Showing papers on "Representation theory published in 1992"

Algebraic groups and number theory

Abstract: (Chapter Heading): Algebraic Number Theory. Algebraic Groups. Algebraic Groups over Locally Compact Fields. Arithmetic Groups and Reduction Theory. Adeles. Galois Cohomology. Approximation in Algebraic Groups. Class Numbers andClass Groups of Algebraic Groups. Normal Structure of Groups of Rational Points of Algebraic Groups. Appendix A. Appendix B: Basic Notation. Algebraic Number Theory: Algebraic Number Fields, Valuations, and Completions. Adeles and Ideles Strong and Weak Approximation The Local-Global Principle. Cohomology. Simple Algebras over Local Fields. Simple Algebras over Algebraic Number Fields. Algebraic Groups: Structural Properties of Algebraic Groups. Classification of K-Forms Using Galois Cohomology. The Classical Groups. Some Results from Algebraic Geometry. Algebraic Groups over Locally Compact Fields: Topology and Analytic Structure. The Archimedean Case. The Non-Archimedean Case. Elements of Bruhat-Tits Theory. Results Needed from Measure Theory. Arithmetic Groups and Reduction Theory: Arithmetic Groups. Overview of Reduction Theory: Reduction in GLn(R).Reduction in Arbitrary Groups. Group-Theoretic Properties of Arithmetic Groups. Compactness of GR/GZ. The Finiteness of the Volume of GR/GZ. Concluding Remarks on Reduction Theory. Finite Arithmetic Groups. Adeles: Basic Definitions. Reduction Theory for GA Relative to GK. Criteria for the Compactness and the Finiteness of Volume of GA/GK. Reduction Theory for S-Arithmetic Subgroups. Galois Cohomology: Statement of the Main Results. Cohomology of Algebraic Groups over Finite Fields. Galois Cohomology of Algebraic Tori. Finiteness Theorems for Galios Cohomology. Cohomology of Semisimple Algebraic Groups over Local Fields and Number Fields. Galois Cohomology and Quadratic, Hermitian, and Other Forms. Proof of Theorems 6.4 and 6.6: Classical Groups. Proof of Theorems 6.4 and 6.6: Exceptional Groups. Approximation in Algebraic Groups: Strong and Weak Approximation in Algebraic Varieties. The Kneser-Tits Conjecture. Weak Approximation in Algebraic Groups. The Strong Approximation Theorem. Generalization of the Strong Approximation Theorem. Class Numbers and Class Groups of Algebraic Groups: Class Numbers of Algebraic Groups and Number of Classes in a Genus. Class Numbers and Class Groups of Semisimple Groups of Noncompact Type The Realization Theorem. Class Numbers of Algebraic Groups of Compact Type. Estimating the Class Number for Reductive Groups. The Genus Problem. Normal Subgroup Structure of Groups of Rational Points of Algebraic Groups: Main Conjecture and Results. Groups of Type An. The Classical Groups. Groups Split over a Quadratic Extension. The Congruence Subgroup Problem (A Survey). Appendices: Basic Notation. Bibliography. Index.

Vertex operator algebras associated to representations of affine and Virasoro Algebras

Abstract: The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac i-K] was greatly inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M]. Though the Macdonald identity found its natural context in representation theory, its mysterious modular invariance was not understood until the work of Witten [W-I on the geometric realization of representations of the loop groups corresponding to loop algebras. The work of Witten clearly indicated that the representations of loop groups possess a very rich structure of conformal field theory which appeared in physics literature in the work of Belavin, Polyakov, and Zamolodchikov [BPZ-I. Independently (though two years later), Borcherds, in an attempt to find a conceptual understanding of a certain algebra of vertex operators invariant under the Monster [FLM1], introduced in [B-I a new algebraic structure. We call vertex operator algebras a slightly modified version of Borcherd’s new algebras [FLM2].

