SOME STUDIES ON DYNKIN DIAGRAMS ASSOCIATED WITH KAC-MOODY ALGEBRA

A PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS SUBMITTED TO NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA BY AMIT KUMAR SINGH ROLL NO. 409MA2075 UNDER THE SUPERVISION OF PROF. K.C. PATI

DEPARTMENT OF MATHEMATICS NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA-769008

May 10, 2011

NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA

DECLARATION I hereby certify that the work which is being presented in the thesis entitled Some studies on Dynkin Diagram associated with Kac-Moody Algebra in partial fulfillment of the requirement for the award of the degree of Master of Science, submitted in the Department of Mathematics, National Institute of Technology, Rourkela is an authentic record of my own work carried out under the supervision of Prof. K.C. Pati. The matter embodied in this thesis has not been submitted by me for the award of any other degree. This is to certify that the above statement made by the candidate is correct to the best of my knowledge.

Prof. K.C. Pati Department of Mathematics National Institute of Technology Rourkela 769 008 Orissa, India.

ACKNOWLEDGEMENT
With immense pleasure I wish to express my profound sense of reverence and gratitude to my supervisor Dr. Kishor Chandra Pati, Department of Mathematics, N.I.T., Rourkela, for his constant encouragement, constructive guidance and thought-provoking discussions throughout the period of my research work, which enabled me to complete the project report so smoothly. I am thankful to Mr. B. Ransingh for his guidance and support and also for his caring attitude towards me. Words at my command are inadequate to convey the profound to my parents whose love, affection and blessings has inspired me the most.

Amit Kumar Singh

ABSTRACT
In the present project report, a sincere report has been made to construct and study the basic information related to Simple Lie Algebras, Kac-Moody algebras and their corresponding Dynkin Diagrams. In chapter-1, I have given the definitions of Lie Algebra and some of the terms related to Lie algebra, i.e. subalgebras, ideals, Abelian, solvability, nilpotency etc. Also, I have done the classifications of Classical Lie algebras. In chapter-2, I addressed the basics of Representation Theory, i.e. structure constants, modules, reflections in a Euclidean space, root systems (simple roots) and their corresponding root diagrams. Then I have discussed the formation of Dynkin Diagrams and cartan matrices associated with the roots of the simple lie algebras. In chapter-3, I have given the necessary theory based on Kac-Moody lie algebras and their classifications. Then the definition of the extended Dynkin diagrams for Affinization of Kac-Moody algebras and the Dynkin Diagrams associated with the affine Kac-Moody algebras are provided

Contents

Introduction 1.1 Basic Definitions, Examples 1.2 Subalgebras & Ideals 1.3 Abelian, solvable & nilpotent 1.4 Simple & Semi-simple lie algebras 1.5 Classical lie algebras
Representations 2.1 Structure Constants 2.2 Root Systems 2.3 Modules & Representations 2.4 Cartan Matrix 2.5 Killing form 2.6 Coxeter Graphs and Dynkin Diagrams 2.7 Cartan matrices of simple lie algebras Kac Moody Algebra 3.1 Basic Definitions 3.2 Types of Kac-Moody Algebras 3.3 Cartan Matrix 3.4 Dynkin Diagrams of Affine Kac-Moody Algebras

References

1 Introduction
1.1 Basic Definitions, Examples:
Before going to my concerned topic KAC-MOODY ALGEBRA, we need to get a precise definition of LIE ALGEBRA, i.e. A Vector Space L over a field F with an operation L L L denoted by ( , ) [ , ] and called the bracket or commutator of and is called a Lie Algebra if the following axioms are satisfied : 1. The bracket operation is bilinear, i.e. [ ]= [ ]+ [ ] for all scalars in F and all elements in L.

2. The bracket operation is skew-symmetric, i.e. [ ] = 0 for all in L. 3. [ , [ ]] + [ , [ ]] + [ , [ ]] = 0 for all in L.

