### Abstract:

This paper provides an accurate mathematical method for determining SU(N) symmetry group generators in a general N-dimensional quantum state space. We identify the generators as well-defined quantum
state transition and eigenvalue operators occurring in an orderly pattern within a system of (N − 1) focal
state transition spaces specified by focal state vectors |m⟩ , m = 2, 3, ..., N which constitute an SU(N)
generator-spectrum. Each focal state transition space specified by a focal state vector |m⟩ (denoted by
FSTS-|m⟩) contains 2m−1 traceless non-diagonal and diagonal symmetric and antisymmetric generators
plus 1 non-traceless diagonal symmetric generator. The full SU(N) generator-spectrum composed of an
orderly pattern of (N − 1) focal state transition spaces contains a total of N
2 − 1 traceless non-diagonal
and diagonal symmetric and antisymmetric generators plus (N − 1) non-traceless diagonal symmetric
generators. A well-defined weighted sum of the (N −1) non-traceless diagonal symmetric generators constitutes a completeness relation within the N-dimensional quantum state space of the SU(N) symmetry
group. Noting that each focal state transition space FSTS-|m⟩ contains m − 1 two-state subspaces, we
determine an orderly distribution of the 1
2N(N − 1) two-state subspaces among the N − 1 focal state
transition spaces in an SU(N) generator-spectrum. Realizing that the basic SU(N) generator-spectrum
for N ≥ 3 is not algebraically closed, we introduce “Cartan” and “conjugate-Cartan” generators which
provide the desired closed SU(N) algebra.