Showing papers on "Algebra representation published in 1998"

Hopf Algebras, Renormalization and Noncommutative Geometry

Abstract: We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem

Abstract: In this paper we solve a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. We show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific Hopf algebra \(\), associated to each integer codimension. This Hopf algebra reduces transverse geometry, to a universal geometry of affine nature. The structure of this Hopf algebra, its relation with the Lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper, in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the Hochschild cohomology of a Hopf algebra carries a highly nontrivial cyclic structure.

On the Hopf algebra structure of perturbative quantum field theories

Abstract: We show that the process of renormalization encapsules a Hopf algebra structure in a natural manner. This sheds light on the recently proposed connection between knots and renormalization theory.

The q-Schur algebra

Abstract: This book focuses on the representation theory of q-Schur algebras and connections with the representation theory of Hecke algebras and quantum general linear groups. The aim is to present, from a unified point of view, quantum analogues of certain results known already in the classical case. The approach is largely homological, based on Kempf's vanishing theorem for quantum groups and the quasi-hereditary structure of the q-Schur algebras. Beginning with an introductory chapter dealing with the relationship between the ordinary general linear groups and their quantum analogies, the text goes on to discuss the Schur Functor and the 0-Schur algebra. The next chapter considers Steinberg's tensor product and infinitesimal theory. Later sections of the book discuss tilting modules; the Ringel dual of the q-Schur algebra; Specht modules for Hecke algebras; and the global dimension of the q-Schur algebras. An appendix gives a self-contained account of the theory of quasi-hereditary algebras and their associated tilting modules. This volume will be primarily of interest to researchers in algebra and related topics in pure mathematics.

On the Algebras of BPS States

Abstract: We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac–Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi–Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry.

The q-characters of representations of quantum affine algebras and deformations of W-algebras

Abstract: We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck ring of representations of a quantum affine algebra to a polynomial ring. We conjecture that the image of this homomorphism is equal to the intersection of certain "screening operators". We also discuss the connection between q-characters and Bethe Ansatz.

Structure theory of finite conformal algebras

Abstract: After the seminal paper [BPZ] of Belavin, Polyakov and Zamolodchikov, conformal field theory has become by now a large field with many remarkable ramifications to other fields of mathematics and physics. A rigorous mathematical definition of the “chiral part” of a conformal field theory, called a vertex (= chiral) algebra, was proposed by Borcherds [Bo] more than ten years ago and continued in [DL], [FHL], [FLM], [K], [L] and in numerous other works. However, until now a classification of vertex algebras, similar, for example, to the classification of finite-dimensional Lie algebras, seems to be far away. In the present paper we give a solution to the special case of this problem when the chiral algebra is generated by a finite number of quantum fields, closed under the operator product expansion (in the sense that only derivatives of the generating fields may occur). In this situation the adequate tool is the notion of a conformal algebra [K] which, to some extent, is related to a chiral algebra in the same way a Lie algebra is related to its universal enveloping algebra. At the same time, the theory of conformal algebras sheds a new light on the problem of classification of infinite-dimensional Lie algebras. About thirty years ago one of the authors posed (and partially solved) the problem of classification of simple Z-graded Lie algebras of finite Gelfand-Kirillov dimension [K1]. This problem was completely solved by Mathieu [M1]-[M3] in a remarkable tour de force. The point of view of the present paper is that the condition of locality (which is the most basic axiom of quantum field theory) along with a finiteness condition, are more natural conditions, which are also much easier to handle. In this paper we develop a structure theory of finite rank conformal algebras. Applications of this theory are two-fold. On the one hand, the conformal algebra structure is an axiomatic description [K] of the operator product expansion (OPE) of chiral fields in a conformal field theory [BPZ]. Hence the theory of finite conformal algebras provides a classification of finite systems of fields closed under the OPE. On the other hand, the category of finite conformal algebras is (more or less) equivalent to the category of infinite-dimensional Lie algebras spanned by Fourier coefficients of a finite number of pairwise local fields (or rather formal distributions)

Integrable Evolution Equations on Associative Algebras

Abstract: This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found.

Systematic Approach to Cyclic Orbifolds

Abstract: We introduce an orbifold induction procedure which provides a systematic construction of cyclic orbifolds, including their twisted sectors The procedure gives counterparts in the orbifold theory of all the current-algebraic constructions of conformal field theory and enables us to find the orbifold characters and their modular transformation properties

Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras

Abstract: The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup onn generators. We obtain a distance formula to an arbitrary WOT-closed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna-Pick type interpolation theorems

Framed Vertex Operator Algebras, Codes and the Moonshine Module

Abstract: For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ½, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ½ are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.

Some properties of finite-dimensional semisimple Hopf algebras

Abstract: Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use this statement to prove that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily the result of Zhu that Kaplansky's conjecture on prime dimensional Hopf algebras over k is true, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne on characterization of tannakian categories to prove that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.

