Showing papers on "Representation theory published in 2005"

New concept of relativistic invariance in noncommutative space-time: twisted poincare symmetry and its implications.

Abstract: We present a systematic framework for noncommutative (NC) quantum field theory (QFT) within the new concept of relativistic invariance based on the notion of twisted Poincare symmetry, as proposed by Chaichian et al. [Phys. Lett. B 604, 98 (2004)]. This allows us to formulate and investigate all fundamental issues of relativistic QFT and offers a firm frame for the classification of particles according to the representation theory of the twisted Poincare symmetry and as a result for the NC versions of CPT and spin-statistics theorems, among others, discussed earlier in the literature. As a further application of this new concept of relativism we prove the NC analog of Haag's theorem.

Global Analysis on Foliated Spaces

Abstract: Global analysis has as its primary focus the interplay between the local analysis and the global geometry and topology of a manifold. This is seen classicallv in the Gauss-Bonnet theorem and its generalizations. which culminate in the Ativah-Singer Index Theorem [ASI] which places constraints on the solutions of elliptic systems of partial differential equations in terms of the Fredholm index of the associated elliptic operator and characteristic differential forms which are related to global topologie al properties of the manifold. The Ativah-Singer Index Theorem has been generalized in several directions. notably by Atiyah-Singer to an index theorem for families [AS4]. The typical setting here is given by a family of elliptic operators (Pb) on the total space of a fibre bundle P = F_M_B. where is defined the Hilbert space on Pb 2 L 1p -llbl.dvollFll. In this case there is an abstract index class indlPI E ROIBI. Once the problem is properly formulated it turns out that no further deep analvtic information is needed in order to identify the class. These theorems and their equivariant counterparts have been enormously useful in topology. geometry. physics. and in representation theory.

From triangulated categories to cluster algebras

Abstract: The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of the cluster category. This realization provides a natural basis for A. We prove new results and formulate conjectures on `good basis' properties, positivity, denominator theorems and toric degenerations.

Averages of Characteristic Polynomials in Random Matrix Theory

Abstract: We compute averages of products and ratios of characteristic polynomials associated with orthogonal, unitary, and symplectic ensembles of random matrix theory. The Pfaffian/determinantal formulae for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact Pfaffian/determinantal formulae for the discrete averages are proven using standard tools of linear algebra; no application of orthogonal or skew-orthogonal polynomials is needed. © 2005 Wiley Periodicals, Inc.

Duality and equivalence of module categories in noncommutative geometry I

Abstract: This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of categories of modules. In this paper, we develop a general framework needed to describe these dualities. In various geometric contexts, e.g. complex geometry, generalized complex geometry, and noncommutative geometry, the geometric structure is encoded in a certain differential graded algebra. We develop the module theory of such differential graded algebras in such a way that we can recover the derived category of coherent sheaves on a complex manifold. In this paper and ones to follow we apply this to stating and proving the duality statements mentioned above. After developing the general framework, we look at a (complex) Lie algebroid $\A\to T_\cx X$. One can then consider our analogue of the derived category of coherent sheaves, integrable with respect to the Lie algebroid. We then establish a (Serre) duality theorem for "elliptic" Lie algebroids and for noncommutative tori.

Real conjugacy classes in algebraic groups and finite groups of Lie type

Abstract: According to the Berman–Witt theorem, the number of real classes of G is equal to the number of complex irreducible characters whose values are real (such characters are called real ). Each real irreducible character is the character of a real or quaternion representation of G. For this reason Problem 1.1 has attracted considerable attention for various classes of groups; see [9], [12], [13]. In particular, Feit and Zuckerman [9] studied this problem for classical groups extended by a graph automorphism. They also showed that all elements are real in the groups Sp2nðqÞ with q1 1 ðmod 4Þ, and Gow [12] proved this for q even. There are some other results in the literature which are not concerned with quasi-simple finite groups. In particular, Gow [13] proved that all elements in SOnðqÞ are real for n1 0 ðmod 4Þ and for n odd. This is also true for GOnðqÞ with n arbitrary; see [8], [13], [20]. The main result of the paper is the following theorem which completely solves Problem 1.1 for finite quasi-simple groups:

Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture

Abstract: We study the representation theory of the superconformal algebra Wk(g,fθ) associated with a minimal gradation of g. Here, g is a simple finite-dimensional Lie superalgebra with a nondegenerate, even supersymmetric invariant bilinear form. Thus, Wk(g,fθ) can be one of the well-known superconformal algebras including the Virasoro algebra, the Bershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the Bershadsky-Knizhnik algebras, the N=2 superconformal algebra, the N=4 superconformal algebra, the N=3 superconformal algebra, and the big N=4 superconformal algebra. We prove the conjecture of V. G. Kac, S.-S. Roan, and M. Wakimoto [17, Conjecture 3.1B] for Wk(g,fθ). In fact, we show that any irreducible highest-weight character of Wk(g,fθ) at any level k∈ℂ is determined by the corresponding irreducible highest-weight character of the Kac-Moody affinization of g

Representations of Quantum Affinizations and Fusion Product

Abstract: In this paper we study general quantum affinizations $\mathcal{U}_q(\widehat{\mathfrak{g}})$ of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1) highest weight simple integrable representations analog to Drinfel'd-Chari-Presley one. A generalization of the q-characters morphism, introduced by Frenkel-Reshetikhin for quantum affine algebras, appears to be a powerful tool for this investigation. For a large class of quantum affinizations (including quantum affine algebras and quantum toroidal algebras), the combinatorics of q-characters give a ring structure * on the Grothendieck group $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ of the integrable representations that we classified. We propose a new construction of tensor products in a larger category by using the Drinfel'd new coproduct (it cannot directly be used for $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ because it involves infinite sums). In particular, we prove that * is a fusion product (a product of representations is a representation).

A New Approach to the Representation Theory of the Symmetric Groups. II

Abstract: The present paper is a revised Russian translation of the paper “A new approach to representation theory of symmetric groups,” Selecta Math., New Series, 2, No. 4, 581–605 (1996). Numerous modifications to the text were made by the first author for this publication. Bibliography: 35 titles.

Cycle Spaces of Flag Domains: A Complex Geometric Viewpoint

Abstract: * Dedication * Acknowledgments * Introduction Part I: Introduction to Flag Domain Theory Overview * Structure of Complex Flag Manifolds * Real Group Orbits * Orbit Structure for Hermitian Symmetric Spaces * Open Orbits * The Cycle Space of a Flag Domain Part II: Cycle Spaces as Universal Domains Overview * Universal Domains * B-Invariant Hypersurfaces in Mz * Orbit Duality via Momentum Geometry * Schubert Slices in the Context of Duality * Analysis of the Boundary of U * Invariant Kobayashi-Hyperbolic Stein Domains * Cycle Spaces of Lower-Dimensional Orbits * Examples Part III: Analytic and Geometric Concequences Overview * The Double Fibration Transform * Variation of Hodge Structure * Cycles in the K3 Period Domain Part IV: The Full Cycle Space Overview * Combinatorics of Normal Bundles of Base Cycles * Methods for Computing H1(C O(E((q+0q)s))) * Classification for Simple g0 with rank t < rank g * Classification for rank t = rank g * References * Index * Symbol Index

Classification of Solvable Lie Algebras

Abstract: In this paper we describe a simple method for obtaining a classification of small-dimensional solvable Lie algebras Using this method, we obtain the classification of three- and fourdimensional solvable Lie algebras (over fields of any characteristic) Precise conditions for isomorphism are given

Noncommutative Lp modules

Abstract: We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutative L^p spaces. Our abstract characterization is based on an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be nearly trivial.

Enveloping algebras of Slodowy slices and the Joseph ideal

Abstract: We study general properties of quantisations of Slodowy slices and discuss in detail the case of the minimal nilpotent orbit. Associated varieties of related primitive ideals of U(g) are determined, in the general case, and the de Vos-van Driel conjecture on Verma modules for finite W-algebras is proved in the minimal nilpotent case.

Simple singularities and integrable hierarchies

Abstract: The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In [12], it is proved that the total descendent potential corresponding to K. Saito’s Frobenius structure on the parameter space of the miniversal deformation of the A n−1-singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy which is satisfied by the total descendent potential of a simple singularity of ADE type. Our description of the hierarchy is parallel to the vertex operator construction of Kac-Wakimoto [17] except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac-Wakimoto theory are studied on a case-by-case basis and remain, generally speaking, unknown.

Group Gradings on Simple Lie Algebras of Type "A"

Abstract: In this paper we describe all group gradings by a finite abelian group G of any Lie algebra L of the type "A" over algebraically closed field F of characteristic zero.

