In Hubbard's original approach, Q = 1 , but by generalizing this quantity to larger values, higher irreducible representations of SU(2|1) are generated. The slave boson representation can be extended from two component to N component fermions, where the spin index α α ∈ ∈ [ 1 , N ] runs over N values. By allowing N to become large, while maintaining the ratio Q / N , it is possible to develop a controlled large- N expansion.