SL3

sl3, or sl_3, is the complex special linear Lie algebra of 3×3 matrices with trace zero. It is the Lie algebra of the Lie group SL(3, C), the group of determinant-1 complex matrices. As a complex Lie algebra, sl3 is simple, has dimension 8 and rank 2, and is classified as type A2 in the Cartan scheme.

A standard realization is the set of all 3×3 complex matrices with trace zero, with Lie bracket given by [X, Y] = XY − YX. A convenient basis consists of the eight matrices E_ij for i ≠ j, together with diagonal matrices H1 = E11 − E22 and H2 = E22 − E33. The Cartan subalgebra is the space of diagonal trace-zero matrices, and the associated root system is of type A2, with simple roots α1 and α2 and positive roots α1, α2, α1+α2.

Representations of sl3 include the defining 3-dimensional representation on C^3 and its dual 3̄ as fundamental representations. The adjoint representation has dimension 8. In highest-weight language, representations are labeled by two nonnegative integers (λ1, λ2) corresponding to the fundamental weights ω1 and ω2. Notable decompositions include 3 ⊗ 3̄ ≅ 1 ⊕ 8 and 3 ⊗ 3 ≅ 6 ⊕ 3̄.

Relation to groups and real forms: sl3 is the complexification of su(3), the compact real form, and is related to sl(3, R), a real form. The group SL(3, C) has sl3 as its Lie algebra, with the exponential map linking algebra elements to group elements locally. The Killing form is nondegenerate, reflecting sl3’s status as a simple Lie algebra.

In summary, sl3 is a fundamental example in Lie theory, illustrating a simple complex Lie algebra of type A2, its root structure, and its essential representations.