Pdf zgraded lie superalgebras of infinite depth and. This paper is concerned with an attempt at new methods in the problem of classifying finitedimensional simple lie algebras over algebraically closed fields of characteristic. What is usually meant be a classification is a complete list of simple. Formally, a lie superalgebra is a nonassociative z 2graded algebra, or superalgebra, over a commutative ring typically r or c whose product, called the lie superbracket or supercommutator, satisfies the two conditions analogs of the usual lie algebra axioms, with grading. Ordinary lie superalgebras find their origin in algebraic topology and find. We construct new examples of infinitedimensional lie superalgebras with a zgradation of infinite depth and finite growth and clas sify zgraded lie superalgebras of infinite depth and finite. In the case of finite dimensional modular lie superalgebras of cartan type, the invariance of the natural filtration was discussed in 12. Introduction to finite and infinite dimensional lie superalgebras. Therefore, we classify the infinitedimensional modular lie superalgebra in the sense of isomorphism. We find it interesting that the main tool is the theory of lie superalgebras. Representation of the affine superalgebras at40, 2. Contravariant form for infinitedimensional lie algebras and. Lie superalgebras 205 the proof of theorem b makes use of a family of infinitedimensional lie superalgebras described in section 1.
A class of infinitedimensional representations of the lie. Introduction to finite and infinite dimensional lie super. For the above listed lie superalgebras g ga, their even parts g 0 and the g 0modules g 1 are explicitly described in terms of simple and solvable lie algebras and modules over them. This volume begins with an introduction to the structure of finitedimensional simple lie algebras, including the representation of \\widehat \mathfrak sl2, \mathbb c\, root systems, the cartan matrix, and a dynkin diagram of a finitedimensional simple lie algebra. The natural filtration of finite dimensional modular lie. Introduction to lie algebras 2006 studies in lie theory 2006.
This refinement takes into account the second grading on the vassiliev invariants induced by cabling of knots. Geometry and analysis on finite and infinitedimensional lie groups 2002 complex semisimple lie algebras 2001. Jag, h denotes the category of all gmodules that decompose into direct sums of. In mathematics, a lie superalgebra is a generalisation of a lie algebra to include a z 2grading. Infinite dimensional lie algebras an introduction progress in. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Ja a denotes the category of all left amodules, where a is a relie algebra. Let be a complex generally infinitedimensional lie superalgebra. Pdf representations of simple finite lie conformal. Infinitedimensional primitive linearly compact lie superalgebras.
Introduction to lie algebras, issn 16152085, 2006, 251 pages. We construct new examples of infinite dimensional lie superalgebras with a zgradation of infinite depth and finite growth and clas sify zgraded lie superalgebras of infinite depth and finite. Pdf zgraded lie superalgebras of infinite depth and finite. In this paper, we study the finite representation theory of the map lie conformal algebra g. We classify open maximal subalgebras of all infinitedimensional linearly compact simple lie superalgebras. Pdf infinitedimensional groups and their representations. In the same chapter we also give an alternative proof of the result that nite dimensional representations of a ne lie algebras decompose into tensor products of evaluation. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions but this is not always true. Several folklore results are rigorously proved and occasionally corrected in detail, sometimes with new proofs. Introduction hopf superalgebras are a generalization of the supergroup notion. Infinite dimensional primitive linearly compact lie superalgebras. It was found that finite dimensional representations of simple lie algebras are most simply described within the framework of the category g consisting of infinite dimensional modules satisfying some natural conditions 15, 93.
A, where g is a finite simple lie conformal algebra and a is a commutative associative algebra with unity over in particular, we give a complete classification of nontrivial finite irreducible conformal modules of g. The classification of modular lie superalgebras of type m. Mixed cohomology of lie superalgebras pdf free download. Flie algebras are natural generalizations of lie algebras f1 and lie superalgebras f2. Classification of finite dimensional modular lie superalgebras 3 3. Finite irreducible representations of map lie conformal. Part a is mainly concerned with the structure and representation theory of infinite dimensional lie algebras and contains articles on the structure of directlimit lie algebras, extended affine lie algebras and loop algebras, as well as representations of loop algebras and kacmoody superalgebras. The starting point of our studies is to construct a class of finite dimensional modular lie superalgebras of special type, which is denoted by. Infinitedimensional lie superalgebras and hook schur functions authors. An introduction lecture notes in mathematics movie to download full. Extended affine lie algebras an introduction to their structure theory, erhard neher, university of ottawa, canada. The classical lie algebras are finite dimensional lie algebras over a field which can be classified into four types, and, where for the general linear lie algebra and the.
