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4 Lie Algebras 4 Lie Algebras Contents 4.1 Why Bother? 61 4.2 How to Linearize a Lie Group 63 4.3 Inversion of the Linearization Map: EXP 64 4.4 Properties of a Lie Algebra 66 4.5 Structure Constants 68 4.6 Regular Representation 69 4.7 Structure of a Lie Algebra 70 4.8 Inner Product 71 4.9 Invariant Metric and Measure on a Lie Group 74 4.10 Conclusion 76 4.11 Problems 76 The study of Lie groups can be greatly facilitated by linearizing the group in the neighborhood of its identity. This results in a structure called a Lie algebra. The Lie algebra retains most, but not quite all, of the properties of the original Lie group. Moreover, most of the Lie group properties can be recovered by the inverse of the linearization operation, carried out by the EXPonential mapping. Since the Lie algebra is a linear vector space, it can be studied using all the standard tools available for linear vector spaces. In particular, we can define convenient inner products and make standard choices of basis vectors. The properties of a Lie algebra in the neighborhood of the origin are identified with the properties of the original Lie group in the neighborhood of the identity. These structures, such as inner product and volume element, are extended over the entire group manifold using the group multiplication operation. 4.1 Why Bother? Two Lie groups are isomorphic if: (i) Their underlying manifolds are topologically equivalent; 61 62 Lie Algebras (ii) The functions defining the group composition laws are equivalent. Two manifolds are topologically equivalent if they can be smoothly deformed into each other. This requires that all their topological in- dices, such as dimension, Betti numbers, connectivity properties, etc., are equal. Two group composition laws are equivalent if there is a smooth change of variables that deforms one function into the other. Showing the topological equivalence of two manifolds is not neces- sarily an easy job. Showing the equivalence of two composition laws is typically a much more difficult task. It is difficult because the group composition law is generally nonlinear, and working with nonlinear func- tions is notoriously difficult. The study of Lie groups would simplify greatly if the group composi- tion law could somehow be linearized, and this linearization retained a substantial part of the information inherent in the original group com- position law. This in fact can be done. Lie algebras are constructed by linearizing Lie groups. A Lie group can be linearized in the neighborhood of any of its points, or group operations. Linearization amounts to Taylor series expansion about the coordinates that define the group operation. What is being Taylor expanded is the group composition function. This function can be expanded in the neighborhoods of any group operations. A Lie group is homogeneous — every point looks locally like every other point. This can be seen as follows. The neighborhood of group element a can be mapped into the neighborhood of group element b by multiplying a, and every element in its neighborhood, on the left by group element ba−1 (or on the right by a−1b). This maps a into b and points near a into points near b. It is therefore necessary to study the neighborhood of only one group operation in detail. Although geometrically all points are equivalent, al- gebraically one is special — the identity. It is very useful and convenient to study the neighborhood of this special group element. Linearization of a Lie group about the identity generates a new set of operators. These operators form a Lie algebra. A Lie algebra is a linear vector space, by virtue of the linearization process. The composition of two group operations in the neighborhood of the identity reduces to vector addition. The construction of more compli- cated group products, such as the commutator, and the linearization of these products introduces additional structure in this linear vector 4.2 How to Linearize a Lie Group 63 space. This additional structure, the commutation relations, carries in- formation about the original group composition law. In short, the linearization of a Lie group in the neighborhood of the identity to form a Lie algebra brings about an enormous simplification in the study of Lie groups. 