1 Continuous Groups

As evident from the name, Continuous groups are, (i) groups, of course, and (ii)continuous, in some sense. So let us first recall the definition of a group.

1.1 Group

A group is a set G, with a binary operation⊗

defined in such a way that thefollowing are satisfied:

1. ∀x, y ∈ G, x⊗y ∈ G, i.e., the set G is closed under the binary operation.

2. The set G is associative w.r.t to the binary operation.

3. Identity in G exists.

4. Each element of G has an inverse.

1.2 Continuous Groups

A continuous group is a set of elements R such that each R depends on somefinite number of real continuous parameters.

R(a) ≡ R(a1, . . . ar). (1)

This is called an r−parameter group. It satisfies all the properties needed tobe a group. Now let us see how continuity gets incorporated. Since this is agroup, it must be closed w.r.t to the binary operation. We now check the groupcondition on this set.

• Let us take three arbitrary elements from the set R(a), R(b), R(c), . . .. Theproduct of any two, say R(a) and R(b) should yield a member within thegroup, say R(c). By definition, if R(c) = R(a)⊗R(b) then there must bea continuous real function f(a, b) such that,

c = f(a, b) (2)

• If product of R(a), R(b), = R(a)(b) (R(b), R(c), = R(b)(c)) impliesthe existence of a function f(a, b) (f(b, c)), then from the associativity ofgroup elements , we have

R(a)[R(b)R(c)] = [R(a)R(b)]R(c) (3)

Or f(a, f(b, c)) = f(f(a, b), c)

• Existence of identity implies

R(a)R(a0) = R(a) = R(a0)R(a) (4)

which can be equivalently expressed as f(a, a0) = a = f((a0, a)

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• Each group element must have an inverse, such that

R(a)R(a′) = R(a0) = R(a′)R(a) (5)

or equivalently, f(a, a′) = a0 = f(a′, a).

If the function f , is analytic, viz., it has a convergent Taylor series (in thedomain defined by the parameters), then this group is said to be r−parameterLie group.

1.3 Application

It would not be inappropriate to say that the mathematics equivalent of theword ”application” is transformation. Whenever we wish to apply math tosolve some physical problem, we will have to use transformation of some kindor the other. The case of Lie groups is no different, and we will be interested inthe tranformations of the d−dimensional spaces. These will simply be transfor-mations over, say, the Euclidean space for instance. In general they will be ofthe form,

x′i = f(x1, . . . , xd; a1, . . . , ar). (6)

As said earlier, if f is analytic, then this defines r−parameter Lie group oftransformations. Example Let us illustrate the previous discussion with a simpleexample, wherein we consider the example of scaling transformation, viz.,

x′ = ax (7)

The primed coordinate is given by scaling the unprimed one by factor of a,where a belongs to set of scalars (except 0). Let us verify the group axioms andcontinuity.

• Suppose the binary operation is composition. Let x′1 = a1x, and x′2 = a2x.Then x′1 ◦ x′2 = a1a2x. We can write a1a2 = a3, so that x′1xn

′2 = a3x, is

same form as (7). Hence the group is closed w.r.t to the binary operation.Furthermore a3 = a1a2 = f(a1, a2) is continuous.

• Associativity condition, x′1 ◦ (x′2 ◦ x′3) = (x′1 ◦ x′2) ◦ x′3 is satisfied, eithertrust me or check yourselves.

• Identity element x′0 = x is also there, such that x′ixn′0 = aix.

• For each x′i,∃x′−1i = a−1i x such that x’i ◦ x′−1i = x = x′0.

Hence this forms an 1−parameter (abelian) Lie group.

1.4 Example

This example is a group of transformations defined by x′i = aix+ bi. This formsa group can be verified, what is important is that one should notice that it is a2−parameter (non-Abelian) Lie group.

