The Simplest Properties of Universal Enveloping Algebras- 11 Definition and Construction- 12 The Poincare-Birkhoff-Witt Theorem- 13 Symmetrization- 14 The Center of the Universal Enveloping Algebra- 15 The Skew-Field of Fractions of the Universal Enveloping Algebra- 2 Bialgebras Associated with Lie Algebras and Lie Groups- 21 Bialgebras- 22 Right Invariant Differential Operators on a Lie Group- 23 Bialgebras Associated with a Lie Group- 3 The Campbell-Hausdorff Formula- 31 Free Lie Algebras- 32 The Campbell-Hausdorff Series- 33 Convergence of the Campbell-Hausdorff Series- 4 Generalizations of Lie Groups- 1 Lie Groups over Complete Valued Fields- 11 Valued Fields- 12 Basic Definitions and Examples- 13 Actions of Lie Groups- 14 Standard Lie Groups over a Non-archimedean Field- 15 Tangent Algebras of Lie Groups- 2 Formal Groups- 21 Definition and Simplest Properties- 22 The Tangent Algebra of a Formal Group- 23 The Bialgebra Associated with a Formal Group- 3 Infinite-Dimensional Lie Groups- 31 Banach Lie Groups- 32 The Correspondence Between Banach Lie Groups and Banach Lie Algebras- 33 Actions of Banach Lie Groups on Finite-Dimensional Manifolds- 34 Lie-Frechet Groups- 35 ILB- and ILH-Lie Groups- 4 Lie Groups and Topological Groups- 41 Continuous Homomorphisms of Lie Groups- 42 Hilbert's 5-th Problem- 5 Analytic Loops- 51 Basic Definitions and Examples- 52 The Tangent Algebra of an Analytic Loop- 53 The Tangent Algebra of a Diassociative Loop- 54 The Tangent Algebra of a Bol Loop- References- II Lie Transformation Groups- 1 Lie Group Actions on Manifolds- 1 Introductory Concepts- 11 Basic Definitions- 12 Some Examples and Special Cases- 13 Local Actions- 14 Orbits and Stabilizers- 15 Representation in the Space of Functions- 2 Infinitesimal Study of Actions- 21 Flows and Vector Fields- 22 Infinitesimal Description of Actions and Morphisms- 23 Existence Theorems- 24 Groups of Automorphisms of Certain Geometric Structures- 3 Fibre Bundles- 31 Fibre Bundles with a Structure Group- 32 Examples of Fibre Bundles- 33 G-bundles- 34 Induced Bundles and the Classification Theorem- 2 Transitive Actions- 1 Group Models- 11 Definitions and Examples- 12 Basic Problems- 13 The Group of Automorphisms- 14 Primitive Actions- 2 Some Facts Concerning Topology of Homogeneous Spaces- 21 Covering Spaces- 22 Real Cohomology of Lie Groups- 23 Subgroups with Maximal Exponent in Simple Lie Groups- 24 Some Homotopy Invariants of Homogeneous Spaces- 3 Homogeneous Bundles- 31 Invariant Sections and Classification of Homogeneous Bundles- 32 Homogeneous Vector Bundles The Frobenius Duality- 33 The Linear Isotropy Representation and Invariant Vector Fields- 34 Invariant A-structures- 35 Invariant Integration- 36 Karpelevich-Mostow Bundles- 4 Inclusions Among Transitive Actions- 41 Reductions of Transitive Actions and Factorization of Groups- 42 The Natural Enlargement of an Action- 43 Some Inclusions Among Transitive Actions on Spheres- 44 Factorizations of Lie Groups and Lie Algebras- 45 Factorizations of Compact Lie Groups- 46 Compact Enlargements of Transitive Actions of Simple Lie Groups- 47 Groups of Isometries of Riemannian Homogeneous Spaces of Simple Compact Lie Groups- 48 Groups of Automorphisms of Simply Connected Homogeneous Compact Complex Manifolds- 3 Actions of Compact Lie Groups- 1 The General Theory of Compact Lie Transformation Groups- 11 Proper Actions- 12 Existence of Slices- 13 Two Fiberings of an Equi-orbital G-space- 14 Principal Orbits- 15 Orbit Structure- 16 Linearization of Actions- 17 Lifting of Actions- 2 Invariants