Representations of the holonomy algebras of gravity and nonAbelian gauge theories

Abstract: Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a nonperturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C*-algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollary of these general results is a precise formulation of the 'loop transform' proposed by Rovelli and Smolin (1990). Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C*-algebra. The structure of this space is investigated and it is shown how observables labelled by 'strips' arise naturally.

Representations of the holonomy algebras of gravity and non-Abelian gauge theories

Abstract: Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C-star algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollory of these general results is a precise formulation of the ``loop transform'' proposed by Rovelli and Smolin. Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C-star algebra. The structure of this space is investigated and it is shown how observables labelled by ``strips'' arise naturally.

Infinite Dimensional Lie Superalgebras

Abstract: Publisher Summary This chapter describes an area which is comparatively new and presents the results of the few references used for recovery.. One of the main sources is the joint monograph [BMPZ]. An extensive use of the monograph is made. All algebras to be considered are vector spaces over K unless otherwise specified. These vector spaces are graded by elements of a commutative additive group G. For other pairs of elements, the commutator is defined by linearity. Ordinary Lie superalgebras find their origin in algebraic topology and find applications in various fields of mathematics. By modifying the operation of a colour Lie superalgebra, there can be a reduction of questions about these algebras to those about ordinary Lie superalgebras. One of the most important concepts, which is a powerful tool in the study of Lie superalgebras is the (universal) enveloping algebra U (L) of a colour Lie superalgebra L. A common way of obtaining superalgebras is through semidirect products. The problem of finding linear bases is important in the case of free Lie algebras, and also in the case of free algebras in varieties other than the variety of all Lie superalgebras.

Twisted yangians and infinite-dimensional classical Lie algebras

Abstract: The Yangians are quantized enveloping algebras of polynomial current Lie algebras and twisted Yangians should be their analogs for twisted polynomial current Lie algebras. We define and study certain examples of twisted Yangians and describe their relationship to a problem which arises in representation theory of infinite-dimensional classical groups.

Non-Abelian Harmonic Analysis : Applications of SL (2,R)

Abstract: This book discusses the representation theory of the group SL(2, R), and some applications of this theory. The emphasis is in fact on the applications, some of which are outside representation theory and some are to representation theory itself. The topics outside representation theory are mostly of substantial classical importance (Fourier analysis, Laplace equation, Huyghen's Principle, Ergodic theory), while those inside representation theory mostly concern themes that have been central to Harish-Chandra's development of harmonic analysis on semisimple groups. This mix of topics should appeal to non-specialists in representation theory by illustrating how the theory can offer new perspectives on familiar topics, and by offering some insight into some important themes in representation theory itself.

Categorical Frameworks for Stone Representation Theories

Abstract: The duality between topological spaces and lattices, first exploited by the famous Stone Representation Theorems [Stone 1936, 1937a, 1937b], is rightly regarded as one of the fundamental developments of twentieth century mathematics [Johnstone 1982]. The full expression of this duality in classical mathematics is seen in the relationship between topological spaces and frames/locales, culminating in the adjunction between topological spaces and locales [Papert-Papert 1958, Isbell 1972] and the resultant equivalence of sober topological spaces and spatial locales. For convenience, we dub this adjunction the Stone adjunction and the resulting (sometimes dual) categorical equivalences the Stone representation theory. The most complete account of these matters, and related issues such as the Stone-tech Compactification, may be found in [Johnstone 1982].

Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications

Abstract: Basics of Representation Theory. Commutative Harmonic Analysis. Representations of Compact and Finite Groups. Lie Groups SU(2) and SO(3). Classical Compact Lie Groups and Algebras. The Heisenberg Group and Semidirect Products. Representations of SL 2 . Lie Groups and Hamiltonian Mechanics. Appendices: Spectral Decomposition of Selfadjoint Operators. Integral Operators. A Primer on Riemannian Geometry: Geodesics, Connection, Curvature. References. List of Frequently Used Notations. Index.

