sl3C

sl3C commonly denotes the special linear group SL(3, C), the group of all 3x3 complex matrices with determinant equal to 1, under matrix multiplication. It is a complex Lie group, non-compact, and connected. As a real manifold, SL(3, C) has dimension 16. Its maximal compact subgroup is SU(3), and SL(3, C) can be viewed as a complexification of SU(3).

The associated Lie algebra is sl(3, C), the set of all 3x3 complex matrices with trace zero. This is an eight-dimensional complex Lie algebra ( sixteen real dimensions) with the commutator as its bracket. The algebra has rank 2 and root system of type A2, reflecting its underlying symmetry structure. Representations of SL(3, C) are classified by highest weights, and its finite-dimensional irreducible representations are labeled by two nonnegative integers corresponding to the two fundamental weights. The two fundamental representations have dimensions 3 and 3̄ (the standard and its dual).

In addition to its intrinsic algebraic structure, SL(3, C) appears in various areas of mathematics and theoretical physics. In representation theory, it serves as a central example of a complex semisimple Lie group and is closely related to the study of algebraic groups, flag varieties, and geometric representation theory. In physics, complexified gauge groups such as SL(3, C) arise in certain formulations of gauge theory and in the study of symmetry breaking, though the physically realized compact form often centers on SU(3).