The group of determinant-one matrices over a commutative ring.
For a nonzero commutative ringR with identity and n≥2, the special linear group over R is
SLn(R)={A∈Mn(R):detA=1}
under matrix multiplication. Here Mn(R) denotes the n-by-n matrices with entries in R, and the determinant is given by its usual signed-permutation polynomial.
Why it is a groupOpen
The determinant is multiplicative, and A−1=adj(A) when detA=1. The adjugate still has entries in R, so inversion stays in the set.
Rank twoOpen
An element is (acbd) with ad−bc=1; its inverse is (d−c−ba). For R=R or C, the Lie-group version supplies the topology and smooth structure. No Lie-group structure is part of the ring-valued definition.
A commutative ring is a ringR such that ab=ba for all a,b∈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).