The axiom is called Jacobis Identity. The followings are some of the examples related to Lie algebra.[1] Example 1 : Let A be an algebra over F (a vector space with an associative multiplication x y ). A is a Lie algebra AL (also called A as Lie algebra) by defining [x , y] = x y y x. Example 2 : If A the algebra of all operators (endomorphisms) of a vector space V ; the corresponding AL is called the general Lie algebra of V , gl(V ). Concretely, taking number space Rn as V, this is the general linear Lie algebra gl(n,R) of all n n real matrices, with [x , y ] = x . y y .x. Similarly gl(n,C). Example 3 : The special linear Lie algebra sl(n,R) consists of all n n real matrices with trace 0 (and has the same linear and bracket operations as gl(n,R)it is a sub Lie algebra); similarly for C. For any vector space V we have sl(V ), the special linear Lie algebra of V , consisting of the operators on V of trace 0.[2]

1.2 Subalgebras, Ideals:

A subset K of a Lie Algebra L is called a sub-algebra of L if for all F, one has in K , [ ] in K. in K and all in

An ideal I of a Lie Algebra L is a sub-algebra of L with the property [I,L] is a subset of I, i.e. for all in I and in L one has [ ] in I. Every (non-zero) Lie Algebra has at least two

Ideals, namely the Lie Algebra L itself and the sub-algebra 0 consisting of the zero element only. Both these ideals are called Trivial. All non-trivial ideals are called Proper ideals.[1]

1.3 Abelian, Solvable & Nilpotent:

The Lie Algebra L is called abelian or commutative if [ ] = 0 for all in L.

More generally, a Lie Algebra L is said to be solvable if in the sequence of ideals of L(the derived series) L(0) = L, L(1) = [L,L], L(2)=[L(1),L(1)], L(3)=[L(2),L(2)], ,L(i) = [L(i-1),L(i-1)] L(n) = 0 for some n. e.g. Abelian implies solvable whereas simple algebras are definitely non-solvable. A lie algebra t+(n) of the upper triangular matrices is a prototype of solvable algebras. A Lie Algebra L is said to be nilpotent if in the lower central series of L L0 = L, L1 = [L,L], L2 = [L,L1], , Li = [L,Li-1] n L = 0 for some n. e.g. Any abelian algebra is nilpotent. A lie algebra t++(n) of strictly upper triangular matrices is the prototype of nilpotent algebras. Li for all I, so nilpotent algebras are solvable. But the converse is false.[1]

Clearly, L(i)

1.4 Simple & Semisimple Lie Algebra:

A Lie algebra L is simple if it has no proper ideals and is not abelian. A Lie Algbera L is said to be semisimple if its radical is zero. Equivalently, L is semisimple if it does not contain any non-zero abelian ideals. In particular, a simple Lie algebra is semisimple.[1]

1.5 Classical Lie Algebras:

V is a finite-dimensional vector space over F and denotes End V, the set of linear transformations V V (endomorphisms of V). : Special linear lie algebra, denoted by (V) or ( trace 0. The dimension of is at most ( )2 1. : Orthogonal lie algebra, denoted by (V) or ( the following property : ( ( ), ) = - ( , ( )) or, =- t , F). It is the End V having

, F). It consists of End V satisfying

where = = + l.

)and = (

)The dimension of
(2 , F). The End V satisfies the

is 2

: Sympletic Lie algebra, denoted by (V) or following property : ( ( ), ) = - ( ,, ( )) or, =- t where =

)and = (
is 2 2 + l.

)
is

The dimension of

: Orthogonal lie algebra, denoted by (V) or (2l,F). The construction of identical that for , except that dim V = 2 is even and s has the simpler form

).

[1]

2 Representations
In representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.

Structure Constants:
Let be a lie algebra and take a basis {X1, X2 , . . . , Xn} for (the vector space) . By bilinearity [ , ] operation in is completely determined once the values [Xi , Xj] are known. We know them by writing them as linear combinations of Xi. The coefficients in the relations [Xi , Xj] = Xk (sum [1] over repeated indices !) are called the structure constants of (relative to the given basis).

Modules:
Let be a lie algebra. It is often convenient to use the language of modules along with the (equivalent) language of representations. A vector space V, endowed with an operation VV(denoted by (x,v) x.v ) is called an module if the following conditions are satisfied: ( = ( . )+ ( . ) .( + )= ( )+ ( ) [ ]. = , where + ).

