Geometric invariant decomposition of SU(3)
Martin Roelfs
TL;DR
This work introduces an invariant decomposition for $\mathbf{B}\in\mathfrak{su}(3)$ into at most three commuting pieces $\mathbf{b}_i$ with $\mathbf{b}_i^2=\lambda_i\mathbb{1}$, enabling a factorized exponential $U=\exp(\mathbf{B})=\prod_{i=1}^3\big(\cos\beta_i\,\mathbb{1}+\hat{\mathbf{b}}_i\sin\beta_i\big)$ and a principal logarithm built from the corresponding factors. It develops explicit constructions, including a non-diagonalization formula when eigenvalues are distinct, and extends to higher dimensions with degenerate cases. The paper then provides a practical SU(3) factorization $U=U_1U_2U_3$, derives invariant blocks to compute each $U_i$, and derives the principal log $\operatorname{Ln}U=\sum_i\operatorname{Ln}U_i$ with non-uniqueness analogous to the complex logarithm. Applications include a decomposition of Gell-Mann matrices and connections to geometric algebra, offering an intuitive, Abelian-like handle on a non-Abelian group and potential computational benefits in SU(3) analyses.
Abstract
A novel invariant decomposition of diagonalizable $n \times n$ matrices into $n$ commuting matrices is presented. This decomposition is subsequently used to split the fundamental representation of $\mathfrak{su}(3)$ Lie algebra elements into at most three commuting elements of $\mathfrak{u}(3)$. As a result, the exponential of an $\mathfrak{su}(3)$ Lie algebra element can be split into three commuting generalized Euler's formulas, or conversely, a Lie group element can be factorized into at most three generalized Euler's formulas. After the factorization has been performed, the logarithm follows immediately.