Definitions

Special_linear_group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group, given by the kernel of the determinant

detcolon operatorname{GL}(n, F) to F^times.

where we write F× for the multiplicative group of F (that is, excluding 0).

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

Geometric interpretation

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.

Lie subgroup

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

Topology

Any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues. The determinant of the unitary matrix is on the unit circle while that of the hermitian matrix is real and positive, and since in the case of a matrix from the special linear group the product of these two determinants must be 1, then each of them must be 1. Therefore, a special linear matrix can be written as the product of a special unitary matrix (or special orthogonal matrix in the real case) and a positive definite hermitian matrix (or symmetric matrix in the real case) having determinant 1.

Thus the topology of the group SL(n, C) is the product of the topology of SU(n) and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues. 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 n^2-1 dimensional Euclidean space.

The topology of SL(n, R) is the product of the topology of SO(n) and the topology of the group of symmetric matrices with positive eigenvalues. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of (n+2)(n-1)/2 dimensional Euclidean space.

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

Relations to other subgroups of GL(n,A)

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. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so [operatorname{GL},operatorname{GL}]leqoperatorname{SL}), but in general do not coincide with it.

The group generated by transvections is denoted operatorname{E}_n(A) (for elementary matrices) or operatorname{TV}_n(A). By the second Steinberg relation, for ngeq 3, transvections are commutators, so for ngeq3, operatorname{E}_n(A) leq [operatorname{GL}_n(A),operatorname{GL}_n(A)]. For n=2, transvections need not be commutators (of 2×2 matrices), as seen for example when A is the field of two elements, then operatorname{Alt}(3) cong [operatorname{GL}_2(mathbb{Z}/2mathbb{Z}),operatorname{GL}_2(mathbb{Z}/2mathbb{Z})] < operatorname{E}_2(mathbb{Z}/2mathbb{Z}) = operatorname{SL}_2(mathbb{Z}/2mathbb{Z}) = operatorname{GL}_2(mathbb{Z}/2mathbb{Z}) cong operatorname{Sym}(3).

In some circumstances these coincide: the special linear group over a field or the integers is generated by transvections, and the stable special linear group over a Dedekind domain is generated by transvections. For more general rings the stable difference is measured by the special Whitehead group SK_1(A) := operatorname{SL}(A)/operatorname{E}(A), where operatorname{SL}(A) and operatorname{E}(A) are the stable groups of the special linear group and elementary matrices.

Generators and relations

If working over a ring where SL is generated by transvections (such as a ring or the integers), one can give a presentation of SL using transvections with some relations. Transvections satisfy the Steinberg relations, but these are not sufficient: the resulting group is the Steinberg group, which is not the special linear group, but rather the universal central extension of the commutator subgroup of GL.

A sufficient set of relations for operatorname{SL}(n,mathbf{Z}) for ngeq 3 is given by two of the Steinberg relations, plus a third relation . Let T_{ij} := e_{ij}(1) be the elementary matrix with 1's on the diagonal and in the ij position, and 0's elsewhere (and ineq j). Then

begin{align}
left[T_{ij},T_{jk} right] &= T_{ik} && mbox{for } i neq k left[T_{ij},T_{kl} right] &= mathbf{1} && mbox{for } i neq l, j neq k (T_{12}T_{21}^{-1}T_{12})^4 &= mathbf{1} end{align} are a complete set of relations for operatorname{SL}(n,mathbf{Z}), ngeq 3.

Structure of GL(n,F)

The group operatorname{GL}(n, F) splits over its determinant (we use F^times stackrel{sim}{to} operatorname{GL}(1,F) hookrightarrow operatorname{GL}(n,F) as the monomorphism from F^times to operatorname{GL}(n,F), see semidirect product), and therefore GL(n, F) can be written as a semidirect product of SL(n, F) by F×:
GL(n, F) = SL(n, F) ⋊ F×.

See also

References

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