SU(2) Spin-1/2 Representation Question

[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.

 

[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~