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Classification of Semisimple Lie Algebras

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Classification of Semisimple Lie Algebras Classification of Semisimple Lie Algebras John Austin Charters Advisor: Dennis Snow Co-advisor: Antonio Delgado 1 Contents 1 Introduction 3 2 Lie groups 3 3 Lie algebras 7 3.1 Solvable and nilpotent Lie algebras . 10 3.2 Killing form . 14 4 Root spaces 17 4.1 Irreducible representations of sl(2; C)................ 17 4.2 Semisimple Lie algebras . 19 5 Root systems 24 5.1 Dynkin diagrams . 28 6 An application to physics 33 6.1 Lorentz group . 33 6.2 Lorentz algebra . 35 2 1 Introduction My thesis describes the theory of Lie groups and Lie algebras, named after the Norwegian mathematician Sophus Lie. Lie was interested in groups whose ele- ments depend smoothly on continuous parameters. His work principally focused on transformation groups, differential equations, and differential geometry. I will focus instead on the algebraic theory. The approach to learning more about Lie groups is to study the linearization of the group at the identity. Such a linearization is called the Lie algebra associated to the Lie group. It is far easier to analyze the algebra, as it takes the structure of a vector space. I will then explain what it means for a Lie algebra to contain a semisimple Lie algebra. The semisimple components are described using geometric structures called root systems, whose classification was completed by the French mathematician Elie´ Cartan. I will introduce root systems and describe the details of the classification. The principal result of my thesis is the list of diagrams on page 31. The theory of continuous groups has many applications to physics and other areas of mathematics. I conclude with an introduction to the Lorentz group and the Lorentz algebra, which arise in physics. 2 Lie groups Definition 1. A Lie group is a smooth manifold G endowed with a group structure, such that the group operation and the inverse map are smooth. In general, Lie groups can be defined abstractly. For simplicity, we will only consider matrix Lie groups, which are Lie groups that can be expressed as matrices. Denote the set of n × n matrices with complex entries by M(n; C). The general linear group GL(n; C) is the subset of M(n; C) consisting of n × n invertible matrices. That is, GL(n; C) = fX 2 M(n; C) : det X 6= 0g. To verify that GL(n; C) is a group, it follows from the identity det XY = det X det Y that GL(n; C) is closed under multiplication and inversion. Certainly GL(n; C) contains the identity, and matrix multiplication is always associative. Further- more, the operations of multiplication and inversion are given by linear func- tions, so they are smooth. To see why GL(n; C) is also a manifold, first observe that the subset of matrices in M(n; C) with vanishing determinant is closed in 2 M(n; C) ' Cn . This implies that GL(n; C) is open in M(n; C) and thus it 2 inherits a manifold structure from Cn . Therefore, GL(n; C) is a Lie group. There is a powerful result that any closed subgroup of GL(n; C) is a manifold, hence a Lie group. For a proof, see [3]. So all of the matrix Lie groups we will consider are to be thought of as closed subgroups of GL(n; C). For example, consider the special linear group SL(n; C), defined as the subset fX 2 GL(n; C): det X = 1g. It is easy to verify that SL(n; C) is a group. Also, SL(n; C) is closed because the determinant is a continuous function, so any convergent sequence in SL(n; C) must converge inside SL(n; C). Therefore, SL(n; C) is a matrix Lie group. 3 Another example of a matrix Lie group is the orthogonal group O(n), which is the set of real matrices that preserve the standard inner product on Rn, hence preserving lengths and relative angles of vectors. This yields the condition O(n) = fX 2 GL(n; R): XT = X−1g. Furthermore, the special orthogonal group SO(n) is the subset of O(n) with determinant unity, i.e. SO(n) = fX 2 O(n) : det X = 1g. It is a matrix Lie group. The group SO(3) is generated by the following 3 × 3 matrices that rotate R3 about the standard coordinate axes: 01 0 0 1 R1(θ) = @0 cos θ − sin θA ; 0 sin θ cos θ 0 cos φ 0 sin φ1 R2(φ) = @ 0 1 0 A ; − sin φ 0 cos φ 0cos − sin 01 R3( ) = @sin cos 0A : 0 0 1 We would like to examine what SO(3) looks like near the identity. By j expanding the rotation matrices up to first order only, say Rj(θ) = I + iθτ , we easily discover the following matrices 00 0 01 00 0 −11 0 0 1 01 1 2 3 τ = i @0 0 1A ; τ = i @0 0 0 A ; τ = i @−1 0 0A : 0 −1 0 1 0 0 0 0 0 Let [X; Y ] denote the commutator XY − YX. Then the τ j satisfy the com- i j k mutation relation [τ ; τ ] = iijkτ . These matrices are called the infinitesimal generators of SO(3), as they are said to generate the group as follows. For any n × n matrix X, we define the matrix exponential of X as the power series 1 X Xk eX ≡ ; (1) k! k=0 which always converges entrywise. It is clear from the definition that the matrix exponential has the property that d etX = XetX = etX X; dt and in particular, d tX e = X: (2) dt t=0 Remarkably, the matrix exponential of the infinitesimal generators of SO(3) recovers the rotation matrices: iθτ j e = Rj(θ): (3) 4 Apparently, linear deviations from the identity were enough to retain all the information in the generators of SO(3). Our next examples of matrix Lie groups are the unitary group U(n) and the special unitary group SU(n), which are analagous to O(n) and SO(n). They are defined as complex matrices that preserve the standard Hermitian inner product on Cn. This gives the condition that U(n) = fX 2 GL(n; C): Xy = X−1g, where Xy represents the adjoint|or conjugate transpose|of a matrix X. Likewise, SU(n) = fX 2 U(n) : det X = 1g. It turns out that the Pauli matrices, 0 1 0 −i 1 0 σ = ; σ = ; σ = ; 1 1 0 2 i 0 3 0 −1 ijk which satisfy the commutation relation [σi=2; σj=2] = i σk=2, are the in- finitesimal generators of SU(2). To see this, when we exponentiate the Pauli matrices, we get cos θ=2 i sin θ=2 U (θ) = eiθσ1=2 = ; 1 i sin θ=2 cos θ=2 cos φ/2 sin φ/2 U (φ) = eiφσ2=2 = ; 2 − sin φ/2 cos φ/2 ei =2 0 U ( ) = ei σ3=2 = : 3 0 e−i =2 Note that these matrices have determinant 1 and satisfy U y = U −1, which means they are indeed elements of SU(2). Furthermore, fU1;U2;U3g are independent, so they generate SU(2). Working backwards, it is easy to check that the Pauli matrices are the first order corrections of the Uj, namely Uj(θ) = I+i(θ=2)σj. So once again, linearizing the generators has retained all the original information. The commutator relations for the infinitesimal generators of SO(3) and SU(2) are identical. This suggests a relationship between SO(3) and SU(2), which we will now introduce. Consider the matrix 3 X x3 x1 − ix2 M = xiσi = : x1 + ix2 −x3 i=1 We see that M has trace zero and is self-adjoint, i.e. M is equal to its adjoint M y. Let V denote the space consisting of matrices of this form. We can identify 3 V with R using the coordinates (x1; x2; x3) and the inner product hX; Y i = 1 2 trace XY . For each U 2 SU(2), define an operator ΦU : V ! V by ΦU (X) = −1 UXU . Direct computation shows that each ΦU preserves the inner product. Indeed, 1 hΦ (X); Φ (Y )i = trace UXU −1UYU −1 = U U 2 1 1 trace UXYU −1 = trace XY = hX; Y i; 2 2 5 where the second to last equality follows from the cyclic property of the trace. So ΦU is an orthogonal operator by definition. Moreover, ΦU1 ΦU2 = ΦU1U2 , which implies the map Φ that sends U to ΦU is a homomorphism. Since an arbitrary matrix in SU(2) is of the form α β ; jαj2 + jβj2 = 1; (4) −β α we have that SU(2) is homeomorphic to S3. It follows that SU(2) is simply connected. And since Φ is continuous, we conclude that the image of Φ lies in the identity connected component of O(3), which is SO(3). To see that Φ: SU(2) ! SO(3) is surjective, take any rotation of R3 and express the axis of rotation P as x3 0 −1 P = U0 U0 0 −x3 for some U0 2 U(2). Then the plane orthogonal to P consists of the matrices 0 x1 − ix2 −1 W = U0 U0 : x1 + ix2 0 Let eiθ=2 0 U = U U −1: 0 0 e−iθ=2 0 −1 −1 Then we find that UPU = X and UWU rotates the x1 and x2 components of W by angle θ. Therefore ΦU agrees with our rotation.
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