Representation theory of finite W algebras

Abstract: In this paper we study the finitely generated algebras underlying $W$ algebras. These so called 'finite $W$ algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of $sl_2$ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite $W$ algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite $W$ symmetry. In the second part we BRST quantize the finite $W$ algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite $W$ algebras in one stroke. Explicit results for $sl_3$ and $sl_4$ are given. In the last part of the paper we study the representation theory of finite $W$ algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finite $W$ algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite $W$ algebras.

Affine Lie algebras and quantum groups : an introduction, with applications in conformal field theory

Abstract: 1 Semisimple Lie algebras 2 Affine Lie algebras 3 WZW theories 4 Quantum groups 5 Duality, fusion rules, and modular invariance Bibliography Index

From representations of quivers via Hall and Loewy algebras to quantum groups

Abstract: Let ∆ be a symmetric generalized Cartan matrix, and g = g(∆) the corresponding Kac–Moody Lie–algebra with triangular decomposition g = n−⊕h⊕n+ (see [K]. We denote by b+ = b+(∆) = h ⊕ n+ the Borel subalgebra. Let Uq(b+) be the quantization of the universal enveloping algebra of b+, it is defined by generators and relations as we will recall below. For ∆ of finite or affine type, we want to survey a construction of Uq(b+) using the representation theory of quivers, following [R2], [R3], [R4] and [R5].

Lattice Wess-Zumino-Witten Model and Quantum Groups

Abstract: Quantum groups play a role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left- and right-moving degrees of freedom. On one hand side, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in the Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice $sl(2)$-based current algebra and show how to use it to rigorously construct an exact solution of the quantum $SL(2)$ WZW model on lattice.

CONFORMAL FIELDS: A CLASS OF REPRESENTATIONS OF Vect(N)

Abstract: Vect(N), the algebra of vector fields in N dimensions, is studied. Some aspects of local differential geometry are formulated as Vect(N) representation theory. There is a new class of modules, conformal fields, whose restrictions to the subalgebra sl(N+1)⊂Vect(N) are finite-dimensional sl(N+1) representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match.

One-sided division absolute valued algebras

Abstract: We develop a structure theory for left divsion absolute valued algebras which shows, among other things, that the norm of such an algebra comes from an inner product. Moreover, we prove the existence of left division complete absolute valued algebras with left unit of arbitrary infinite hilbertian division and with the additional property that they have nonzero proper closed left ideals. Our construction involves results from the representation theory of the so called "Canonical Anticommutation Relations" in Quantum Mechanics. We also show that homomorphisms from complete normed algebras into arbitrary absolute valued algebras are contractive, hence automatically continuous.

An introduction to quantized Lie groups and algebras

Abstract: We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

Conformal fields: a class of representations of Vect(N)

Abstract: $Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose restrictions to the subalgebra $sl(N+1) \subset Vect(N)$ are finite-dimensional $sl(N+1)$ representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match.

Rank of quantum groups and braided groups in dual form

Abstract: We give a dual formulation of recent work on the representation theory of general quantum groups. These form a rigid quasitensor category C to which is associated a braided group Aut(C) of braided-commutative “co-ordinate functions” analogous to the ring of functions on a group or supergroup. Every dual quasitriangular Hopf algebra A gives rise to such a braided group A. We give the example of the braided group BSL(2) in detail. We also give the rank of quantum groups in dual form and explain its connection with the partition function of simple quantum systems.

