Table Of Content

i Semi-Simple Lie Algebras and Their Representations Robert N. Cahn Lawrence Berkeley Laboratory University of California Berkeley, California 1984 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY Advanced Book Program Menlo Park, California · Reading, Massachusetts London Amsterdam Don Mills, Ontario Sydney · · · · ii Preface iii Preface Particle physics has been revolutionized by the development of a new “paradigm”,thatofgaugetheories. TheSU(2)xU(1)theoryofelectroweakin- teractionsandthecolorSU(3)theoryofstronginteractionsprovidethepresent explanationofthreeofthe fourpreviouslydistinctforces. Fornearlytenyears physicists have sought to unify the SU(3) x SU(2) x U(1) theory into a single group. This has led to studies of the representations of SU(5), O(10), and E . 6 Effortstounderstandthereplicationoffermionsingenerationshaveprompted discussions of even larger groups. The present volume is intended to meet the need of particle physicists for a book which is accessible to non-mathematicians. The focus is on the semi-simple Lie algebras, and especially on their representations since it is they,andnotjustthealgebrasthemselves,whichareofgreatestinteresttothe physicist. If the gauge theory paradigm is eventually successful in describing the fundamental particles, then some representation will encompass all those particles. The sources of this book are the classical exposition of Jacobson in his LieAlgebrasandthreegreatpapersofE.B.Dynkin. Alistingofthereferences isgivenintheBibliography. Inaddition,attheendofeachchapter,references iv Preface aregiven,withtheauthors’namesincapitalletterscorrespondingtothelisting in the bibliography. Thereaderisexpectedtobefamiliarwiththerotationgroupasitarises inquantummechanics. Areviewofthismaterialbeginsthebook. Afamiliarity with SU(3) is extremely useful and this is reviewed as well. The structure of semi-simple Lie algebras is developed, mostly heuristically, in Chapters III - VII, culminating with the introduction of Dynkin diagrams. The classical Lie algebras are presented in Chapter VIII and the exceptional ones in Chapter IX. Properties of representations are explored in the next two chapters. The Weylgroupis developedin ChapterXIII andexploitedin Chapter XIVin the proofofWeyl’sdimensionformula. Thefinalthreechapterspresenttechniques forthreepracticaltasks: findingthedecompositionofproductrepresentations, determining the subalgebras of a simple algebra, and establishing branching rules for representations. Although this is a book intended for physicists, it contains almost none of the particle physics to which it is germane. An elementary account of some of this physics is given in H. Georgi’s title in this same series. ThisbookwasdevelopedinseminarsattheUniversityofMichiganand the University of California, Berkeley. I benefited from the students in those seminars,especially H.Haber andD. PetersoninAnn Arbor andS. Sharpe in Berkeley. Sharpe, and H.F. Smith, also at Berkeley, are responsible for many improvements in the text. Their assistance is gratefully acknowledged. v Table of Contents I. SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . 9 III. The Killing Form . . . . . . . . . . . . . . . . . . . . 17 IV. The Structure of Simple Lie Algebras . . . . . . . . . . . 25 V. A Little about Representations . . . . . . . . . . . . . . 31 VI. More on the Structure of Simple Lie Algebras . . . . . . . . 39 VII. Simple Roots and the Cartan Matrix . . . . . . . . . . . 43 VIII. The Classical Lie Algebras . . . . . . . . . . . . . . . . 52 IX. The Exceptional Lie Algebras . . . . . . . . . . . . . . . 64 X. More on Representations . . . . . . . . . . . . . . . . . 73 XI. Casimir Operators and Freudenthal’s Formula . . . . . . . 84 XII. The Weyl Group . . . . . . . . . . . . . . . . . . . . 98 XIII. Weyl’s Dimension Formula . . . . . . . . . . . . . . . 102 XIV. Reducing Product Representations . . . . . . . . . . . 116 XV. Subalgebras . . . . . . . . . . . . . . . . . . . . . 128 XVI. Branching Rules . . . . . . . . . . . . . . . . . . . 144 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 154 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 vi I. SU(2) 1 I. SU(2) A geometrical vector in three-dimensional space can be represented by a column vector whose entries are the x, y, and z components of the vector. A rotationofthe vector canbe representedby a three-by-three matrix. In particular, a rotation by φ about the z-axis is given by cosφ sinφ 0 − sinφ cosφ 0 . (I.1)   0 0 1     For small rotations, cosφ sinφ 0 − sinφ cosφ 0 I iφT , (I.2)   z ≈ − 0 0 1     where T is the matrix z 2 I. SU(2) 0 i 0 − i 0 0 . (I.3)   0 0 0     In a similar fashion we find T and T : x y 0 0 0 0 0 i T = 0 0 i , T = 0 0 0 . (I.4) x   y   − 0 i 0 i 0 0       −  By direct computation we find that the finite rotations are given as exponentials of the matrices T , T , and T . Thus we have x y z cosφ sinφ 0 − exp( iφT )= sinφ cosφ 0 . (I.5) z   − 0 0 1     The product of two rotations like exp( iθT )exp( iφT ) can always be written as y z − − a single exponential, say exp( iα T) where α T =α T +α T +α T . Suppose x x y y z z − · · we set exp( iα T)exp( iβ T)=exp( iγ T) and try to calculate γ in terms of − · − · − · α and β. If we expand the exponentials we find [1 iα t 1(α t)2+ ][1 iβ t 1(β t)2+ ] − · −2 · ··· − · − 2 · ··· = 1 i(α+β) t 1((α+β) t)2 1[α t,β t]+ − · − 2 · − 2 · · ··· =(cid:2)exp i(α+β) t 1[α t,β t]+ . (cid:3) (I.6) {− · − 2 · · ···} To this order in the expansion, to calculate γ we need to know the value of the commutatorslike [T ,T ], but not ordinaryproducts likeT T . In fact, this is true x y x y to all orders (and is known as the Campbell-Baker-Hausdorff theorem1). It is for this reason that we can learn most of what we need to know about Lie groups by studying the commutation relations of the generators (here, the T’s). By direct computation we can find the commutation relations for the T’s: I. SU(2) 3 [T ,T ]=iT , [T ,T ]=iT , [T ,T ]=iT . (I.7) x y z y z x z x y These commutation relations which we obtained by considering geometrical rotations can now be used to form an abstractLie algebra. We suppose there are three quantities t , t , and t with a Lie product indicated by [ , ] x y z [t ,t ]=it , [t ,t ]=it , [t ,t ]=it . (I.8) x y z y z x z x y We consider all linear combinations of the t’s and make the Lie product linear in each of its factors and anti-symmetric: [a t+b t,c t]=[a t,c t]+[b t,c t], (I.9) · · · · · · · [a t,b t]= [b t,a t]. (I.10) · · − · · It is easy to show that the Jacobi identity follows from Eq. (I.8): [a t,[b t,c t]]+[b t,[c t,a t]]+[c t,[a t,b t]]=0 . (I.11) · · · · · · · · · When we speak of the abstract Lie algebra, the product [a t,b t] is not to be · · thought of as a t b t b t a t , since the product a tb t has not been defined. · · − · · · · Whenwerepresentthealgebrabymatrices(aswedidattheoutset),thenofcourse the ordinaryproducthasawell-definedmeaning. Nevertheless,bycustomweoften refer to the Lie product as a commutator. The abstractLie algebraderivedabovefromthe rotationgroupdisplays the features which define Lie algebras in general. A Lie algebra is a vector space, L, (above, the linear combinations of the t’s) together with a bilinear operation(from L L into L ) satisfying × 4 I. SU(2) [x +x ,y]=[x ,y]+[x ,y] , x ,x ,y L 1 2 1 2 1 2 ∈ [ax,y]=a[x,y] , a F, x,y L ∈ ∈ [x,y]= [y,x] , x,y L − ∈ 0=[x,[y,z]]+[y,[z,x]]+[z,[x,y]] , x,y,z L . (I.12) ∈ Here F is the field over which L is a vector space. We shall always take F to be the field of real numbers, , or the field of complex numbers, . R C Having motivated the formal definition of a Lie algebra, let us return to the specific example provided by the rotation group. We seek the representations of the Lie algebra defined by Eq. (I.8). By a representation we mean a set of linear transformations (that is, matrices) T , T , and T with the same commutation x y z relations as the t’s. The T’s of Eqs. (I.3) and (I.4) are an example in which the matrices are 3 3 and the representation is said to be of dimension three. × We recall here the construction which is familiar from standard quantum mechanics texts. It is convenient to define t =t +it , t =t it , (I.13) + x y − x− y so that the commutation relations become [t ,t ]=t , [t ,t ]= t , [t ,t ]=2t . (I.14) z + + z − − − + − z We now suppose that the t’s are to be representedby some linear transformations: t T ,t T , t T . The T’s act on some vector space, V. We shall in x x y y z z → → → fact constructthis spaceand the T’sdirectly. We startwith a single vector,v and j define the actions of T and T on it by z + T v =jv , T v =0 . (I.15) z j j + j Now consider the vector T v . This vector is an eigenvector of T with eigenvalue − j z j 1 as we see from −