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.