General Linear Group - Special Linear Group

Special Linear Group

The special linear group, SL(n,F), is the group of all matrices with determinant 1. They are special in that they lie on a subvariety — they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. SL(n, F) is a normal subgroup of GL(n,F).

If we write F× for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism

det: GL(n,F) → F×.

that is surjective and its kernel is the special linear group. Therefore, by the first isomorphism theorem, GL(n,F)/SL(n,F) is isomorphic to F×. In fact, GL(n,F) can be written as a semidirect product:

GL(n,F) = SL(n,F) ⋊ F×

When F is R or C, SL(n,F) is a Lie subgroup of GL(n,F) of dimension n2 − 1. The Lie algebra of SL(n,F) consists of all n×n matrices over F with vanishing trace. The Lie bracket is given by the commutator.

The special linear group SL(n,R) can be characterized as the group of volume and orientation preserving linear transformations of Rn.

The group SL(n,C) is simply connected while SL(n,R) is not. SL(n,R) has the same fundamental group as GL+(n, R), that is, Z for n=2 and Z₂ for n>2.