This is applied to the classification of infinitedimensional lie superalgebras of vector fields, acting transitively and primitively in a formal neighborhood of a point of a finite dimensional supermanifold. In this article we construct finitedimensional flie algebras f2 by an inductive process starting from lie algebras and lie superalgebras. Finitedimensional lie algebra an overview sciencedirect. Shmelev, irreducible representations of infinite dimensional hamiltonian and poisson lie superalgebras and invariant differential operators, serdika blg. On vassiliev invariants not coming from semisimple lie algebras. For the above listed lie superalgebras g ga, the existence of restrictedness is explicitly. Petrogradskyinfinite dimensional lie superalgebras. We prove a refinement of vogels statement that the vassiliev invariants of knots coming from semisimple lie algebras do not generate all vassiliev invariants. Continuing on, the main subjects of the book are the structure real and imaginary root systems of and the.
Introduction to lie algebras, issn 16152085, 2006, 251. The classical lie algebras are finitedimensional lie algebras over a field which can be classified into four types. Abstractwe classify open maximal subalgebras of all infinite dimensional linearly compact simple lie superalgebras. Lie algebra modules with finite dimensional weight spaces, i 759 i denotes an adhstable reductive complement to u in p. We introduce the class of classically semisimple locally finite lie superalgebras over an algebraically closed field k of characteristic 0 and classify all lie superalgebras in this class. Bryanton the fixed points of a finite group acting on a free lie algebra. Representations of lie superalgebras with fusion flags. Cominuscule parabolic subalgebras of simple finite. Explicit expressions for the transformation of the basis under the action of algebra generators are found. The similar result was obtained for the infinitedimensional case. It features lie superalgebras both of finite and of infinite dimension, kacmoody and borcherdskacmoody superalgebras, and spreads this material out over almost 500 pages. Homology, homotopy and applications project euclid.
Finite and infinite dimensional lie algebras and lie superalgebras. For any finitedimensionai gmodule f which admits a nondegenerate symmetric invariant bilinear form, we dctine. For a large class of finitedimensional lie superalge. I will also discuss the classes of integrable modules and bounded modules. Read cominuscule parabolic subalgebras of simple finite dimensional lie superalgebras, journal of pure and applied algebra on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Infinitedimensional lie superalgebras finite wsuperalgebras for basic lie superalgebras mixed schurweylsergeev duality for queer lie superalgebras discrete series of lie superalgebras. In chapter 1, we find explicit formulas for the top singular vectors in any verma module of an affine kacmoody lie superalgebraor called a superloop algebra whose irreducible quotient is a unitary integrable representation.
Part a is mainly concerned with the structure and representation theory of infinitedimensional lie algebras and contains articles on the structure of directlimit lie algebras, extended affine lie algebras and loop algebras, as well as representations of loop algebras and kacmoody superalgebras. N sthanumoorthy lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. The original motivation for this paper comes from the researches of structures for the finite dimensional modular lie superalgebras and, which were first introduced in 20, 21, respectively. Exposure to published, written works has greatly influenced his writing, as other writers rhythm, flow and observed style of writing, always affect your work. Application to the classification of finite order automorphisms. Finitedimensional lie subalgebras of the weyl algebra. Kacmoody superalgebras and integrability vera serganova dept. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. Finite versus infinite, making writing skills better.
Representations of simple finite lie conformal superalgebras of type w and s article pdf available in journal of mathematical physics 474 march 2010 with 80 reads how we measure reads. History of the development of finitedimensional lie algebras is described in the preface itself. Any finitedimensional lie algebra over c is the semidirect sum of a semlsimple algebra and a maximal solvable. But the complete classification of the simple modular lie superalgebras remains an open problem. The natural filtrations of finite dimensional modular lie algebras of cartan type were proved to be invariant in 9, 10. Infinitedimensional lie algebras, theta functions and. Infinite dimensional lie algebras an introduction progress.
