Showing papers on "Algebra representation published in 2006"

Calabi-Yau algebras

Abstract: We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution In dimension 3, the resolution is determined by a noncommutative potential Representation varieties of the Calabi-Yau algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential Numerical invariants, like ranks of cyclic homology groups, are expected to be given by `matrix integrals' over representation varieties We discuss examples of Calabi-Yau algebras involving quivers, 3-dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-Simons Examples related to quantum Del Pezzo surfaces will be discussed in [EtGi]

Approximations and Endomorphism Algebras of Modules

Abstract: This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

Twisted representations of vertex operator algebras

Abstract: This paper gives an analogue of Ag(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and Ag(V)-modules is established. It is proved that if V is g-rational, then Ag(V) is finite-dimensional semi-simple associative algebra and there are only finitely many irreducible g-twisted V-modules.

Cluster-tilted algebras

Abstract: We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Abstract: In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

Rigid modules over preprojective algebras

Abstract: Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.

Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

Abstract: The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise'' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .

Logarithmic minimal models

Abstract: Working in the dense loop representation, we use the planar Temperley?Lieb algebra to build integrable lattice models called logarithmic minimal models . Specifically, we construct Yang?Baxter integrable Temperley?Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley?Lieb algebra are inherently non-local and not (time-reversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1?(6(p?p')2/pp'), where p, p' = 1, 2, ... are coprime. The first few members of the principal series are critical dense polymers (m = 1, c = ?2), critical percolation (m = 2, c = 0) and the logarithmic Ising model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights ?r,s = (((m+1)r?ms)2?1)/4m(m+1), r, s = 1, 2, .... The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.

Finite vs affine W-algebras

Abstract: In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra ZhuΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established.

The Weyl groupoid of a Nichols algebra of diagonal type

Abstract: The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding.

New realizations of Lie algebra kappa-deformed Euclidean space

Abstract: We study Lie algebra κ-deformed Euclidean space with undeformed rotation algebra SOa(n) and commuting vectorlike derivatives. Infinitely many realizations in terms of commuting coordinates are constructed and a corresponding star product is found for each of them. The κ-deformed noncommutative space of the Lie algebra type with undeformed Poincare algebra and with the corresponding deformed coalgebra is constructed in a unified way.

Max-Plus Algebra

Abstract: Max-plus algebra has been discovered more or less independently by several schools, in relation with various mathematical fields. This chapter is limited to finite dimensional linear algebra. For more information, the reader may consult the books [CG79], [Zim81], [CKR84], [BCOQ92], [KM97], [GM02], and [HOvdW06]. The collections of articles [MS92], [Gun98], and [LM05] give a good idea of current developments.

Techniques of representation theory

Abstract: Introduction 1. Algebras and modules 2. Quivers and algebras 3. Representations and modules 4. Auslander-Reiten theory 5. Nakayama algebras and representation-finite group algebras 6. Tilting theory 7. Representation-finite hereditary algebras 8. Tilted algebras 9. Directing modules and postprojective components Appendix: categories, functors and homology.

The magnon kinematics of the AdS/CFT correspondence

Abstract: The planar dilatation operator of N = 4 supersymmetric Yang-Mills is the hamiltonian of an integrable spin chain whose length is allowed to fluctuate. We will identify the dynamics of length fluctuations of planar N = 4 Yang-Mills with the existence of an abelian Hopf algebra Z symmetry with non-trivial co-multiplication and antipode. The intertwiner conditions for this Hopf algebra will restrict the allowed magnon irreps to those leading to the magnon dispersion relation. We will discuss magnon kinematics and crossing symmetry on the spectrum of Z. We also consider general features of the underlying Hopf algebra with Z as central Hopf subalgebra, and discuss the giant magnon semiclassical regime.

Quivers with Relations and Cluster Tilted Algebras

Abstract: Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.

Representation dimension of exterior algebras

Abstract: We determine the representation dimension of exterior algebras. This provides the first known examples of representation dimension > 3. We deduce that the Loewy length of the group algebra over F2 of a finite group is strictly bounded below by the 2-rank of the group (a conjecture of Benson). A key tool is the use of the concept of dimension of a triangulated category.

