SU(2) Spin-1/2 Representation Question

1. May 20, 2014

Xenosum

In Howard Georgi's Lie Algebras in Particle Physics (and other texts I'm sure), it is determined that the Pauli matrices, $\sigma_1$ $\sigma_2$ and $\sigma_3$, in 2 dimensions form an irreducible representation of the SU(2) algebra.

This is a bit confusion to me. The SU(2) algebra is given by
$$\left[ J_j, J_k \right] = i\epsilon_{jkl}J_l ,$$
where $J_i$ are the generators. Meanwhile, the Pauli matrices satisfy:
$$\sigma_a \sigma_b = \delta_{ab} + i\epsilon_{abc} \sigma_c .$$
But this implies that the zero generator gets mapped to the identity operator in the spin-1/2 SU(2) representation, because $\left[ J_a, J_a \right] = 0$, while $\sigma_a \sigma_a = 1$.

Isn't it a condition for any representation of an algebra for the identity element in the algebra to get mapped to the identity operator?

Thanks for any help.

2. May 20, 2014

Xenosum

Ah, nevermind.

Firstly it's not exactly the Pauli matrices which form a representation of the SU(2) algebra; rather it is the Pauli matrices multiplied by a factor of 1/2.

Moreover I was mistaken in generalizing the concept of a representation of a group to the concept of a representation of an algebra. In a group, the binary operation always gets mapped to matrix multiplication, no matter what it is. Apparently for an algebra it's a bit different: the binary operation gets mapped to itself, and the matrices are taken to preserve the relationship which is defined on the algebra. Since the Pauli matrices satisfy
$$\left[ \frac{1}{2}\sigma_a, \frac{1}{2}\sigma_b \right] = i \epsilon_{abc}\frac{1}{2} \sigma_c ,$$
they form a viable representation of the SU(2) algebra.

Sorry~