Drazen Adamovic

Bio: Drazen Adamovic is an academic researcher from University of Zagreb. The author has contributed to research in topics: Vertex operator algebra & Vertex (graph theory). The author has an hindex of 12, co-authored 30 publications receiving 551 citations.

Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

Abstract: We investigate vertex operator algebras L(k,0) associated with modular-invariant representations for an affine Lie algebra A (1) , where k is an'admissible' rational number. We show that VOA L(k,0) is rational in the category O and find all irreducible representations in the category of weight modules.

Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

Abstract: We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal O$ and find all irreducible representations in the category of weight modules.

Representations of N=2 superconformal vertex algebra

Abstract: Let $L_c$ be simple vertex operator superalgebra(SVOA) associated to the vacuum representation of N=2 superconformal algebra with the central charge $c$. Let $c_m = {3m}/{m+2}$. We classify all irreducible modules for the SVOA $L_{c_m}$. When $m$ is an integer we prove that the set of all unitary representations of N=2 superconformal algebra with the central charge $c_m$ provides all irreducible $L_{c_m}$-modules. When $m otin {\N} $ and $m$ is an admissible rational number we show that irreducible $L_{c_m}$-modules are parameterized with the union of one finite set and union of finitely many rational curves.

Some Rational Vertex Algebras

Abstract: Let $L((n-\tfrac 3 2)\Lambda_0)$, $n \in \Bbb N$, be a vertex operator algebra associated to the irreducible highest weight module $L((n-\tfrac 3 2)\Lambda_0)$ for a symplectic affine Lie algebra. We find a complete set of irreducible modules for $L((n-\tfrac 3 2)\Lambda_0)$ and show that every module for $L((n-\tfrac 3 2)\Lambda_0)$ from the category $\Cal O$ is completely reducible.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

Logarithmic conformal field theory: beyond an introduction

Abstract: This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with the remarkable observation of Cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory. Cardy’s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators. For this, a short introduction to the fusion algorithm of Nahm, Gaberdiel and Kausch is provided. While the percolation logarithmic conformal field theory is still not completely understood, there are several examples for which the formalism familiar from rational conformal field theory, including bulk partition functions, correlation functions, modular transformations, fusion rules and the Verlinde formula, has been successfully generalized. This is illustrated for three examples: the singlet model , related to the triplet model , symplectic fermions and the fermionic bc ghost system; the fractional level Wess–Zumino–Witten model based on at , related to the bosonic βγ ghost system; and the Wess–Zumino–Witten model for the Lie supergroup , related to at and 1, the Bershadsky–Polyakov algebra and the Feigin–Semikhatov algebras . These examples have been chosen because they represent the most accessible, and most useful, members of the three best-understood families of logarithmic conformal field theories. The logarithmic minimal models , the fractional level Wess–Zumino–Witten models, and the Wess–Zumino–Witten models on Lie supergroups (excluding ). In this review, the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum, which turns out to be continuous, as well as a selection of natural reducible, but indecomposable, modules. This is followed by a detailed description of how to obtain character formulae for each irreducible, a derivation of the action of the modular group on the characters, and an application of the Verlinde formula to compute the Grothendieck fusion rules. In each case, the (genuine) fusion rules are known, so comparisons can be made and favourable conclusions drawn. In addition, each example admits an infinite set of simple currents, hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed. The spectrum of such an extended theory is typically discrete and this is how the triplet model arises, for example. Moreover, simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory. Finally, each example is concluded by a brief description of the computation of some bulk correlators, a discussion of the structure of the bulk state space, and remarks concerning more advanced developments and generalizations. The final part gives a very short account of the theory of staggered modules, the (simplest class of) representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins. These modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven. Then, the important quantities known as logarithmic couplings are reviewed for Virasoro staggered modules and their role as fundamentally important parameters, akin to the three-point constants of rational conformal field theory, is discussed. An appendix is also provided in order to introduce some of the necessary, but perhaps unfamiliar, language of homological algebra.

Annihilating fields of standard modules of Sl(2, C)〓 and combinatorial identities

Abstract: Introduction Formal Laurent series and rational functions Generating fields The vertex operator algebra $N(k\Lambda_0)$ Modules over $N(k\Lambda_0)$ Relations on standard modules Colored partitions, leading terms and the main results Colored partitions allowing at least two embeddings Relations among relations Relations among relations for two embeddings Linear independence of bases of standard modules Some combinatorial identities of Rogers-Ramanujan type Bibliography.

Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and their Generalized Modules

Abstract: This is the first part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. This theory generalizes the tensor category theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a “conformal vertex algebra” or even more generally, for a “Mobius vertex algebra.” We do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. As in the earlier series of papers, our tensor product functors depend on a complex variable, but in the present generality, the logarithm of the complex variable is required; the general representation theory of vertex operator algebras requires logarithmic structure. This work includes the complete proofs in the present generality and can be read independently of the earlier series of papers. Since this is a new theory, we present it in detail, including the necessary new foundational material. In addition, with a view toward anticipated applications, we develop and present the various stages of the theory in the natural, general settings in which the proofs hold, settings that are sometimes more general than what we need for the main conclusions. In this paper (Part I), we give a detailed overview of our theory, state our main results and introduce the basic objects that we shall study in this work. We include a brief discussion of some of the recent applications of this theory, and also a discussion of some recent literature.