An analogous category of modules can also be defined over kacmoody algebras. This is applied to the classification of infinite dimensional lie superalgebras of vector fields, acting transitively and primitively in a formal neighborhood of a point of a finite dimensional supermanifold. Methods from the theory of representations of infinitedimensional lie algebras of vector fields the special vectors of rudakov 53, 54 and analogues of poincares lemma occupy. As an application we get an amelioration of the actually known lower bounds for the dimensions of the space of vassiliev. Quantum stochastic calculus and representations of lie superalgebras 1998. For illustration, we explicitly give presentations in terms of analogs of chevalley generators of finite dimensional lie superalgebras with indecomposable cartan matrix in characteristic 2 and in the arxiv version of the paper in other characteristics for completeness of the. Without further explanations connected, compact lie group with trivial center 1. Infinitedimensional lie superalgebras sciencedirect. An important category of supermodules is the category of finitedimensional supermodules which are completely reducible over the category has enough projectives and is in general not semisimple. Developments and trends in infinitedimensional lie theory. Simple lie algebras in characteristic 2 recovered from. The theory of modular lie superalgebras has obtained many important results during the last twenty years e. Pdf dualities and representations of lie superalgebras. It is based on courses given over a number of years at mit and in paris, and is sufficiently selfcontained and detailed to be used for graduate courses.
For modular lie superalgebras, new notions are introduced. Berezinintroduction to algebra and analysis with anticommuting. Divided power homology and divided power cohomology. In physics, infinite dimensional hopf superalgebras based on lie superalgebras turn out to be related to the integrable smatrix of the adscft correspondence. While methods and constructions from the theory of weight modules over simple nite dimensional lie algebras apply, interesting phenomena speci c to gl1 arise. This book gives a systematic account of the structure and representation theory of finite dimensional complex lie superalgebras of classical type and serves as a good introduction to representation theory of lie superalgebras. Representations of infiniterank kacmoody lie superalgebras of classical types and super duality. Introduction to finite and infinite dimensional lie algebras and superalgebras introduces the theory of lie superalgebras, their algebras, and their representations. Finite and infinite dimensional lie algebras and applications in physics studies in mathematical physics pt.
Lie theory has its name from the work of sophus lie 6, who studied certain transformation groups, that is, the groups of symmetries of algebraic or geometric objects that are now called lie. Descargar infinitedimensional lie algebras en pdf libros. For type i lie superalgebras, the category is a highest weight category as defined by cline, parshall, and scott. Therefore, the theory of finitedimensional lie superalgebras is reduced in a certain sense to the. Lectures on infinitedimensional lie algebra minoru wakimoto kyushu university, japan. Three important dualities are presented in the book. Every nite dimensional complex lie algebra which is a direct sum of simple lie algebras is called semisimple.
Representation theory of finite dimensional lie algebras. Sthanumoorthy, in introduction to finite and infinite dimensional lie superalgebras, 2016. Complexity and module varieties for classical lie superalgebras. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. Introduction to lie algebras covers the core material required for almost all other work in lie theory and provides a selfstudy guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics. When f2 not many finitedimensional examples are known. Lie algebras of lie groups, kacmoody groups, supergroups and some specialized topics in finite and infinite dimensional lie algebras. I will discuss the classi cation of irreducible weight modules with nite dimensional weight spaces over gl1. These representations are particular lowest weight representations vp, with a lowest weight of the form, p being a positive integer. We now give a brief account of the main features of the theory of finite dimensional lie superalgebras. An application of lie superalgebras to affine lie algebras. By definition, a countably dimensional locally finite lie superalgebra g.
The starting point of our studies is to construct a class of finite dimensional. C rlieg where lieg is the tangent space of g at the. Representations of vertex operator algebras and superalgebras are studied. Pdf this article provides an introduction to the representation theory of banach lie groups of. History of the development of finite dimensional lie algebras is described in the preface itself. Its accordingly rather an encyclopedic business, at the same time sharply focused, and this requires a sophisticated reading audience. Classically semisimple locally finite lie superalgebras. Shunjen cheng, ngau lam submitted on 5 jun 2002 v1, last revised 17 jan 2003 this version, v2. The moyal algebra is an infinitedimensional lie algebra that contains all classical lie algebras as subalgebras. Highest weightrepresentations ofinfinite dimensional liealgebras xi 9. Introduction to finite and infinite dimensional lie superalgebras 1st edition isbn.1323 466 590 281 871 1519 146 577 559 466 865 1059 584 976 886 959 786 312 1215 491 826 96 810 473 568 1389 109 1168 1383 1323 743 480 301 1060 1238 1365 237 1299 1366 266 855 632 612 1265 1346 563 69 832