4.2 How to Linearize a Lie Group We illustrate how to construct a Lie algebra for a Lie group in this section. The construction is relatively straightforward once an explicit parameterization of the underlying manifold and an expression for the group composition law is available. In particular, for the matrix groups the group composition law is matrix multiplication, and one can con- struct the Lie algebra immediately for the matrix Lie groups. We carry this construction out for SL(2; R). It is both customary and convenient to parameterize a Lie group so that the origin of the coordinate system maps to the identity of the group. Accordingly, we parameterize SL(2; R) as follows 1+ a b (a,b,c) −→ M(a,b,c)= (4.1) c (1 + bc)/(1 + a) The group is linearized by investigating the neighborhood of the identity. This is done by allowing the parameters (a,b,c) to become infinitesimals and expanding the group operation in terms of these infinitesimals to first order 1+ δa δb (a,b,c) → (δa,δb,δc) → M(δa,δb,δc)= δc (1 + δbδc)/(1 + δa) (4.2) The basis vectors in the Lie algebra are the coefficients of the first order infinitesimals. In the present case the basis vectors are 2 × 2 matrices 1+ δa δb (δa,δb,δc) → I + δaX + δbX + δcX = (4.3) 2 a b c δc 1 − δa 64 Lie Algebras 1 0 ∂M(a,b,c) X = = a 0 −1 ∂a (a,b,c)=(0,0,0) 0 1 ∂M(a,b,c) Xb = = (4.4) 0 0 ∂b (a,b,c)=(0,0,0) 0 0 ∂M(a,b,c) X = = c 1 0 ∂c (a,b,c)=(0,0,0) Lie groups that are isomorphic have Lie algebras that are isomorphic. Remark: The group composition function φ(x, y) is usually linearized in one of its arguments, say φ(x, y) → φ(x, 0+ δy). This generates a left-invariant vector field. The commutators of two left-invariant vector fields at a point x is independent of x, so that x can be taken in the neighborhood of the identity. It is for this reason that the linearization of the group in the neighborhood of the identity is so powerful. 4.3 Inversion of the Linearization Map: EXP Linearization of a Lie group in the neighborhood of the identity to form a Lie algebra preserves the local group properties but destroys the global properties — that is, what happens far from the identity. It is important to know whether the linearization process can be reversed — can one recover the Lie group from its Lie algebra? To answer this question, assume X is some operator in a Lie algebra — such as a linear combination of the three matrices spanning the Lie algebra of SL(2; R) given in (4.4). Then if ǫ is a small real number, I + ǫX represents an element in the Lie group close to the identity. We can attempt to move far from the identity by iterating this group operation many times ∞ 1 Xn lim (I + X)k = = EXP (X) (4.5) k→∞ k n! n=0 X The limiting and rearrangement procedures leading to this result are valid not only for real and complex numbers, but for n × n matrices and bounded operators as well. Example: We take an arbitrary vector X in the three-dimensional linear vector space of traceless 2 × 2 matrices spanned by the generators 4.3 Inversion of the Linearization Map: EXP 65 Xa,Xb,Xc of SL(2; R) given in (4.4) a b X = aX + bX + cX = (4.6) a b c c −a The exponential of this matrix is ∞ n 1 a b sinh θ EXP (X)= EXP (aX +bX +cX )= = I cosh θ+X a b c n! c −a 2 θ n=0 X cosh θ + a sinh(θ)/θ b sinh(θ)/θ = (4.7) c sinh(θ)/θ cosh θ − a sinh(θ)/θ θ2 = a2 + bc The actual computation can be carried out either using brute force or finesse. With brute force, each of the matrices Xn is computed explicitly, a pattern is recognized, and the sum is carried out. The first few powers 0 1 2 2 2 are X = I2, X = X [given in (4.6)], and X = θ I2. Since X is a multiple of the identity, X3 = X2X1 must be proportional to X (= θ2X), X4 is proportional to the identity, and so on. Finesse involves use of the Cayley-Hamilton theorem, that every ma- trix satisfies its secular equation. This means that a 2 × 2 matrix must satisfy a polynomial equation of degree 2. Thus we can replace X2 by 0 1 3 a function of X = I2 and X = X. Similarly, X can be replaced by 2 2 a linear combination of X and X, and then X replaced by I2 and X. By induction, any function of the 2 × 2 matrix X can be written in the form 0 1 F (X)= f0(a,b,c)X + f1(a,b,c)X (4.8) Furthermore, the functions f0,f1 are not arbitrary functions of the three parameters (a,b,c), but rather functions of the invariants of the matrix X. These invariants are the coefficients of the secular equation.
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