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Representations of Lie Algebras, with Applications to Particle Physics REPRESENTATIONS OF LIE ALGEBRAS, WITH APPLICATIONS TO PARTICLE PHYSICS JAMES MARRONE UNIVERSITY OF CHICAGO MATHEMATICS REU, AUGUST 2007 Abstract. The structure of Lie groups and the classication of their representations are subjects undertaken by many an author of mathematics textbooks; Lie algebras are always considered as an indispensable component of such a study. Yet every author approaches Lie algebras dierently - some begin axiomatically, some derive the algebra from prior principles, and some barely connect them to Lie groups at all. The ensuing problem for the student is that the importance of the Lie algebra can only be deduced by reading between the lines, so to speak; the relationship between a Lie algebra and a Lie group is not always illuminated (in fact, there isn't always a relationship in the rst place) and so considering the representations of Lie algebras may appear redundant or at best cloudy. The approach to Lie algebras in this paper should clarify such problems. Lie algebras are historically a consequence of Lie groups (namely, they are the tangent space of a Lie group at the identity) which may be axiomatized so that they stand alone as an algebraic concept. The important correspondence between representations of Lie algebras and Lie groups, however, makes Lie algebras indispensable to the study of Lie groups; one example of this is the Eight-Fold Way of particle physics, which is actually just an 8-dimensional representation of but which has the enlightening property of sl3(C) corresponding to the strong nuclear interaction, giving a physical manifestation of the action of a representation and showing just one of the many interesting ramications of mathematics on science within the past 50 years. 1. Lie Groups and Representations We begin with Denition 1. A Lie group is a group such that the map ω : G × G → G sending (x, y) 7→ xy−1 is C ∞. The important consequence of this is that a Lie group is endowed with a manifold structure. Without looking very far, we can easily nd Lie groups - just take groups of non-singular n ×n matrices over a complete metric space, such as R or C. Then this is locally equivalent to working in the nite- dimensional space Rm or Cm , depending on the number of degrees of freedom characterizing the matrix group. For example, , the group of non-singular real matrices with determinant 1, has SLn (R) dimension n2 − 1. These are elementary facts from linear algebra, and we might create a complete list 1 REPRESENTATIONS OF LIE ALGEBRAS, WITH APPLICATIONS TO PARTICLE PHYSICS 2 of all the classical groups , , and the corresponding complex Lie groups with SOn (R) SUn (R) Spp,q (R) their corresponding dimensions. To see how these might have a manifold structure, consider the map ! a b above as a continous map on a vector space. For example, consider any matrix . ∈ SL2 (R) c d Since ad−cb = 1, we have 3 free parameters but the last element is then xed by the other three in order to get determinant 1. This group consists of many 'components,' one of which is the set of matrices ! a b such that . We can map such a matrix to the element 1+cb 4. This 1+cb a 6= 0 a, b, c, a ∈ R c a denes a subspace of R4 that gives the local structure that denes a manifold; to get dierentiability, the map ω :(x, y) 7→ xy−1 can be dened by ! !! ! 1+fg ! a+afg c+cfg g+gcb ! a b e f a b e −f e − bg e − a 1+cb , 1+fg 7→ 1+cb = e+ecb c a g e c a −g e be − af a − cf corresponding to the map 1 + cb 1 + fg a + afg c + cfg g + gcb e + ecb ω : (a, b, c, ), (e, f, g, ) 7→ ( − bg, − , be − af, − cf). R a e e e a a ∂ω Since any of the six parameters a, b, c, e, f, g can be continuously varied in the reals, we can take R ∂xi ∂ω 1+fg gb for etc. Hence R eb and so forth, and we have a smooth map on xi = a, b, c, ∂c = (0, e − a , 0, a − f) the group. This map gives a manifold structure because it provides a local coordinatization for each element by multiplying The next important concept is Denition 2. A representation of a group G is a smooth group homomorphism ϕ : G → GLn(V ) for some nite-dimensional vector space V . Hence, a representation assigns to each g ∈ G a linear map ϕg : V → V , and one can say for short that V is a representation of G. Given V a nite-dimensional vector space over a eld F , it is clear that if V is an FG-module, V is also a representation of G since we can dene ϕg(v) = g · v. Conversely, if V is a representation of G we can turn it into an FG-module by dening P P ( αigi) · v = αiϕgi (v) so that there is a one-to-one correspondence between nite-dimensional representations of G and FG- modules. This is useful in creating a conceptual basis for the idea of a representation, especially since nite-dimensional representations are the only ones in which we will be interested. Without going into too many details, two important examples are stated, as these will reappear later: REPRESENTATIONS OF LIE ALGEBRAS, WITH APPLICATIONS TO PARTICLE PHYSICS 3 ∼ Example 1. (The Standard Representation): Let G be a permutation group, i.e. G = Sn for some n. n Then g ∈ G acts on elements of F : let e1 , e2 ,..., en be a basis; then g · ei = eg(i). Then the vector space is a representation of G, as each element of the group is associated with a linear map on the vector space. - Example 2. (The Dual Representation): If V is an arbitrary vector space, take ψ : V → V with its dual ψ∗ : V ∗ → V ∗ dened by ψ∗(ρ) = ρ ◦ ψ in the normal way for a vector space. In matrix form, ψ∗ is t ψ and since t (ψ · φ) =t φ ·t ψ for any two matrices ψ, φ, we have (ψ · φ)∗(ρ) = φ∗(ψ∗(ρ)). If is a representation of a group , we have for each and we see ∗ ∗ ∗so this is an V G ϕg g ∈ G ϕg·h = ϕh ◦ ϕg anti-homomorphism. By dening, in the case of group representations, ∗ t , we get ϕg = ϕg−1 ∗ t t t t ∗ ∗ ϕg·h = ϕ(g·h)−1 = ϕh−1 · ϕg−1 = ϕg−1 · ϕh−1 = ϕg ◦ ϕh as desired. We have thus described V ∗as a representation of G. We now turn away from the group structure of a Lie group and explore the implications of its manifold structure. 2. Lie Algebras 2.1. The Tangent Space at the Identity. Since a Lie group is a nite-dimensional manifold, given a point go on its surface we may nd the tangent space at go . That is, for G a Lie group we have a space of all analytic functions (not necessarily linear) G → G. We may take the partial derivatives of any of these functions at our specied point go . We do this by xing a chart (U , ϕ) and considering n i ∂ o the space of operators ξ , applying them to analytic functions at go . The direction of the partial ∂xi derivative is determined by ξi (or, specically, the ratio of ξi to ξj for various coordinates i,j), and the space of such operators analyzed at is the tangent space at , . If we have a tangent vector at go go Tgo G each point of the manifold, we have a vector eld. We wish to invest the space of analytic vector elds A with the structure of a vector space, like its corresponding Lie group G, and so we need to have a i ∂ i ∂ multiplication. Merely taking two tangent vectors ξ and η at a point go and composing them ∂xi ∂xi as operators yields ∂ ∂ ∂2 ∂ηj ∂ ξi ηj = ξiηj + ξi ∂xi ∂xj ∂xi∂xj ∂xi ∂xj which is not a tangent vector because of the double partial derivative. However, if we take two such operators as before, we see that ∂ ∂ ∂ ∂ ∂ηj ∂ξi ∂ ξi ηj − ηj ξi = ξi − ηj ∂xi ∂xj ∂xj ∂xi ∂xj ∂xj ∂xi REPRESENTATIONS OF LIE ALGEBRAS, WITH APPLICATIONS TO PARTICLE PHYSICS 4 which is itself another tangent vector. Hence, we can dene multiplication of two vector elds X, Y ∈ A to be [X, Y] = XY − YX, which is also known as the commutator. This is a repetition of results in dierential geometry, but there are some useful results for Lie groups. In particular, as with any group, we consider the action of a Lie group on itself by right (or left) 0 0 multiplication, i.e. for g ∈ G consider the map Φg (g ) = g · g . Any homomorphism of groups ρ : G → H must preserve this action, since 0 0 0 0 ρ(g · g ) = ρ(g) · ρ(g ) = ρ(g) · ρ(g ) = Φρ(g)(g ). That a map respects multiplication on the right or left is equivalent to saying that it is a homomorphism. Hence, nding group representations amounts in some sense to nding maps between groups that are compatible with right multiplication. Since right multiplication is everywhere continuous, we can take its dierential at some point . Let i ∂ be a tangent vector at and consider go X = ξ ∂x go i i ∂(ϕ◦Φg )(v) ∂((ϕ◦Φg )(v))k k,i XΦg (v) = ξ .
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