Constructing the Cubic Interaction Vertex of Higher Spin Gauge Fields

Abstract: We propose a method of construction of a cubic interaction in massless higher spin gauge theory both in flat and in AdS space-times of arbitrary dimensions. We consider a triplet formulation of the higher spin gauge theory and generalize the higher spin symmetry algebra of the free model to the corresponding algebra for the case of cubic interaction. The generators of this new algebra carry indexes which label the three higher spin fields involved into the cubic interaction. The method is based on the use of oscillator formalism and on the Becchi-Rouet-Stora-Tyutin (BRST) technique. We derive general conditions on the form of cubic interaction vertex and discuss the ambiguities of the vertex which result from field redefinitions. This method can in principle be applied for constructing the higher spin interaction vertex at any order. Our results are a first step towards the construction of a Lagrangian for interacting higher spin gauge fields that can be holographically studied.

Lorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillator

Abstract: The D-dimensional (β, β')-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a (D + 1)-dimensional quantized spacetime. In the D = 3 and β = 0 case, the latter reproduces Snyder algebra. The deformed Poincare transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a (1 + 1)-dimensional spacetime described by such an algebra is studied in the case where β' = 0. Extending supersymmetric quantum mechanical and shape-invariance methods to energy-dependent Hamiltonians provides exact bound-state energies and wavefunctions. Physically acceptable states exist for β < 1/(m2c2). A new interesting outcome is that, in contrast with the conventional Dirac oscillator, the energy spectrum is bounded.

Newton-Hooke algebras, nonrelativistic branes, and generalized pp-wave metrics

Abstract: The Newton-Hooke algebras in d dimensions are constructed as contractions of dS(AdS) algebras. Nonrelativistic brane actions are WZ terms of these Newton-Hooke algebras. The NH algebras appear also ! ~

Invariant Means and Finite Representation Theory of $c*$-Algebras

Abstract: Introduction Notation, definitions and useful facts Amenable traces and stronger approximation properties Examples and special cases Finite representations Applications and connections with other areas Bibliography.

Cyclotomic Nazarov-Wenzl algebras

Abstract: Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain "cyclotomic quotients" of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank $r^{n}(2n-1)!!$ (when $\Omega$ is $\mathbf{u}$-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

Virasoro representations and fusion for general augmented minimal models

Abstract: In this paper we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models, the augmented c2,3 = 0 model as well as the augmented Yang–Lee model at c2,5 = −22/5. We analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra. We also describe the first few higher rank representations in detail. In particular, we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure. Knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented cp,q models.

IIA Ten-forms and the Gauge Algebras of Maximal Supergravity Theories

Abstract: We show that IIA supergravity can be extended with two independent 10-form potentials. These give rise to a single BPS IIA 9-brane. We investigate the bosonic gauge algebra of both IIA and IIB supergravity in the presence of 10-form potentials and point out an intriguing relation with the symmetry algebra E11, which has been conjectured to be the underlying symmetry of string theory/M-theory.

Unfolded Form of Conformal Equations in M Dimensions and ?(M + 2)-MODULES

Abstract: A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra 𝔣. Under certain conditions, 𝔣-invariant systems of differential equations are shown to be associated with 𝔣-modules that are integrable with respect to some parabolic subalgebra of 𝔣. The suggested construction is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to the nonlinear case. It is applied to the conformal algebra 𝔬(M, 2) to classify all linear conformally invariant differential equations in the Minkowski space. Numerous examples of conformal equations are discussed from this perspective.

Extended E8 invariance of 11-dimensional supergravity

Abstract: The hyperbolic Kac-Moody algebra E10 has repeatedly been suggested to play a crucial role in the symmetry structure of M-theory. Recently, following the analysis of the asymptotic behaviour of the supergravity fields near a cosmological singularity, this question has received a new impulse. It has been argued that one way to exhibit the symmetry was to rewrite the supergravity equations as the equations of motion of the non-linear sigma model E10/K(E10). This attempt, in line with the established result that the scalar fields which appear in the toroidal compactification down to three spacetime dimensions form the coset E8/SO({16}), was verified for the first bosonic levels in a level expansion of the theory. We show that the same features remain valid when one includes the gravitino field.

Cluster Algebras of Finite Type and Positive Symmetrizable Matrices

Abstract: The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However, the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

The Restricted Kirillov–Reshetikhin Modules for the Current and Twisted Current Algebras

Abstract: We define a family of graded restricted modules for the polynomial current algebra associated to a simple Lie algebra. We study the graded character of these modules and show that they are the same as the graded characters of certain Demazure modules. In particular, we see that the specialized characters are the same as those of the Kirillov Reshetikhin modules for quantum affine algebras.