and Almost-Invariants- 21 Applications of Invariant Integration- 22 Finiteness Theorems for Invariants- 23 Finiteness Theorems for Almost Invariants- 3 Applications to Homogeneous Spaces of Reductive Groups- 31 Complexification of Homogeneous Spaces- 32 Factorization of Reductive Algebraic Groups and Lie Algebras- 4 Homogeneous Spaces of Nilpotent and Solvable Groups- 1 Nilmanifolds- 11 Examples of Nilmanifolds- 12 Topology of Arbitrary Nilmanifolds- 13 Structure of Compact Nilmanifolds- 14 Compact Nilmanifolds as Towers of Principal Bundles with Fibre T1- 2 Solvmanifolds- 21 Examples of Solvmanifolds- 22 Solvmanifolds and Vector Bundles- 23 Compact Solvmanifolds (The Structure Theorem)- 24 The Fundamental Group of a Solvmanifold- 25 The Tangent Bundle of a Compact Solvmanifold- 26 Transitive Actions of Lie Groups on Compact Solvmanifolds- 27 The Case of Discrete Stabilizers- 28 Homogeneous Spaces of Solvable Lie Groups of Type (I)- 29 Complex Compact Solvmanifolds- 5 Compact Homogeneous Spaces- 1 Uniform Subgroups- 11 Algebraic Uniform Subgroups- 12 Tits Bundles- 13 Uniform Subgroups of Semi-simple Lie Groups- 14 Connected Uniform Subgroups- 15 Reductions of Transitive Actions of Reductive Lie Groups- 2 Transitive Actions on Compact Homogeneous Spaces with Finite Fundamental Groups- 21 Three Lemmas on Transitive Actions- 22 Radical Enlargements- 23 A Sufficient Condition for the Radical to be Abelian- 24 Passage from Compact Groups to Non-Compact Semi-simple Groups- 25 Compact Homogeneous Spaces of Rank 1- 26 Transitive Actions of Non-Compact Lie Groups on Spheres- 27 Existence of Maximal and Largest Enlargements- 3 The Natural Bundle- 31 Orbits of the Action of a Maximal Compact Subgroup- 32 Construction of the Natural Bundle and Its Properties- 33 Some Examples of Natural Bundles- 34 On the Uniqueness of the Natural Bundle- 35 The Case of Low Dimension of Fibre and Basis- 4 The Structure Bundle- 41 Regular Transitive Actions of Lie Groups- 42 The Structure of the Base of the Natural Bundle- 43 Some Examples of Structure Bundles- 5 The Fundamental Group- 51 On the Concept of Commensurability of Groups- 52 Embedding of the Fundamental Group in a Lie Group- 53 Solvable and Semi-simple Components- 54 Cohomological Dimension- 55 The Euler Characteristic- 56 The Number of Ends- 6 Some Classes of Compact Homogeneous Spaces- 61 Three Components of a Compact Homogeneous Space and the Case when Two of them Are Trivial- 62 The Case of One Trivial Component- 7 Aspherical Compact Homogeneous Spaces- 71 Group Models of Aspherical Compact Homogeneous Spaces- 72 On the Fundamental Group- 8 Semi-simple Compact Homogeneous Spaces- 81 Transitivity of a Semi-simple Subgroup- 82 The Fundamental Group- 83 On the Fibre of the Natural Bundle- 9 Solvable Compact Homogeneous Spaces- 91 Properties of the Natural Bundle- 92 Elementary Solvable Homogeneous Spaces- 10 Compact Homogeneous Spaces with Discrete Stabilizers- 6 Actions of Lie Groups on Low-dimensional Manifolds- 1 Classification of Local Actions- 11 Notes on Local Actions- 12 Classification of Local Actions of Lie Groups on ?1, ?1- 13 Classification of Local Actions of Lie Groups on ?2 and ?2- 2 Homogeneous Spaces of Dimension ?3- 21 One-dimensional Homogeneous Spaces- 22 Two-dimensional Homogeneous Spaces (Homogeneous Surfaces)- 23 Three-dimensional Manifolds- 3 Compact Homogeneous Manifolds of Low Dimension- 31 On Four-dimensional Compact Homogeneous Manifolds- 32 Compact Homogeneous Manifolds of Dimension ?6- 33 On Compact Homogeneous Manifolds of Dimension ?7- References