V;

F.[1]

Representations:
A representation of a lie algebra on a vector space V is a homomorphism (say ) of into the general linear lie algebra (V) of V. assigns to each X in an operator depending ] linearly on X (thus, ) and satisfying [ [ ]. A vector space V, together with the representation , is called an -space or -module. For example, the adjoint representation ad : bracket as follows : [ad , ad ] (z) = ad = ad ad ( ) ad ad ( ) , where ad ( ) = [ , ]. It preserves the

([ , ]) ad

([ , ]

= [ , [ , ]] [ , [ , ]] = [ , [ , ]] + [[ , ] , ] = [[ , ] , ] = ad [ , ] ( ).[1]

Reflections in a Euclidean Space:

We are here concerned with a fixed Euclidean space E, i.e. a finite dimensional vector space over R equipped with a positive definite symmetric bilinear form ( , ). Geometrically, a reflection in E is an invertible linear transformation leaving pointwise fixed some hyperplane (subspace of codimension one) and sending any vector orthogonal to that hyperplane into its negative. Ultimately, a reflection is orthogonal, i.e. preserves the inner product on E. Any non-zero determines a reflection , with reflecting hyperplane P = { E: ) = }. An . (This is because it sends to and fixes all points in

vector P ).

explicit formula for

We replace

by

and is linear only in the first variable .[1]

Root Systems:
A subset satisfied: of the Euclidean space E is called a root system in E if the following axioms are

is finite, spans E, and doesnt contain . If , the only multiples of in are . If , the reflection leaves invariant . If , , then Z.[1]

The simple roots can be defined as are the positive roots that cannot be written as the linear combination of other positive roots. If there are r simple roots for an algebra L of rank r and they form a basis of the root system.[7] Algebra Root system , , 2
[4]

, , , , , , ,

Root Diagrams of Lie Algebras[1]

Root Diagram of

Root Diagram of

Root Diagram of

Root Diagram of Cartan Matrix:

A generalised Cartan matrix is a square matrix A = ( For diagonal entries, = 2. For non-diagonal entries, = 0 if and only if = 0. ) with integer entries such that

0.

A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix.

We can always choose D with positive diagonal entries and if S is positive definite, then A is said to be a Cartan Matrix. The Cartan Matrix of a simple lie algebra is the matrix whose elements are the scalar products = .[4]

Killing Form:
Let be any lie algebra. If x , y , define bilinear form on , called the Killing form. is also associative, i.e. It follows from vector space.[1] [ ] [ ] [ [ ] . of a finite dimensional . Then is a symmetric

] for endomorphisms

Coxeter Graphs and Dynkin Diagrams:

are distinct positive roots, then we know that Define the Coxeter graph of to be a graph having l vertices, the edges.

If

= 0, 1, 2 or 3. joined to the ( ) by

Examples[1]:

A1 A1 A2 B2 G2
in case all roots have equal lengths, since then = . In case more than one root length occurs(Ex. B2 or G2), the graph fails to tell us which of a pair of vertices should correspond to a short simple root, which to alone. When a double or triple edge occurs in the coxeter graph, we can add an arrow pointing to the shorter of the two roots. This additional information gives us the cartan integers, and resulting figure Dynkin Diagram.[1] To a cartan matrix is associated a Dynkin Diagram, consisting of vertices representing the simple roots and (oriented) lines connecting them. The Dynkin Diagram an algebra L of rank r is constructed using the following rules: 1. Draw r vertices, one for each simple root . 2. Connect the vertices and with number of lines equal to max{| | , | |}, or equivalently to the product . 3. If | | | |, then draw an arrow pointing towards from , i.e. from the biggest to the smallest root.[7]

The coxeter graph determines the numbers

Examples[1]: B2 G2 Dynkin Diagrams of Simple Lie Algebra[1]

Cartan matrices[1]

3 Kac-Moody Algebra
3.1 Basic Definition
A Kac-Moody algebra can be defined as follows: A generalised cartan matrix C = ( ) of rank .

A vector space W over the complex numbers of dimension . A set of linearly independent elements of W and a set of linearly independent elments of the dual space, such that ( ) = . The are known as coroots and are known as roots. and and the elements of W

The Kac-Moody algebra is the lie algebra defined by the generators and the relations [ [ [ [ [ ad ad , ]= , ] = 0 for ]= for W ]= for W ] = 0 for W ( )=0 ( )=0 End( ), ad =[

where ad :

] is the adjoint representation of .

