THE CLASSIFICATION OF SIMPLE COMPLEX LIE …

1 THE CLASSIFICATION OF SIMPLE COMPLEX LIE ALGEBRASJOSHUA paper introduces Lie groups and their associated Lie the goal of describing SIMPLE Lie groups, we analyze semisimple complexLie algebras by their root systems to classify SIMPLE Lie algebras. We assumea background in linear algebra , differential manifolds, and covering Introduction12. Lie Groups and Lie Algebras23. The Exponential Map and Adjoint Representation44. Covering Groups85. Fundamentals of Lie Algebras106. Semisimple Lie Algebras and the Killing Form127. Representations ofsl2(C)158. Root Space Decomposition of Semisimple Lie Algebras179.

2 Root Systems2010. The CLASSIFICATION the intersection of differential topology and algebra , Lie groups are smoothmanifolds with a compatible group structure. In the first section we introduce Liegroups with several canonical examples. While many classical Lie groups are matrixgroups, we shall develop the theory in its full generality. The primary aim of thispaper will be to describe SIMPLE Lie groups, or connected Lie groups possessing noproper, nontrivial analytic normal subgroups. In contrast to the colossal projectconstituting the CLASSIFICATION of SIMPLE finite groups, the CLASSIFICATION of simpleLie groups is greatly simplified by exploiting the manifold structure.

3 In particular,every Lie group has an associated Lie algebra consisting of its tangent space atthe identity equipped with a bracket operation. A theme of this paper will be toinvestigate the relationship between Lie groups and their Lie algebras. We developtools like the exponential map and adjoint representation to translate informationbetween the two objects. For example, we show that a Lie group is SIMPLE if andonly if its Lie algebra is SIMPLE , or contains no proper, nontrivial : August 24, BOSSHARDTW hile an arbitrary Lie group is not completely determined by its Lie algebra ,there is a bijective correspondence between Lie algebras and simply-connected Liegroups.

4 Moreover, any Lie group can be realized as the quotient of its universalcovering group by a discrete central subgroup. This reduces the problem of findingsimple Lie groups to classifying SIMPLE simply-connected Lie groups and therebyto classifying SIMPLE Lie algebras. With this end in mind, the rest of the paperfocuses on developing the theory of Lie algebras. We introduce representations ofsemisimple Lie algebras to decompose them into their root spaces. A root sys-tem, encoded in its associated Dynkin diagram, bears all the information aboutits Lie algebra . As the roots of semisimple Lie algebras satisfy several restrictivegeometrical properties, we can classify all irreducible root systems by a brief seriesof combinatorial arguments.

5 After unwinding the equivalences between this cast ofobjects, the result will finally provide a CLASSIFICATION of SIMPLE COMPLEX Lie algebrasand a major step in classifying SIMPLE Lie Groups and Lie AlgebrasDefinition groupGis a smooth manifold with a group operation suchthat multiplicationm:G G Gand inversioni:G Gare most straightforward example of a Lie group isRnwith smoothstructure given by the identity and group operation given by vector addition. Sim-ilarly,Cnis a Lie group of dimension Lie group is the torusTn=Rn/Znwith smooth structure given byprojections of small neighborhoods inRnand group operation given by additionmodulo the integer linear groupGLn(R) ={X Mn n(R)|det(X)6= 0}representinglinear automorphisms ofRnis an open subset ofRn2and therefore a manifold ofdimensionn2.

6 Matrix multiplication and inversion are rational functions in thecoordinates that are well-defined onGLn(R), so the group operations are ,GLn(C) ={X Mn n(C)|det(X)6= 0}is a Lie group of dimension of2n2. Many classical Lie groups are closed subgroups ofGLn(R) orGLn(C).Thespecial linear groupSLn(R) ={X GLn(R)|det(X) = 1}representsvolume and orientation preserving automorphisms ofRn. Using elementary meth-ods from the theory of smooth manifolds, one can showSLn(R) is a Lie group ofdimensionn2 groupOn={X GLn(R)|XXt=In}representing automor-phisms ofRnwhich preserve the standard inner product is a closed subgroup ofdimensionn(n 1)2.

7 The group consists of two connected components depending onthe sign of the determinant. Thespecial orthogonal groupSOn=On SLn(R) isa connected Lie group of dimensionn(n 1) course, we also have the corresponding Lie groups overC, such asSLn(C),SOn(C), andSp2n(C). They can be embedded as subgroups ofGL2n(R).Theunitary groupU(n) ={X GLn(C)|XX =In}representing auto-morphisms ofCnwhich preserve the Hermitian inner-product is a Lie group ofdimensionn2. Thespecial unitary groupSUn=SLn(C) U(n) is a Lie group ofdimensionn2 CLASSIFICATION OF SIMPLE COMPLEX LIE ALGEBRAS3 LetJ=(0In In0).Thesymplectic groupSp2n(R) ={X GL2n(R)|XtJX=J}represents automorphisms ofR2npreserving the nondegenerate skew-symmetric inner-product represented by the matrixJ.

8 It is a Lie group of dimension2n2+ first observe the interaction of the algebraic and topological properties ofa Lie groupGby considering its tangent bundle. For anyxinGthe left actionLx:G Gsending an elementyto the productxyis a diffeomorphism. Hence thecollection of differential mapsd1Lx:T1(G) Tx(G) smoothly rotate the tangentspace at the identity to any point alongG. The tangent space at the identitytherefore describes the entire tangent bundle. For example, the mapsd1 Lxrotateany fixed vectorv T1(G) alongGto define a vector field. We say that a vectorfieldXisleft-invariantifXx=d1Lx(v) for some fixedv T1(G).

9 Clearly thesubspace of left-invariant vector fields is isomorphic toT1(G).A vector fieldXonGacts on a smooth real-valued functionf:G Rat a pointx GbyXx(f) =dxf(Xx). As this action defines a new smooth functionX(f), weapply another vector fieldYto define the smooth functionY X(f). There generallydoes not exist a vector field corresponding to the action ofY X. However, an easycalculation in coordinates shows that there is a unique vector field corresponding tothe commutatorXY Y X. Define a bracket operation on vector fields by choosing[X,Y] to be the unique vector field satisfying [X,Y](f) = (XY Y X)(f) for allsmooth left-invariant vector fields onG, then so is the bracket [X,Y].

10 Using the correspondence between left-invariant vector fields and vectors inT1(G),we obtain a bracket operation onT1(G).Definition a Lie groupG, the tangent space at the identityT1(G)equipped with the described bracket [ , ] :T1(G) T1(G) T1(G) is theLiealgebra of G. We denote the Lie algebra inspection, for any Lie groupGthe bracket operation on the Lie algebra ofGsatisfies the following properties:(1) Bilinearity(2) Antisymmetry: [x,x] = 0 for anyx g.(3) The Jacobi identity: [[x,y],z] + [[z,x],y] + [[y,z],x] = 0 for allx,y,z value of the above relations is that they characterize general properties of thebracket operation expressible purely in terms of vectors ing.