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, [itex]\sigma_1[/itex] [itex]\sigma_2[/itex] and [itex]\sigma_3[/itex], 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
[tex] \left[ J_j, J_k \right] = i\epsilon_{jkl}J_l , [/tex]
where [itex]J_i[/itex] are the generators. Meanwhile, the Pauli matrices satisfy:
[tex] \sigma_a \sigma_b = \delta_{ab} + i\epsilon_{abc} \sigma_c . [/tex]
But this implies that the zero generator gets mapped to the identity operator in the spin-1/2 SU(2) representation, because [itex]\left[ J_a, J_a \right] = 0[/itex], while [itex]\sigma_a \sigma_a = 1[/itex].

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
[tex] \left[ \frac{1}{2}\sigma_a, \frac{1}{2}\sigma_b \right] = i \epsilon_{abc}\frac{1}{2} \sigma_c , [/tex]
they form a viable representation of the SU(2) algebra.

Sorry~