Discrete Gauge Theory: From Lattices to Tqft

Abstract: # Quantization of Discretized Gauge Theories # Topology: Decomposition of Manifolds # Categories and Diagrams # Representation Theory: Groups and Hopf Algebras # Cellular Gauge Theory # Topological Quantum Field Theory # Related Constructions # Applications to Lattice Models and Quantum Gravity

Poisson geometry, deformation quantisation and group representations

Abstract: Poisson geometry lies at the cusp of noncommutative algebra and differential geometry, with natural and important links to classical physics and quantum mechanics. This book presents an introduction to the subject from a small group of leading researchers, and the result is a volume accessible to graduate students or experts from other fields. The contributions are: Poisson Geometry and Morita Equivalence by Bursztyn and Weinstein; Formality and Star Products by Cattaneo; Lie Groupoids, Sheaves and Cohomology by Moerdijk and Mrcun; Geometric Methods in Representation Theory by Schmid; Deformation Theory: A Powerful Tool in Physics Modelling by Sternheimer.

Modularity in orbifold theory for vertex operator superalgebras

Abstract: This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C2-cofinite and g-rational for any g ∈ G.

Cohomology ring of n-Lie algebras

Abstract: Natural graded Lie brackets on the space of cochains of n-Leibniz and n-Lie algebras are introduced. It turns out that these brackets agree under the natural embedding introduced by Gautheron. Moreover, n-Leibniz and n-Lie algebras turn to be canonical structures for these brackets in a similar way in which associative algebras (respectively, Lie algebras) are canonical structures for the Gerstenhaber bracket (respectively, Nijenhuis-Richardson bracket). This allows to define the corresponding cohomology operators and graded Lie algebra structures on the cohomology spaces in an uniform simple way by means of square zero elements.

From triangulated categories to Lie algebras: A theorem of Peng and Xiao

Abstract: We present a simplified and more intuitive proof of a theorem of Peng and Xiao, which constructs a Lie algebra from any 2-periodic triangulated k-category (satisfying some finiteness assumptions).

Mckay's observation and vertex operator algebras generated by two conformal vectors of central charge 1/2

Abstract: This paper is a continuation of (33) at which several coset subalgebras of the lattice VOA Vp 2E8 were constructed and the relationship between such algebras with the famous McKay observation on the extended E8 diagram and the Monster simple group were discussed In this article, we shall provide the technical details We completely determine the structure of the coset subalgebras constructed and show that they are all generated by two conformal vectors of central charge 1/2 We also study the represen- tation theory of these coset subalgebras and show that the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subal- gebra U is actually contained in the Moonshine VOA V ♮ The existence of U inside the Moonshine VOA V ♮ for the cases of 1A,2A,2B and 4A is also established Moreover, the cases for 3A, 5A and 3C are discussed and the Monster simple group were discussed In this article, we shall provide the technical details We shall determine the structure of the coset subalgebras and show that they are all generated by two conformal vectors of central charge 1/2 We also study the representation theory of these coset subalgebras and show that the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subalgebra U is actually contained in the Moonshine

Rouquier's theorem on representation dimension

Abstract: Based on work of Rouquier, some bounds for Aulander's representation dimension are discussed. More specifically, if X is a reduced projective scheme of dimension n over some field, and T is a tilting complex of coherentOX-modules, then the representation dimension of the endomorphism algebra EndOX(T) is at least n.

States and representations in deformation quantization

Abstract: In this review we discuss various aspects of representation theory in deformation quantization starting with a detailed introduction to the concepts of states as positive functionals and the GNS construction. Rieffel induction of representations as well as strong Morita equivalence, Dirac monopole and strong Picard Groupoid are also discussed.

The twisted Drinfeld double of a finite group via gerbes and finite groupoids

Abstract: The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3-dimensional topological quantum field theory. In particular the representation category of the twisted Drinfeld double is viewed as the `space of sections' associated to a transgressed gerbe over the loop groupoid.

Linear Free Divisors and Quiver Representations

Abstract: Linear free divisors are free divisors, in the sense of K.Saito, with linear presentation matrix (example: normal crossing divisors). Using techniques of deformation theory on representations of quivers, we exhibit families of linear free divisors as discriminants in representation spaces for real Schur roots of a finite quiver. We review some basic material on quiver representations, and explain in detail how to verify whether the discriminant is a free divisor and how to determine its components and their equations, using techniques of A. Schofield. As an illustration, the linear free divisors that arise as the discriminant from the highest roots of Dynkin quivers of type E7 and E8 are treated explicitly.

The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution

Abstract: In this paper, we compute all the moments of the real Wishart distribution. To do so, we use the Gelfand pair (S2k,H), where H is the hyperoctahedral group, the representation theory of H and some techniques based on graphs.