Lie semigroups and their applications

Abstract: Lie semigroups and their tangent wedges.- Examples.- Geometry and topology of Lie semigroups.- Ordered homogeneous spaces.- Applications of ordered spaces to Lie semigroups.- Maximal semigroups in groups with cocompact radical.- Invariant Cones and Ol'shanskii semigroups.- Compression semigroups.- Representation theory.- The theory for Sl(2).

Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n))

Abstract: Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction. The representation theory is rather similar to that of gl(m/n) at generic q, but drastically different when q is a root of unity. In the latter case, atypicality conditions of highest weight irreducible representations (irreps) are substantially altered, and such finite‐dimensional irreps arise that do not have highest weight and/or lowest weight vectors. As concrete examples, the irreps of Uq(gl(2/1)) are classified.

Relative homology and representation theory 1

Abstract: (1993). Relative homology and representation theory III. Communications in Algebra: Vol. 21, No. 9, pp. 3081-3097.

A quiver quantum group

Abstract: We construct quantum groups at a root of unity and we describe their monoidal module category using techniques from the representation theory of finite dimensional associative algebras.

Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups

Abstract: Preface Part I. Semi-Simple Modules: Part II. Projective Modules: Part III. Modules and Subgroups: Part IV. Blocks: Part V. Cyclic Blocks.

7-dimensional nilpotent Lie algebras

Abstract: All 7-dimensional nilpotent Lie algebras over C are determined by elementary methods. A multiplication table is given for each isomorphism class. Distinguishing features are given, proving that the algebras are pairwise nonisomorphic. Moduli are given for the infinite families which are indexed by the value of a complex parameter

Some quantum analogues of solvable Lie groups

Abstract: In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commutative rings. In particular, we find maximal dimensions of their irreducible representations. Our results confirm the validity of the general philosophy that the representation theory is intimately connected to the Poisson geometry.

Macdonald's polynomials and representations of quantum groups

Abstract: In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining operators between certain modules over quantum sl(n). We also describe the commutative system of Macdonald's difference operators using the generators of the center of the quantum universal enveloping algebra, and use this description to prove a trace formula for generic eigenfunctions of these operators. These functions are generalized q-hypergeometric functions which are related to solutions of the quantum Knizhnik-Zamolodchikov equations.

On the theory of Frobenius extensions and its application to Lie superalgebras

Abstract: By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory

Algebraic integrability for the Schrödinger equation and finite reflection groups

Abstract: Algebraic integrability of ann-dimensional Schrodinger equation means that it has more thann independent quantum integrals. Forn=1, the problem of describing such equations arose in the theory of finite-gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrodinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero—Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero—Sutherland problem for a special value of the coupling constant.

Irreducible unitary representations of quantum Lorentz group

Abstract: A complete classification of irreducible unitary representations of a one parameter deformationSqL(2,C) (0

Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups

Abstract: The Steenrod algebra structures of H*(BG; Z/p) for compact Lie groups are studied Using these, Brown-Peterson cohomology and Morava K-theory are computed for many concrete cases All these cases have properties similar as torsion free Lie groups or finite groups, eg, BP odd (BG) = 0

Sums of adjoint orbits

Abstract: We investigate a natural generalization of the problem of the description of the eigenvalues of the sum of two Hermitian matrices both of whose eigenvalues are known We describe more generally the convolution of the invariant probability measures supported on any two adjoint orbits of a compact Lie group Our techniques utilize the convexity results of Guillemin and Sternberg and Kirwan on the one hand, and the character formulae of Weyl and Kirillov on the other Applications to representation theory are discussed

Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space

Abstract: This is an expanded and updated version of a talk given at the Conference on Topics in Geometry and Physics at the University of Southern California, November 6, 1992. It is a survey talk, aimed at mathematicians AND physicists, which attempts to bring together the topics in the title without assuming much background in any of them. Closed string field theory leads to a (strong homotopy) generalization of Lie algebra, which is strongly related to the way the moduli spaces $\Cal M_{0,N+1}$ fit together as an ``operad''. The latter in turn plays an important role in the understanding of vertex operator algebras.