A real (possibly infinite dimensional) Lie algebra is also considered as a Kac-Moody algebra if its complexification is a Kac-Moody algebra.[3]

3.2 Types of Kac-Moody Algebra

Properties of Kac-Moody algebra depend on the algebraic properties of its generalised cartan matrix C. If C is indecomposable, i.e. assume that there is no decomposition of the set of indices I into a disjoint union of non-empty subsets and such that = 0 for all and . An important subclass of Kac-Moody algebras corresponds to symmetrizable generalised cartan matrices C, which can be decomposed in DS, where D is a diagonal matrix with positive integer entries and S is the symmetric matrix. The Kac-Moody algebras are broadly divided into three classes A positive definite matrix S gives a finite-dimensional simple lie algebra. A positive semidefinite matrix S gives an infinite-dimensional Kac-Moody algebra of affine lie algebra.

An indefinite matrix S gives rise to a Kac-moody algebra of indefinite type. Since the diagonal entries of C and S are positive, S cant be negative definite or negative semidefinite. An indefinite matrix S, but for each proper subset of I, the corresponding submatrix is positive definite or positive semidefinite gives rise to a Kac-moody algebra of hyperbolic type.[3]

3.3 Cartan Matrix of Kac-Moody Algebra

A generalised cartan matrix C = ( ) is defined as follows:

The diagonal entries are all 2. The off-diagonal entries are all either non-positive, with

= 0 if and only if

= 0.

The Cartan matrix is indecomposable. The Cartan matrix is symmetrizable, and the symmetrized matrix is positive definite. if all the entries are negative.

For any column matrix if all the entries are positive, and We now define an cartan matrix C as

If . C is finite if and only if C is symmetric and the symmetrized matrix has signature ( ), If . C is affine if and only if C is symmetric and the symmetrized matrix has signature ( ), If . C is hyperbolic if and only if det C < 0 and deletion of any row and the corresponding column gives a direct sum of affine or finite matrices. matrix .[7]

for some

3.4 Affine Lie Algebra

An affine Lie Algebra is constructed out of a affine cartan matrix C, that has the following conditions = 2, The off diagonal elements are non-positive integers and =0 if and only if =0, det C=0 and deletion of any row corresponding column gives the direct sum of finite cartan matrices.

The matrix C is thus positive semidefinite.[3]

Dynkin diagrams of affine Kac-Moody algebra:

Consider a generalised cartan matrix C = Here ( . . ., = .

) are independent vectors in dimensional Euclidean space. cartan matrix C is obtained by the following rules: . . ., .

The Dynkin diagram associated with the

The diagram has vertices , which correspond to the simple roots

When o . 4 the vertices with . If | | [3] pointing from to .

and are connected by lines and | | , the lines are equipped with an arrow

The possible links between and ( ) are restricted by the above rules. The vertices for generalised cartan matrix(GCM) of finite or affine type are given in the following table[3]: | 0 1 1 2 1 3 1 4 2 | | 0 1 2 1 3 1 4 1 2 |

Dynkin diagram[4]

References
1. Humphreys James E., Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York Hiedelberg Berlin. 2. Samelson Hans, Notes on Lie Algebras, Hans Samelson ,Stanford,Sep 1989. 3. Tripathy L.K., SOME STUDIES ON DYNKIN DIAGRAMS AND SATAKE DIAGRAMS ASSOCIATED WITH KAC-MOODY SUPER) ALGEBRAS, Berhampur University, Orissa (Dec 2006). 4. Frappath L., Sciarrino A., Sorba P., Distionary on Lie Algebras and Superalgebras, An Imprint of Elsevier, California, USA. 5. Fulton W., Harris J., Representation Theory: A first course, Springer-Verlag. 6. Kac Victor G., Infinite Dimensional Lie Algebras Third Edition), Cambridge University Press. 7. Jamsin Ella, Palmkvist J., Lie and Kac- Moody Algebras, Physique Th_eorique et Math_ematique, Universit_e Libre de Bruxelles & International Solvay Institutes, ULB-Campus Plaine C.P. 231, B-1050, Bruxelles, Belgium