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Monday, November 9, 2020 | History

3 edition of Some Generalized Kac-Moody Algebras with Known Root Multiplicities found in the catalog.

# Some Generalized Kac-Moody Algebras with Known Root Multiplicities

Written in English

Subjects:
• Fields & rings,
• Mathematics,
• Science/Mathematics,
• Algebra - Linear,
• General,
• Kac-Moody algebras,
• Root systems (Algebra)

• The Physical Object
FormatMass Market Paperback
Number of Pages119
ID Numbers
Open LibraryOL11420011M
ISBN 100821828886
ISBN 109780821828885

The Conference on Generalized Kac-Moody Algebras at the Mathematical Research Institute at Oberwolfach, Germany, organized by Richard Borcherds and Peter Slodowy, July , The rd Meeting of the AMS, Special Session on Representations of Lie Algebras, April , , State University of New York at Buffalo, NY. It is now known that lying behind monstrous moonshine is a certain conformal field theory having the Monster group as symmetries. The conjectures made by Conway and Norton were proved by Richard Borcherds in using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. It is a Borcherds-Kac-Moody Lie algebra with Lorentzian root lattice; and has an associated automorphic form having a product expansion describing its structure. Lie superalgebras are generalizations of Lie algebras, useful for depicting supersymmetry – the symmetry relating fermions and bosons. Abstract: We discuss a method for estimating root multiplicities for symmetric Kac-Moody algebras. Our work with Baumann and Kamnitzer implies that certain root multiplicities coincide with the number of stable'' irreducible components of a quiver variety. We associate to a component its string data, a word in the nodes of the Dynkin diagram.

methode of Kac-Moody algebras. We will follow very closely Kac’s book [4]. AMS Subject Classiﬁcation: 17B Key words: nilpotent, Kac-Moody algebras, hyperbolic algebras, root spaces. 1 Kac-Moody algebras. Deﬁnition Let A(aij) 1 • i,j • l be a square matrix with entries in Z with the following properties: (i) aii = 2, (ii) aij Author: X. Agrafiotou.

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### Some Generalized Kac-Moody Algebras with Known Root Multiplicities by Peter Niemann Download PDF EPUB FB2

The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice.

As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including $$AE_3$$. Some Generalized Kac-Moody Algebras With Known Root Multiplicities Article in Memoirs of the American Mathematical Society () February with 8 Reads How we measure 'reads'.

Abstract: Starting from Borcherds' fake monster Lie algebra we construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly.

The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. Some generalized Kac-Moody algebras with known root multiplicities.

[Peter Niemann] Home. WorldCat Home About WorldCat Help. Search. Search for Library Items Search for Lists Search for Contacts Search for a Library. Create Generalized Kac-Moody algebras -- 2. Modular forms -- 3. Lattices and their theta-functions -- 4.

Get this from a library. Some generalized Kac-Moody algebras with known root multiplicities. [Peter Niemann] -- Introduction Generalized Kac-Moody algebras Modular forms Lattices and their Theta-functions The proof of Theorem The real simple roots Hyperbolic Lie algebras Appendix A.

However little in known about hyperbo lic Kac-Moo dy algebras, (but more is k nown concerning generalized Some Generalized Kac-Moody Algebras with Known Root Multiplicities book a lgebras).

In particular the root multiplicit ies are k nown only for a small. Root multiplicities, realizations, and homology modules for some extended-hyperbolic Kac-Moody algebras for E H A 1 (1) and E H A 2 (2) [60] are also discussed in detail in this chapter.

For a general study of finite and affine Lie algebras, one can refer Carter [ 94 ]. Using the coset construction, we compute the root multiplicities at level-3 for some hyperbolic Kac–Moody algebras including the basic hyperbolic extension of A 1 1 and E This is a preview of subscription content, log in to check by: 8.

In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple lized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds best known example is the monster Lie algebra.

GEOMETRY OF THE ROOT SYSTEM In this section we prove or state some facts about the root system and Weyl group of a GKM G that generalize results about Kac-Moody algebras.

In particular we prove enough about the root system so that the arguments in Kac [3] can be used to prove that (,)o is almost positive definite, and that G is simple provided Cited by: The lifted root number conjecture and Iwasawa theory - Jürgen Ritter and Alfred Weiss: MEMO/ Almost commuting elements in compact Lie groups - Armand Borel, Robert Friedman and John W.

Morgan: MEMO/ Some generalized Kac-Moody algebras with known root multiplicities - Peter Niemann: MEMO/ In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently discovered them) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such.

Some generalized Kac-Moody algebras with known root multiplicities / Peter Niemann. PUBLISHER: Providence, RI: American Mathematical Society, The lifted root number conjecture and Iwasawa theory / Jürgen Ritter, Alfred Weiss.

PUBLISHER: Providence, RI: American Mathematical Society, Standard integral table algebras. is not usually a Kac-Moody algebra, but is one of these more general algebras. There is also a generalized Kac-Moody algebra associated to any even Lorentzian lattice of dimension at most 26 or any Lorentzian lattice of dimension at m and we give a simple formula for the multiplicities.

DIMENSIONS OF ROOT SPACES OF HYPERBOLIC KAC–MOODY ALGEBRAS 5 3. Works of Feingold and Frenkel, and Kac, Moody and Wakimoto Pioneering works on root multiplicities of hyperbolic Kac–Moody algebras began with the paper by Feingold and Frenkel [FF], where the hyperbolic Kac–Moody algebra Fof type HA(1) 1 was considered.

Using the same method. Vertex algebras, Kac-Moody algebras, and the Monster. Proc Natl. Acad. Sci. USA Vol. 83, pp Richard E. Borcherds, Trinity College, Cambridge CB2 1TQ, England.

Communicated by Walter Feit, Decem ABSTRACT It is known that the adjoint representation of any Kac-Moody. generalized Kac-Moody algebras, because all the positive norm roots of a generalized Kac-Moody algebra with root lattice II s+1,1 must have norm 2, which means that the function f(τ) cannot have any terms in qn for n≤ −2.

We conclude with an example of a generalized Kac-Moody algebra related to one of the modular forms in theorem An Exposition of Generalized Kac-Moody algebras Elizabeth Jurisich Abstract.

We present a detailed exposition of the theory of generalized Kac-Moody algebras associated to symmetrizable matrices. A proof of the character formula for a standard module is given, generalizing the argument of Garland and Lepowsky for the Kac-Moody case.

Assume that $\hat{Q}$ and $\Gamma$ is the generalized Mckay quiver and the valued graph corresponding to $(Q, G)$ respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $\mathfrak{g}(\Gamma)$.

Thus, Kac–Moody algebras are infinite-dimensional analogues of the finite-dimensional semi-simple Lie algebras. A systematic study of Kac–Moody algebras was started independently by V.G. Kac and R.V. Moody, and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac–Moody algebras.

Frenkel [14]. We use some standard results about Kac-Moody algebras which can all be found in Kac’s book [13], and the results about generalized Kac-Moody algebras that we need are in Borcherds [2,3] and are summarized in Section 2. The results about the geometry of the Leech.

For Kac-Moody algebras the root multiplicities of only the finite and affine algebras are explicitly known. In this paper, the third of a series of articles on the structure of non-finite, non-affine Kac-Moody algebras, we study certain indefinite Kac-Moody algebras coming from the special linear Lie algebra A n.

Intuitive meaning. The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed".

One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra. Concerned with Kac-Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations, this is the third revision of an important monograph.

Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems/5(8). Generalized Kac-Moody algebras, automorphic forms and Conway’s group I Nils R.

Scheithauer, Mathematisches Institut, Universit at Heidelberg, D Heidelberg [email protected] The moonshine properties imply that the twisted denominatoridentities coming from the action of the monster group on the monster algebra de ne modular forms.

You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read.

Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Free ebooks since   This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular by: $\begingroup$ It would help to clarify here what you mean by the symbol $\mathfrak{u}(n)$.

Aside from that, your screen name seems to imply that you are looking at Kac-Moody theory from the physics viewpoint. I'm mainly acquainted with some texts taking a more mathematical viewpoint, which might or might not be useful to you: Kac Infinite Dimensional Lie Algebras (3rd ed., Cambridge, algebras, the afﬁne Lie algebras admit a classiﬁcation in terms of Dynkin dia-grams.

For the remainder of the Kac-Moody algebras, those of so-called inde-terminate type, little is known. See, for example, [Moo79]. In this chapter, we describe the structure of Kac-Moody algebras ﬁrst in general, and then focusing on those of afﬁne Size: KB.

The CFT of the SU(N) level k Kac-Moody current algebra has many Kac-Moody primary fields. I wonder if any one has calculated the quantum dimensions of those Kac-Moody primary fields. Dual graded graphs for Kac–Moody algebras Lam, Thomas and Shimozono, Mark, Algebra & Number Theory, ; Quiver varieties and cluster algebras Nakajima, Hiraku, Kyoto Journal of Mathematics, ; Affine Kac-Moody algebras, CHL strings and the classification of tops Bouchard, Vincent and Skarke, Harald, Advances in Theoretical and Mathematical Physics, AN INTRODUCTION TO AFFINE KAC-MOODY ALGEBRAS 5 3.

Affine Kac-Moody algebras A natural problem is to generalize the theory of ﬁnite dimensional semi-simple Lie algebras to inﬁnite dimensional Lie algebras. A class of inﬁnite dimensional Lie algebras called aﬃne Kac-Moody algebra is of particular importance for this : David Hernandez.

Idea. The notion of Kac-Moody Lie algebra is a generalization of that of semisimple Lie algebra to infinite dimension of the underlying vector space. Definition () Examples.

The higher Kac-Moody analogs of the exceptional semisimple Lie algebras E7, E7, E8 are. affine: E9 hyperbolic: E10, Lorentzian: E11, Related concepts. Kac-Moody group. Borcherds algebra. Construction of Kac-Moody Lie algebras.

Intergrable representations of Kac-Moody Lie algebras. Classification of Kac-Moody Lie algebras. Affine Lie algebras. Character formula for the intergrable highest weight modules. If time permit we shall also see some connections with theta functions.

Prerequisites: Basics of linear algebra and Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras.

With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. so as to see Kac-Moody groups as discrete groups { lattices for Fq large enough { of their geometries.

Tools.| Let us talk about tools now, and start with the main di culty: no algebro-geometric structure is known for split Kac-Moody groups as de ned by J. Tits. The idea is to replace this structure by two well-understood Size: KB. Well, you have the theory of Coxeter groups that can be defined in two following ways (see Reflection Groups and Coxeter Groups (Cambridge Studies in Advanced Mathematics): James E.

Humphreys: : Books for full details). We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram.

We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of. Buy Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras Book Condition: Shows some signs of wear, and may have some markings on the inside.

% Money Back Guarantee. Shipped to over one million happy customers. In by: Some de nitions we will need to introduce Kac-Moody algebras: De nition 1.

An (associative) algebra is a vector space Aover k = R or C together with a bilinear map: A A!Acalled multiplication that is associative, i.e. (a b) c= a (b c), 8a;b;c2A.

If, in addition, there exists an e2Asuch that e a= a e= a, 8a2A, eis called the unit and Ais called 1. A group corresponding to a Kac-Moody algebra is a Kac-Moody group. This subsumes the case of affine Lie algebras in which case the corresponding Kac-Moody group is a centrally extended loop group (see there for more) and often the words “Kac-Moody group” are used synonomously with “centrally extended loop group”.

Properties Classifying.Title: A correction factor for Kac-Moody groups and t-deformed root multiplicities. Abstract: We will discuss a correction factor which arises in the theory of p-adic Kac-Moody groups, for example in formulas for Whittaker functions in the infinite dimensional setting.Some appendices recall useful results from other areas, so the work may be considered self-contained, although some familiarity with semi-simple Lie algebras or algebraic groups is helpful.

It is clear that this book is a valuable reference for all those interested in flag varieties and representation theory in the semi-simple or Kac-Moody case."Price: \$