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Lie Groups and Their Algebras

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Lie Groups and Their Algebras 1 LIE GROUPS Lie groups and their algebras 1 Lie groups A topological group is a group that is also a topological space, in which both the left translations λg : G ! G h 7! gh (for g 2G) and the inversion map ιG : G ! G g 7! g-1 are continuous. 1 LIE GROUPS A (real) Lie group is a topological group that is locally Euclidean, i.e. covered by open sets W each homeomorphic to an open subset U⊆Rn (Rn is the set of n-tuples of real numbers with the usual topology associated to the Euclidean metric); the local homeomorphism α: W !U is called a local chart. Since U is a subset of Rn, local charts allow to parametrize (locally) group elements; the group element g 2W will have parameter vector α~ (g) = (α1(g) ;:::; αn(g)) Remark. Overlapping local charts give rise to dierent local parametriza- tions of the same element, related to each other by transition functions. 1 LIE GROUPS If the positive integer n is the same for all local charts (e.g. if G is connected), then it is called the dimension of G (put dierently, G is an n parameter Lie group). Example: 3D space translations form a three parameter Lie group, with (global) coordinates given by the vector components (with respect to a given basis) of the image of the origin ~0. Remark. While Rn has no distinguished point as a topological space, its linear structure distinguishes the origin ~0=(0;:::; 0), hence it is natural (but by no means necessary) to associate this point to the only distin- guished group element, the identity element of G; whenever possible, ~ we shall use the convention α~ (1G)=0. 1 LIE GROUPS Given a local chart α : W ! U in a neighborhood of the identity, the group structure implies the existence of continuous maps µ : U×U ! U and ι: U!U (the structure functions) for which α~ (gh) = µα~ (g) ; α~ (h) α~ g-1 = ι(α~ (g)) µα~ ; β~ is the parameter vector of the product of the group elements with parameter vectors α~ and β~ , while ι(α~ ) is that of the inverse. Associativity of the group product implies the relation µα~ ; µβ~ ; γ~ = µµα~ ; β~ ; γ~ for α~ ; β~ ; γ~ 2U. 1 LIE GROUPS Since ~0 is, by convention, the parameter vector of the identity element µα~ ;~0 = µ~0; α~ = α~ µα~ ; ια~ = µια~ ; α~ = ~0 Remark. Since the parameter vectors are n-tuples of real numbers, the maps µ and ι (that characterize locally the group structure) can be studied by means of calculus in several variables. Gleason-Montgomery-Zippin: the structure functions µ : U×U ! U and ι:U!U are analytic, i.e. their Taylor-series around the origin have a positive radius of convergence. 1 LIE GROUPS Examples 1. The additive group (R; +) of real numbers is a one dimensional Lie group, with structure functions µ(α; β) = α+β and ι(α) = −α in case of the trivial parametrization α(z)=z. 2. The circle group U(1) = fz 2C j jzj=1g of complex phases is again a one dimensional Lie group, with structure functions µ(α; β)= α +β and ι(α) = −α in case of the exponential parametrization α(z)=−i log z. 3. The isospin group y SU(2)= U 2Mat2 (C) j det U =1;U U =1 is a three parameter Lie group. 1 LIE GROUPS 4. The 3D rotation group of rotations around axes having a common point (the center) is a three parameter Lie group, since rotations can be parametrized e.g. by the 3 Euler angles; topologically, it is the interior of a ball, with nontrivial identication of the bound- ary. Since rotations preserve orientation, and the distance from the center is invariant under them, the rotation group can be identied with the group SO(3) of 3-by-3 orthogonal matrices. 5. The Poincaré group P (the symmetry group of 4D Minkowski space- time) is 10 dimensional: 4 parameters correspond to space-time translations, while 6 to 4D rotations, out of which 3 describe 3D rotations, and another 3 the Lorentz boosts. 