Character formulas for tilting modules over Kac-Moody algebras

Abstract: We show how to express the characters of tilting modules in a (possibly parabolic) category O over a Kac-Moody algebra in terms of the characters of simple highest weight modules. This settles, in lots of cases, Conjecture 7.2 of Kazhdan-Lusztig-Polynome and eine Kombinatorik für Kipp-Moduln, Representation Theory (An electronic Journal of the AMS) (1997), by the author, describing the character of tilting modules for quantum groups at roots

Quantum Weyl Reciprocity and Tilting Modules

Abstract: Quantum Weyl reciprocity relates the representation theory of Hecke algebras of type A with that of q-Schur algebras. This paper establishes that Weyl reciprocity holds integrally (i. e., over the ring ℤ[q, q− 1] of Laurent polynomials) and that it behaves well under base-change. A key ingredient in our approach involves the theory of tilting modules for q-Schur algebras. New results obtained in that direction include an explicit determination of the Ringel dual algebra of a q-Schur algebra in all cases. In particular, in the most interesting situation, the Ringel dual identifies with a natural quotient algebra of the Hecke algebra.

On the Use of Local Cohomology in Algebra and Geometry

Abstract: Local cohomology is a useful tool in several branches of commutative algebra and algebraic geometry. The main aim of this series of lectures is to illustrate a few of these techniques. The material presented in the sequel needs some basic knowledge about commutative resp. homological algebra. The basic chapters of the textbooks [9], [28], and [48] are a recommended reading for the preparation. The author’s intention was to present applications of local cohomology in addition to the examples in these textbooks as well as those of [7].

Discrete spectral triples and their symmetries

Abstract: We classify zero-dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure. We investigate also whether such spectral triples admit a symmetry arising from the Hopf algebra structure of the finite algebra. We discuss examples of commutative algebras and group algebras.

Continuous Time and Consistent Histories

Abstract: We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.

Coadjoint orbits of the virasoro algebra and the global liouville equation

Abstract: The classification of the coadjoint orbits of the Virasoro algebra is reviewed and then applied to analyze the so-called global Liouville equation. The review is self-contained, elementary and is tailor-made for the application. It is well known that the Liouville equation for a smooth, real field φ under periodic boundary condition is a reduction of the SL(2,R) WZNW model on the cylinder, where the WZNW field g∈SL(2,R) is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field Q=κg22 where κ≠0 is a constant, what we call the global Liouville equation. Corresponding to the winding number of the SL(2,R) WZNW model, there is a topological invariant in the reduced theory, given by the number of zeros of Q over a period. By the substitution Q=±exp(-φ/2), the Liouville theory for a smooth φ is recovered in the trivial topological sector. The nontrivial topological sectors can be viewed as singular sectors of the Liouville theory that contain blowingup solutions in terms of φ. Since the global Liouville equation is conformally invariant, its solutions can be described by explicitly listing those solutions for which the stress–energy tensor belongs to a set of representatives of the Virasoro coadjoint orbits chosen by convention. This direct method permits to study the "coadjoint orbit content" of the topological sectors as well as the behavior of the energy in the sectors. The analysis confirms that the trivial topological sector contains special orbits with hyperbolic monodromy and shows that the energy is bounded from below in this sector only.

A generalization of the notion of instanton

Abstract: The concept of an instanton in four dimensions can be generalized when one considers the curvature as having values in a general Lie algebra ℷ instead of su (2)+ ≅ Λ+2. For a manifold with a G-structure P, a canonical complex is defined in order to formulate the theory of infinitesimal deformations of generalized instantons. A study is made of the exact conditions imposed on P by the existence of this complex, and an examination of these conditions leads one to distinguish classes of Riemannian manifolds characterized by alternatives to a holonomy reduction.

Higher spin N = 8 supergravity

Abstract: The product of two N = 8 supersingletons yields an infinite tower of massless states of higher spin in four dimensional anti de Sitter space. All the states with spin s ≥ 1 correspond to generators of Vasiliev's super higher spin algebra shsE(8|4) which contains the D = 4,N = 8 anti de Sitter superalgebra OSp(8|4). Gauging the higher spin algebra and introducing a matter multiplet in a quasi-adjoint representation leads to a consistent and fully nonlinear equations of motion as shown sometime ago by Vasiliev. We show the embedding of the N = 8 AdS supergravity equations of motion in the full system at the linearized level and discuss the implications for the embedding of the interacting theory. We furthermore speculate that the boundary N = 8 singleton field theory yields the dynamics of the N = 8 AdS supergravity in the bulk, including all higher spin massless fields, in an unbroken phase of M-theory.

Geometric methods in the representation theory of Hecke algebras and quantum groups

Abstract: These lectures are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the Iwahori-Hecke algebra of bi-invariant functions (under convolution) on a p-adic group, are considered. We give a uniform geometric construction of these algebras in terms of homology of an appropriate “Steinberg-type” variety Z (or its modification, such as K-theory or elliptic cohomology of Z, or an equivariant version thereof). We then explain how to obtain a complete classification of finite dimensional irreducible representations of the algebras in question, using our geometric construction and perverse sheaves methods.

Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra

Abstract: We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical -algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL 2 . The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL 2 , and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra.