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10.15 Generators 429 10.15 Generators To study continuous groups, we will use calculus and algebra, and we will focus on the simplest part of the group—the elements g(↵) for ↵ 0which ⇡ are near the identity e = g(0) for which all ↵a = 0. Each element g(↵) of the group is represented by a matrix D(↵) D(g(↵)) in the D representation of ⌘ the group and by another matrix D (↵) D (g(↵)) in any other D repre- 0 ⌘ 0 0 sentation of the group. Every representation respects the multiplication law of the group. So if g(β)g(↵)=g(γ), then the matrices of the D representa- tion must satisfy D(β)D(↵)=D(γ), and those of any other representation D0 must satisfy D0(β)D0(↵)=D0(γ). A generator ta of a representation D is the partial derivative of the matrix D(↵) with respect to the component ↵a of ↵ evaluated at ↵ =0 @D(↵) t = i . (10.57) a − @↵ a ↵=0 When all the parameters ↵ are infinitesimal, ↵ 1, the matrix D(↵)is a | a|⌧ very close to the identity matrix I D(↵) I + i ↵ t . (10.58) ' a a a X Replacing ↵ by ↵/n, we get a relation that becomes exact as n !1 ↵ ↵ D = I + i a t . (10.59) n n a a ⇣ ⌘ X The nth power of this equation is the matrix D(↵) that represents the group element g(↵)intheexponential parametrization n ↵ n ↵a D(↵)=D =lim I + i ta =exp i ↵a ta . (10.60) n n n !1 a ! a ! ⇣ ⌘ X X The i’s appear in these equations so that when the generators ta are hermitian matrices, (ta)† = ta, and the ↵’s are real, the matrices D(↵) are unitary 1 D− (↵)= exp i ↵ t = D†(↵)=exp i ↵ t . (10.61) − a a − a a a ! a ! X X Compact groups have finite-dimensional, unitary representations with her- mitian generators. 430 Group theory 10.16 Lie algebra If ta and tb are any two generators of a representation D, then the matrices D(↵)=ei✏ta and D(β)=ei✏tb (10.62) represent the group elements g(↵) and g(β) with infinitesimal exponential parameters ↵i = ✏ia and βi = ✏ib. The inverses of these group elements g 1(↵)=g( ↵) and g 1(β)=g( β) are represented by the matrices − − − − D( ↵)=e i✏ta and D( β)=e i✏tb . The multiplication law of the group − − − − determines the parameters γ(↵,β) of the product g(β) g(↵) g( β) g( ↵)=g(γ(↵,β)). (10.63) − − The matrices of any two representations D with generators ta and D0 with generators ta0 obey the same multiplication law D(β) D(↵) D( β) D( ↵)=D(γ(↵,β)) − − (10.64) D0(β) D0(↵) D0( β) D0( ↵)=D0(γ(↵,β)) − − with the same infinitesimal exponential parameters ↵,β, and γ(↵,β). To order ✏2, the product of the four D’s is 2 2 i✏t i✏ta i✏t i✏ta ✏ 2 ✏ 2 e b e e− b e− (1 + i✏t t )(1 + i✏t t ) ⇡ b − 2 b a − 2 a ✏2 ✏2 (10.65) (1 i✏t t2)(1 i✏t t2) ⇥ − b − 2 b − a − 2 a 1+✏2(t t t t )=1+✏2[t ,t ]. ⇡ a b − b a a b The other representation gives the same result but with primes i✏t i✏t i✏t i✏t 2 e b0 e a0 e− b0 e− a0 1+✏ [t0 ,t0 ]. (10.66) ⇡ a b The products (10.65 & 10.66) represent the same group element g(γ(↵,β)), so they have the same infinitesimal parameters γ(↵,β) and therefore are the same linear combinations of their respective generators tc and tc0 n D(γ(↵,β)) 1+✏2[t ,t ]=1+i✏2 f c t ⇡ a b ab c c=1 Xn (10.67) 2 2 c D0(γ(↵,β)) 1+✏ [t0 ,t0 ]=1+i✏ f t0 . ⇡ a b ab c c=1 X which in turn imply the Lie algebra formulas n n c c [ta,tb]= fab tc and [ta0 ,tb0 ]= fab tc0 . (10.68) c=1 c=1 X X 10.16 Lie algebra 431 The commutator of any two generators is a linear combination of c the generators. The coefficients f ab are the structure constants of the group. They are the same for all representations of the group. Unless the parameters ↵a are redundant, the generators are linearly in- dependent. They span a vector space, and any linear combination may be called a generator. By using the Gram-Schmidt procedure (section 1.10), we may make the generators ta orthogonal with respect to the inner product (1.91) (ta,tb)=Tr ta† tb = kδab (10.69) ⇣ ⌘ in which k is a nonnegative normalization constant that depends upon the representation. We can’t normalize the generators, making k unity, because c the structure constants fab are the same in all representations. In what follows, I will often omit the summation symbol when an index is repeated. In this notation, the structure-constant formulas (10.68) are P c c [ta,tb]=fab tc and [ta0 ,tb0 ]=fab tc0 . (10.70) This summation convention avoids unnecessary summation symbols. By multiplying both sides of the first of the two Lie algebra formulas (10.68) by td† and using the orthogonality (10.69) of the generators, we find c c d Tr [ta,tb] td† = if ab Tr tc td† = if ab kδcd = ik f ab (10.71) ⇣ ⌘ ⇣ ⌘ c which implies that the structure constant f ab is the trace c i f = Tr [t ,t ] t† . (10.72) ab −k a b c ⇣ ⌘ Because of the antisymmetry of the commutator [ta,tb], structure constants are antisymmetric in their lower indices f c = f c . (10.73) ab − ba From any n n matrix A, one may make a hermitian matrix A + A and ⇥ † an antihermitian one A A†. Thus, one may separate the nG generators − (h) (ah) into a set that are hermitian ta and a set that are antihermitian ta . The exponential of any imaginary linear combination of n n hermitian (h) ⇥ generators D(↵)=exp i↵ ta is an n n unitary matrix since a ⇥ ⇣ ⌘ (h) (h) 1 D†(↵)=exp i↵ t† =exp i↵ t = D− (↵). (10.74) − a a − a a ⇣ ⌘ ⇣ ⌘ 432 Group theory A group with only hermitian generators is compact and has finite-dimensional unitary representations. On the other hand, the exponential of any imaginary linear combination (ah) of antihermitian generators D(↵)=exp i↵a ta is a real exponential of (ah) their hermitian counterparts ita whose⇣ squared⌘ norm 2 (ah) D(↵) =Tr D(↵)†D(↵) =Tr exp 2↵ it (10.75) k k a a h i h ⇣ ⌘i grows exponentially and without limit as the parameters ↵ .A a !±1 group with some antihermitian generators is noncompact and does not have finite-dimensional unitary representations. (The unitary representa- tions of the translations and of the Lorentz and Poincar´egroups are infinite dimensional.) Compact Lie groups have hermitian generators, and so the structure- constant formula (10.72) reduces in this case to c f =( i/k)Tr [t ,t ] t† =( i/k)Tr ([t ,t ] t ) . (10.76) ab − a b c − a b c ⇣ ⌘ Now, since the trace is cyclic, we have f b =( i/k)Tr ([t ,t ] t )=( i/k)Tr (t t t t t t ) ac − a c b − a c b − c a b =( i/k)Tr (t t t t t t ) − b a c − a b c =( i/k)Tr ([t ,t ] t )=f c = f c . (10.77) − b a c ba − ab Interchanging a and b, we get f a = f c = f c . (10.78) bc ab − ba Finally, interchanging b and c in (10.77) gives f c = f b = f b . (10.79) ab ca − ac Combining (10.77, 10.78, & 10.79), we see that the structure constants of a compact Lie group are totally antisymmetric f b = f b = f c = f c = f a = f a . (10.80) ac − ca ba − ab − bc cb Because of this antisymmetry, it is usual to lower the upper index c f ab = fcab = fabc (10.81) and write the antisymmetry of the structure constants of compact Lie groups as f = f = f = f = f = f . (10.82) acb − cab bac − abc − bca cba 10.17 The Rotation Group 433 For compact Lie groups, the generators are hermitian, and so the struc- ture constants fabc are real, as we may see by taking the complex con- jugate of the formula (10.76) for fabc f ⇤ =(i/k)Tr (t [t ,t ]) = ( i/k)Tr ([t ,t ] t )=f . (10.83) abc c b a − a b c abc It follows from (10.68 & 10.81–10.83) that the commutator of any two generators of a Lie group is a linear combination c [ta,tb]=ifab tc (10.84) of its generators t , and that the structure constants f f c are c abc ⌘ ab real and totally antisymmetric if the group is compact. Example 10.19 (Gauge Transformation) The action density of a Yang- Mills theory is unchanged when a spacetime dependent unitary matrix U(x) changes a vector (x) of matter fields to 0(x)=U(x) (x). Terms like † are invariant because †(x)U †(x)U(x) (x)= †(x) (x), but how can i kinetic terms like @i † @ be made invariant? Yang and Mills introduced matrices Ai of gauge fields, replaced ordinary derivatives @i by covariant derivatives D @ + A , and required that D = UD or that i ⌘ i i i0 0 i @i + Ai0 U = @iU + U@i + Ai0 U = U (@i + Ai) .
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  • Lie Algebras from Lie Groups

    Lie Algebras from Lie Groups

    Preprint typeset in JHEP style - HYPER VERSION Lecture 8: Lie Algebras from Lie Groups Gregory W. Moore Abstract: Not updated since November 2009. March 27, 2018 -TOC- Contents 1. Introduction 2 2. Geometrical approach to the Lie algebra associated to a Lie group 2 2.1 Lie's approach 2 2.2 Left-invariant vector fields and the Lie algebra 4 2.2.1 Review of some definitions from differential geometry 4 2.2.2 The geometrical definition of a Lie algebra 5 3. The exponential map 8 4. Baker-Campbell-Hausdorff formula 11 4.1 Statement and derivation 11 4.2 Two Important Special Cases 17 4.2.1 The Heisenberg algebra 17 4.2.2 All orders in B, first order in A 18 4.3 Region of convergence 19 5. Abstract Lie Algebras 19 5.1 Basic Definitions 19 5.2 Examples: Lie algebras of dimensions 1; 2; 3 23 5.3 Structure constants 25 5.4 Representations of Lie algebras and Ado's Theorem 26 6. Lie's theorem 28 7. Lie Algebras for the Classical Groups 34 7.1 A useful identity 35 7.2 GL(n; k) and SL(n; k) 35 7.3 O(n; k) 38 7.4 More general orthogonal groups 38 7.4.1 Lie algebra of SO∗(2n) 39 7.5 U(n) 39 7.5.1 U(p; q) 42 7.5.2 Lie algebra of SU ∗(2n) 42 7.6 Sp(2n) 42 8. Central extensions of Lie algebras and Lie algebra cohomology 46 8.1 Example: The Heisenberg Lie algebra and the Lie group associated to a symplectic vector space 47 8.2 Lie algebra cohomology 48 { 1 { 9.