1 LIE GROUPS A Lie homomorphism φ: G1 !G2 between the Lie groups G1 and G2 is a continuous (analytic) group homomorphism.A local isomorphism is an analytic map that is a bijective homomorphism when restricted to a suitable neighborhood of the identity (identical structure functions). Remark. Note that Lie (resp. local) isomorphism is a more (resp. less) restrictive notion than ordinary algebraic isomorphism. A closed subgroup of a Lie group is itself a Lie (sub)group. A one-parameter subgroup of the Lie group G is a continuous (Lie) ho- momorphism from the additive group (R; +) of real numbers into G (its image, a Lie subgroup of dimension 1, is a continuous curve in G). 2 LIE ALGEBRAS 2 Lie algebras A Lie algebra is a linear space L endowed with a binary operation, the Lie bracket [; ]: L×L!L, that is bilinear and skew symmetric, i.e. [λa+µb, c]=λ[a; c]+µ[b; c] [a; λb+µc]=λ[a; b]+µ[a; c] [b; a]=− [a; b] and satises the Jacobi identity [a; [b; c]] + [b; [c; a]] + [c; [a; b]] = 0 for all a; b; c2L and arbitrary scalars λ, µ. 2 LIE ALGEBRAS Examples: 1. R3 with the cross product as Lie bracket; 2. with the commutator ; Matn (R) [A;B]=AB−BA 3. general linear algebra gl(V ) of all linear operators A: V ! V with the commutator [A;B]=AB−BA; 4. observable quantities (i.e. functions on phase space) of a Hamilto- nian system with the Poisson bracket as Lie bracket; 5. angular momentum operators in quantum mechanics. 2 LIE ALGEBRAS A Lie algebra morphism is a linear map φ:L1 !L2 such that φ[a; b]=[φ(a); φ(b)] Given a basis B=fb1; : : : ; bng of L, the Lie brackets n X ij [bi; bj] = ck bk k=1 of the basis vectors completely determine the algebra. The coecients ij, the structure constants of , satisfy ck L jk kj skew symmetry ci + ci = 0 X jm kl km lj lm jk Jacobi identity ci cm + ci cm + ci cm = 0 m and characterize the Lie algebra up to isomorphism. 3 THE LIE ALGEBRA OF A LIE GROUP 3 The Lie algebra of a Lie group The analytic functions on a Lie group G form a ring A(G) with the pointwise operations of addition and multiplication. A derivation of A(G) is a map r: A(G)!A(G) that is additive r(a + b) = ra + rb vanishes on constant functions, and satises the Leibnitz rule r(ab) = (ra) b + arb Derivations of A(G) form a Lie algebra Der(A(G)) with the commutator [r1; r2] = r1 ◦r2 − r2 ◦r1 as Lie bracket. 3 THE LIE ALGEBRA OF A LIE GROUP For a2A(G) and g 2G, the function g a : G ! R h 7! ag-1h is analytic, ag 2A(G), and gives rise to an additive map λg : A(G) ! A(G) a 7! ag for which λgh = λg ◦λh A (left) invariant derivation r2Der(A(G)) is one that satises r◦λg = λg ◦r for all g 2G. 3 THE LIE ALGEBRA OF A LIE GROUP The sum and commutator of invariant derivations is itself invariant invariant derivations form a Lie algebra Lie(G) Every Lie group has a canonical parametrization such that n 1 X µα~ ; β~ = α + β + cjkα β + higher order terms i i i 2 i j k j;k=1 and ια~ =−α~ . Lie's theorem: the coecients jk are the structure constants of ci Lie(G) (with respect to a suitable basis), and the higher order terms are deter- mined by the quadratic ones. Local structure of G reected by the linear object Lie(G)! 3 THE LIE ALGEBRA OF A LIE GROUP Q: how can we compute the Lie algebra? Lie transformation group: continuous group of dierentiable coordinate transformations 0 of m, where n. xi 7!xi(x1; : : : ; xmjα~ ) R α~ 2U⊆R Innitesimal generators( i=1; : : : ; n) m 0 X @xj @ Ti = @α α~ =~0 @x j=1 i j are rst order partial dierential operators, with commutators n X ij [Ti;Tj] = Ti ◦Tj − Tj ◦Ti = ck Tk k=1 with ij the structure constants of the Lie algebra. ck 3 THE LIE ALGEBRA OF A LIE GROUP Examples 1. The group of 3D translations 3 3 tα~ : R ! R x~ 7! x~ + α~ with α~ 2R3. The innitesimal generators read X @(xj +αj) @ @ T = = i @α @x @x j i j i The Lie algebra is commutative, since all Lie brackets vanish: [Ti;Tj]=0. 2. The one-parameter group of 2D rotations x cosα x−sinα y R(α): 7! y sinα x+cosα y 3 THE LIE ALGEBRA OF A LIE GROUP for α2[0; 2π]. The only generator reads @(cosα x−sinα y) @ @(sinα x+cosα y) @ @ @ T = + =−y + x @α @x @α @y @x @y 3. The ane group Affn consisting of the transformations a(λ, ~a):x~ 7!λx~ +~a for ~a 2Rn and λ2R. The generators and their commutators read n n X @(λxi +ai) @ X @ D = = x @λ @x i @x i=1 λ=1;~a=~0 i i=1 i n X @(λxi +ai) @ @ T = = j @a @x @x i=1 j λ=1;~a=~0 i j n n ! X @ @ @ X @ @ [D; T ]= x − x =− =−T j i @x @x @x i @x @x j i=1 i j j i=1 i j 3 THE LIE ALGEBRA OF A LIE GROUP If G is a linear Lie group, i.e.
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