Special linear group

In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).

The special linear group SL(n, R) can be characterized as the group of volume and orientation preserving linear transformations of Rn; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.

A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of (n2 − 1)-dimensional Euclidean space.

Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and the group generated by transvections.

The group generated by transvections is denoted E(n, A) (for elementary matrices) or TV(n, A).

For n = 2, transvections need not be commutators (of 2 × 2 matrices), as seen for example when A is F2, the field of two elements, then where Alt(3) and Sym(3) denote the alternating resp.

[dubious – discuss] In some circumstances these coincide: the special linear group over a field or a Euclidean domain is generated by transvections, and the stable special linear group over a Dedekind domain is generated by transvections.

Let Tij := eij(1) be the elementary matrix with 1's on the diagonal and in the ij position, and 0's elsewhere (and i ≠ j).

In characteristic other than 2, the set of matrices with determinant ±1 form another subgroup of GL, with SL as an index 2 subgroup (necessarily normal); in characteristic 2 this is the same as SL.

is in SL±(n,F) but not in SL(n,F) and thus the group splits as an internal direct product

is already in SL(n,F) , SL± does not split, and in general is a non-trivial group extension.

Over the real numbers, SL±(n, R) has two connected components, corresponding to SL(n, R) and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant −1).