Lie algebra of axial charges Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\tau_a$ are pauli matrices)
$L=\bar Ni\gamma^\mu \partial_\mu N+ \frac{1}{2} \partial_\mu\sigma\partial^\mu\sigma+\frac{1}{2}\partial_\mu\pi_a\partial^\mu\pi_a+g\bar N(\sigma+i\gamma_5\pi_a \tau_a)N$
we can construct conserved currents using Noether's Theorem applied to $SU(2)_L\otimes SU(2)_R$ symmetry: we get three currents for every $SU(2)$.
By adding and subtracting them, we obtain vector and axial currents.
We could have obtained vector charges quickly by observing that they are just isospin charges, so nucleons behave as an $SU(2)$ doublet (fundamental 
representation), pions as a triplet (adjoint representation) and sigma as a singlet (so basically it does not transform):
$V_a=-i\int d^3x \,\,[iN^\dagger\frac{\tau_a}{2}N+\dot\pi_b(-i\epsilon_{abc})\pi_c]$
But if I wanted to do the same with axial charges, what Lie algebra/representation must I use for pions and sigma?
I mean, axial charges are
$A_a=-i\int d^3x \,\,[iN^\dagger\frac{\tau_a}{2}\gamma_5N+i(\sigma\dot\pi_a-\dot\sigma\pi_a)]$
and I would like to reproduce the second term using a representation of Lie algebra generators of axial symmetry which act on $\sigma$ and $\pi$, but I don't know the algebra (I think it is $SU(2)$), neither the representation to use.
I tried to reproduce that form using the three matrices
$T^1=\begin{bmatrix} 0&-i&0&0\\i&0&0&0\\0&0&0&0\\0&0&0&0 \end{bmatrix}\quad
T^2=\begin{bmatrix} 0&0&-i&0\\0&0&0&0\\i&0&0&0\\0&0&0&0 \end{bmatrix}\quad
T^3=\begin{bmatrix} 0&0&0&-i\\0&0&0&0\\0&0&0&0\\i&0&0&0 \end{bmatrix}$
which should act on the vector $(\sigma,\pi_1,\pi_2,\pi_3)$, but I calculated their commutator and they don't form an algebra, so I think I'm getting wrong 
somewhere in my reasoning.
 A: In the linear sigma model, the chiral action on the pion fields can be
implemented on the following matrix combination of the fields:
$$U(2) \ni \Sigma = \sigma + i \tau^a  \pi_a $$
An  element $ (U_L = exp(\frac{i}{2}\theta^{(L)}_a \tau^a), U_R = exp(\frac{i}{2}\theta^{(R)}_a \tau^a)) \in SU(2)_L \otimes  SU(2)_R $
acts on \Sigma as follows:
$$\Sigma  \rightarrow \Sigma' = U_L \Sigma U_R^{\dagger}$$
The kinetic term of the Lagrangian in the matrix representation is given by:
$$L_{kin} = \frac{1}{2} \partial_{\mu}\Sigma \partial^{\mu}\Sigma^{\dagger}$$. 
This term is manifestly invariant under all transformations. The interaction term has also a manifestly invariant form:
$$L_{int} = \bar{N}_L \Sigma N_R+ \bar{N}_R \Sigma^{\dagger} N_L$$.
where $N_{L,R} = (1\pm \gamma_5)N $. Thus the whole Lagrangian is invariant under the chiral transformations.
The vector transformation is generated by the subgroup characterized by:
$$\theta^{(L)} = \theta^{(R)} = \theta^{(V)}$$
The axial transformation is generated by the subset characterized by:
$$\theta^{(L)} = -\theta^{(R)} = \theta^{(A)}$$
Substituting in the transformation equations of $\Sigma$ and keeping only the linear terms (this is sufficient for the application of the Noether's theorem), we obtain:
-Vector transformation:
$$ \pi_a' = \pi_a +\epsilon_{abc}\theta^{(V)}_b \pi_c $$
$$ \sigma' = \sigma$$
-Axial transformation:
$$ \pi_a' = \pi_a +\theta^{(A)}_a \sigma $$
$$ \sigma' = \sigma + \theta^{(A)}_a \pi_a$$
Now it is not hard to see that these transformations generate the correct contributions of the pionic fields to the currents.