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2 Linear Transformation Group

Matrices are a wonderful way to transform mathematical objects of any di-mension. We can transform d−dimensional objects using d × d matrices. Forexample, (

x′

y′

)=

(a11 a12a21 a22

)(xy

)(8)

Or x′ = Ax. This group of linear transformations is a group with compositiondefined by usual matrix multiplication. Provided the determinant is non-zero,the group of all 2 × 2 matrices form a 4-parameter Lie group. This is alsodenoted by GL(2,R). In general, we use the notation GL(n,R). If the entriesof the matrix are complex, then the notation is, GL(n,C).

2.1 Orthogonal Transformations

Symmetries in Mathematics lead to conservation laws and hence reduce thevariables/parameters, yielding the problem easier to handle. In our case, ifwe demand that the length of the vector is preserved after applying the lineartransformation, then it will reduce the parameters. Let us see how. First of all,we impose the condition,

x′1 + . . . + x′n = x1 + . . . + xn. (9)

For simplicity, we take n = 2, then (8) implies

x′1 + x′2 = (a11x1 + a12x2)2 + (a21x1 + a22x2)2 (10)

Equation (10) breaks down to the following three conditions:

a211 + a221 = 1, a212 + a222 = 1, a11a12 + a21a22 = 0 (11)

Now if we multiply the first two equations in (11), we get

(a211 + a221)(a212 + a222) = 1 (12)

Using the third equation in (11), we get the following condition

(a11a22 − a21a12)2 = 1 (13)

Or detA = ±1, where A =

(a11 a12a21 a22

)Positive signature leads to rotations

known as proper while negative one is called improper. If positive signature ischosen, which we will, then the group is denotes as SO(n), where S stands forspecial. In summary SO(n) ⊂ O(n) ⊂ GL(n,R).

2.2 Parameterisation

We will use the following parametrisation for the matrix A:

R =

(cosϕ sinϕ− sinϕ cosϕ

)(14)

Notice there is only single parameter in use, viz., ϕ.

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3 Generators

Our Norwegian fellow Mathematician Sophus Lie, put emphasis on generatorsof groups, so that instead of caring about the whole group, you just have tocare for the generator, and everything else can be ”generated” by the genera-tor. So we will also spend considerable time on studying generators of the Liegroups we have discussed. These generators are called infinitesimal generatorsfor the simple reason that rotation is a continuous operation; the parametercan take infinite values, thus the generator of such transformation is also calledinfinitesimal generator.

3.1 Matrix Exponential form of generators

Firstly observe from (14) that R(ϕ1)R(ϕ2) = R(ϕ1 + ϕ2) = R(ϕ2 + ϕ1) =R(ϕ2)R(ϕ1), that is, the group is Abelian.Secondly we will also need the Taylor expansion of R about 0:

R(ϕ) = R(0) + ϕdR

dϕ|ϕ=0 +

1

2ϕ2 d

2R

dϕ2|ϕ=0 + . . . (15)

The coefficients in (15) can be determined using (14), but let us search for somekind of recurrence relation. Start with the fact that

R(ϕ1 + ϕ2) = R(ϕ1)R(ϕ2) (16)

and differentiate the LHS w.r.t to ϕ1 at ϕ1 = 0,

dR(ϕ1 + ϕ2)

dϕ1|ϕ1=0 =

[dR(ϕ1 + ϕ2

d(ϕ1 + ϕ2)

d(ϕ1 + ϕ2)

dϕ1

]|ϕ1=0 (17)

=dR(ϕ2)

dϕ2.I (18)

Now equating above result to the derivative of RHS of (16) gives us,

dR(ϕ2)

dϕ2=

dR(ϕ1)

dϕ1|ϕ1=0R(ϕ2) (19)

dR(ϕ1)dϕ1

can be calculated from (14), which gives,(− sinϕ1 − cosϕ1

cosϕ1 − sinϕ1

)|ϕ1=0 =

(0 −11 0

)≡ X (20)

So finally we have finally obtained the useful relationship for determining thecoefficients in (15),

dR(ϕ)

dϕ= XR(ϕ) (21)

(where we have omitted the subscripts since ϕ’s are arbitrary).

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