A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices

Abstract: We exhibit a nice Frobenius splitting σ on G× b where b is the Lie algebra of the Borel group B of upper triangular matrices in the general linear group G = Gln. What is nice about it, is that it descends along familiar maps and specializes to familiar subvarieties in a manner that is useful for the study of the singularities of closures of conjugacy classes of nilpotent n by n matrices. In particular, we show that these closures are simultaneously Frobenius split, are normal, and have rational singularities. The result on rational singularities is derived from a general vanishing theorem that will be proved in our paper [15]. Note that normality has already been proved by Donkin in [3]. His method uses a lot of representation theory and employs resolutions of the closures of conjugacy classes invented by Kraft and Procesi. An alternative approach to these singularities has been given by G. Lusztig. In [11] he showed that the same singularities occur in Schubert varieties for Kac-Moody groups of affine Weyl groups. Now Schubert varieties for such infinite dimensional groups are mastered in Mathieu’s book [12], where Mathieu shows they are normal and have rational singularities. In contrast with this, our work remains in finite dimensions. It relies on explicit formulas. Indeed the formula for our splitting σ is given by a product of principal minors and the specialization of the splitting to subvarieties is based on an inspection of what happens to the determinants. To descend σ to the Lie algebra g of G, (along the natural map G×b → g, cf. Grothendieck’s “simultaneous resolution” [2]), we use a Galois theoretic argument. We find that above the generic point of g the action of the Weyl group on σ is trivial. As preparation for that computation we first spell out trivializations of the canonical bundles of G× b and G× t.

Group Rings and Class Groups

Abstract: I Some general facts.- II Some notes on representation theory.- III The leading coefficient of units.- IV Class sum correspondence.- V More on the class sum correspondence.- VI Subgroup rigidity.- VII Global units.- VIII Locally isomorphic group rings.- IX Zassenhaus conjecture.- X Variations of the Zassenhaus conjecture.- XI Group Extensions.- XII Class sums of p-elements.- XIII Clifford theory revisited.- XIV Examples.- I Introduction and Review of the Tame Case.- II Hopf Orders.- III Principal Hornogeneous Spaces.- IV Arithmetic Applications:- The Cyclotomic Case.- V Arithmetic Applications:- The Elliptic Case.- References.

Complex quantum groups and their quantum enveloping algebras

Abstract: We construct complexified versions of the quantum groups associated with the Lie algebras of typeA n−1 ,B n ,C n , andD n . Following the ideas of Faddeev, Reshetikhin and Takhtajan we obtain the Hopf algebras of regular functionals Uℛ on these complexified quantum groups. In the special exampleA 1 we derive theq-deformed enveloping algebraU q (sl(2, ℂ)). In the limitq→1 the undeformedU q (sl(2, ℂ)) is recovered.

New n=1 extended superconformal algebras with two and three generators

Abstract: In this paper we consider extensions of the super Virasoro algebra by one and two super primary fields. Using a nonexplicitly covariant approach we compute all SW algebras with one generator of dimension up to 7 in addition to the super Virasoro field. In complete analogy to W algebras with two generators, most results can be classified using the representation theory of the super Virasoro algebra. Furthermore, we find that the algebra can be realized as a subalgebra of at . We also construct some new SW algebras with three generators, namely , and .

An algebraic approach to vertex models and transfer matrix spectra

Abstract: We analyse the transfer matrix spectra of vertex models associated with the Lie superalgebras gl(P|M) and sl(P|M) using the representation theory of the Hecke algebra Hn(q). We develop a terminology for discussing the Bethe ansatz computation of the spectrum from this perspective. Using representations coming from these vertex models we develop some new methods for dealing with the analysis of Hecke algebras in any specialisation, including roots of unity. We also discuss the construction and spectrum of fusion models from the viewpoint of representation theory, begining a classification of the spectrum and identifying some sectors with trivial spectrum. In particular we show that the spectrum of the sl(P|M) λ-fusion model is trivial if λP+1>M.

Extremal projector and universal R-matrix for quantized contragredient Lie (super)algebras

Abstract: Two basic elements of the representation theory of quantized finite-dimensional contragredient Lie (super)algebras g (U q(g)) are presented. These are the universal R-matrix to be an interwining operator, and the extremal projector which gives a powerful method for decomposition of representations. Properties of Cartan-Weyl basis for U q(g) are discussed. Some Taylor extension of U q(g) and U q(g) ⊗ U q(g) are introduced in terms of this basis. The extremal projector p and the universal R-matrix R are described as unique elements of these extensions. Explicit formulae for p and R are given.