Mathematical Principles of Theoretical Physics

160 CHAPTER 3. MATHEMATICAL FOUNDATIONS

is derived by (2.2.51) and detΩ=1 forΩas in (3.5.18).
In particular, at the unit matrixe=I, we have

(3.5.19)

TeSU(N) ={iτ|τ†=τ,Trτ= 0 },

Ω=eiτ, τ∈TeSU(N),

whereΩ⊂SU(N)is in a neighborhood ofe=I.

3.Coordinate systems on TSU(N).Since the representation (3.5.17)-(3.5.18) ofSU(N)is
essentially the same for allA∈SU(N), it suffices to only consider the representation (3.5.19)
ofSU(N)at the unit matrixe=I.
It is known thatTeSU(N)is anN^2 −1 dimensional linear space. Therefore we can take a
coordinate basis, called the generator basis ofSU(N), denoted by

(3.5.20) τ 1 ,···,τK (K=N^2 − 1 ),

such that for anyτ∈TeSU(N), we have

(3.5.21) τ=θaτa,

andθa( 1 ≤a≤K)are complex numbers, called the coordinate system onTeSU(N). In this
case,Ωin (3.5.19) can be expressed as

Ω=eiθ

aτa

Ω⊂SU(N).

3.5.3 SU(N)tensors

Let(τ 1 ,···,τK)⊂TeSU(N)be a generator basis ofSU(N). If the basis undergoes a linear
transformation:

(3.5.22)

̃τa=xbaτb,

X= (xba) is a complexK-th order matrix,

then the coordinate system(θ^1 ,···,θK)ofTeSU(N)also undergoes a corresponding trans-
formation as follows

(3.5.23)

θ ̃a=yabθb,

Y= (yab) is a complexK-th order matrix.

Since the expression (3.5.21) is independent of the choice of the generator bases ofSU(N),
we have

θ ̃a ̃τa= (θ^1 ,···,θK)YTX







τ 1

..

.

τK





=θaτa,

which requires that

(3.5.24) Y= (X−^1 )T.

Thus, we see that(θ^1 ,···,θK)is a first order contra-variant tensor defined onTeSU(N).
We are now ready to define more generalSU(N)tensors.