with matrix multiplication and the inherited real Lie group structure.
Dimensions and structureOpen
This group has real dimension 0. Differentiating the determinant at the identity gives the trace-zero tangent algebra. The center consists of scalar matrices λI with λ1=1. In size one the defining equation forces the sole matrix to be 1, so this is the trivial group.
The inclusion sends a determinant-one matrix to the same invertible matrix; it is injective and smooth. See GL(1,R).
The scalar-center quotient has kernel {λI:λ1=1}. See PSL(1,R).
Consequently, the underlying real Lie groupSL(n,C)R has real dimension 2(n2−1). An unqualified statement that SL(n,C) has manifold dimension n2−1 is correct only when “dimension” means complex dimension.
Its Lie algebra is the trace-zero matrices, the special linear Lie algebrasln(F), and the exponential map restricts to exp:sln(F)→SL(n,F) (see exponential map). The determinant condition differentiates to the trace condition:
dtdt=0det(I+tX)=tr(X).
The groups SL(n,R) and SL(n,C) are basic examples of connected linear Lie groups, and they play a central role in semisimple theory (compare semisimple Lie algebras and the root-theoretic framework starting at root systems).
A Lie group is a group G equipped with the structure of a smooth manifold such that the group operations are smooth maps:
μ:G×G→G,μ(g,h)=gh,ι:G→G,ι(g)=g−1.
For each g∈G, the left translationLg(h)=gh and the right translationRg(h)=hg are diffeomorphisms of G, with inverses Lg−1 and Rg−1. The tangent space at the identity TeG carries a canonical Lie algebra structure, called the Lie algebra of G, and the exponential map relates this infinitesimal structure to local group behavior near e.