Colour Algebra

1. Fundamental Properties

The $SU(N_c)$ Lie group depends on the real parameters

${\theta }^a\text{ with }a=1,...,N_c^2-1$

Then, any arbitrary element of the group U can be written as

$U_{ij}=[e^{-i{g}_s{\theta }^a{t}^a}{]}_{ij}$ for generators $t^{ij}$ and strong coupling constant $g_s$ with $U\in SU(N_c)$

Here,

$t^a=\frac{{\lambda }^a}{2}$ where ${\lambda }^a$ are the 8 Gell-Mann matrices.

An important relation for generators is

$[t^a,t^b]\equiv f_{abc}{t}^c$

where f are the totally anti-symmetric structure constants.

2. Fundamental Representation

Consider the generators of the fundamental representation, $t_{ij}^a$ where $a=1,...,N_c^2-1$ and $i,j=1,...,N_c$

There are $(N_c^2-1)$ generators that can each be represented by an $N_c\times N_c$ matrix.

These generators have the following properties:

• Hermitian: $(t^a{)}^{†}=t^a$

  • Proof: Using $U=e^{-i{g}_s{\theta }^a{t}^a}$ and since $U\in SU(N_c)$ (unitary),

  $U{U}^{†}=(1-i{g}_s{\theta }^a{t}^a)(1+i{g}_s{\theta }^b(t^b{)}^{†})+...=1+i{g}_s{\theta }^a((t^a{)}^{†}-t^a...=1)⟹(t^a{)}^{†}=t^a$

• Traceless: $tr[t^a]=0$

  • Proof: U is special,

  $det[U]=e^{-i{g}_s{\theta }^{a}tr[t^a]}=1⟹tr[t^a]=0$

• Normalisation: $tr[t^a{t}^b]=T_F{\delta }^{ab}$ (we will pick $T_F=\frac{1}{2}$)

• Commutator: $[t^a,t^b]=i{f}^{abc}{t}^c$

• Fierz Identity: $t_{ij}^a{t}_{kl}^a=T_F({\delta }_{il}{\delta }_{kj}-\frac{1}{N_c}{\delta }_{ij}{\delta }_{kl})$

  • Proof: Let M represent an arbitrary hermitian $N_c\times N_c$ matrix, then:

  $M=m_0{1}_{N_c\times N_c}+m_a{t}^a$

  $tr[M]=m_0\underset{=N_c}{\underbrace{tr[1_{N_c\times N_c}]}}+m_{a}tr[t^a]⟹m_0=tr[M]/N_c$

  $tr[t^{a}M]=m_{0}tr[t^a]+m_{b}tr[t^a{t}^b]⟹m_a=tr[t^{a}M]/T_F$

  Therefore, we can write,

  $M=\frac{1}{N_c}tr[M]1_{N_c\times N_c}+\frac{1}{T_F}tr[t^{a}M]t^a$

  Then,

  $M_{ij}=\frac{1}{N_c}{M}_{kk}{\delta }_{ij}+\frac{1}{T_F}{t}_{kl}^a{M}_{lk}{t}_{ij}^a=M_{lk}\underset{{\delta }_{il}{\delta }_{kj}}{\underbrace{(\frac{1}{N_c}{\delta }_{kl}{\delta }_{ij}+\frac{1}{T_F}{t}_{kl}^a{t}_{ij}^a)}}\forall M$

  $⟹t_{ij}^a{t}_{kl}^a=T_F({\delta }_{il}{\delta }_{kj}-\frac{1}{N_c}{\delta }_{ij}{\delta }_{kl})$

The totally anti-symmetric structure constants $f^{abc}$ have the following properties:

1. Totally anti-symmetric:

$i{f}^{abc}{t}^c=[t^a,t^b]=-[t^b,t^a]=-i{f}^{bac}{t}^c⟹f^{abc}=-f^{bac}$

2. Real:

$i{f}^{abc}{t}^c=[t^a,t^b]=([t^b,t^a]{)}^{†}=-i(f^{bac}{)}^{\ast }{t}^c⟹f^{abc}=(f^{abc}{)}^{\ast }$

3. Obey the Jacobi identity:

$f^{abc}{f}^{cde}+f^{adc}{f}^{ceb}+f^{dec}{f}^{cbd}=0$

3. Adjoint Representation

The totally anti-symmetric structure constants are (up to normalisation) the generators of the adjoint representation,

$(F^a{)}_{bc}\equiv -i{f}^{abc}$ for $a=1,...,N_c^2-1,b,c=1,...,N_c^2-1$

There are $(N_c^2-1)$ generators, each of which can be represented by an $(N_c^2-1)\times (N_c^2-1)$ matrix.

• Commutator: $[F^a,F^b]=i{f}^{abc}{F}^c$

4. Quadratic Casimir Invariants

For each representation of the Lie Algebra, we can construct quadratic Casimir invariants which commute with all generators,

${\mathbb{C}}_2=T^a{T}^a=C_R{1}_R$ where $T^a$ are the generators of a some arbitrary representation R.

• Fundamental Casimir: $(t^a{t}^a{)}_{ij}=C_F{\delta }_{ij}$ with $C_F=T_F\frac{N_c^2-1}{N_c}$
• Adjoint Casimir: $(F^c{F}^c{)}_{ab}=C_A{\delta }_{ab}$ with $C_A=N_c$
  • In terms of structure constants, $(-i{f}^{cad})(-i{f}^{cdb})=f^{acd}{f}^{bcd}=C_A{\delta }^{ab}$

5. Symmetric Constants

If we consider the anti-commutator,

$\{t^a,t^b\}=\frac{2T_F}{N_c}{\delta }_{ab}{1}_{N_c\times N_c}+d^{abc}{t}^c$

The symmetric constants $d^{abc}$ have the following properties:

• $d^{aac}=0$
• $d^{abc}{d}^{abd}=\frac{N_c^2-4}{N_c}{d}^{cd}$