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Inmathematics, thespecial linear groupSL(n,R) of degreen over acommutative ringR is the set ofn ×nmatrices withdeterminant 1, with the group operations of ordinarymatrix multiplication andmatrix inversion. This is thenormal subgroup of thegeneral linear group given by thekernel of thedeterminant

${\displaystyle det:\mathrm{GL}(n,R)\to R^{\times }.}$

whereR^× is themultiplicative group ofR (that is,R excluding 0 whenR is a field).

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

WhenR is thefinite field of orderq, the notationSL(n,q) is sometimes used.

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

WhenF isR orC,SL(n,F) is aLie subgroup ofGL(n,F) of dimensionn² − 1. TheLie algebra${\displaystyle \mathfrak{s}\mathfrak{l}(n,F)}$ of SL(n,F) consists of alln ×n matrices overF with vanishingtrace. TheLie bracket is given by thecommutator.

Any invertible matrix can be uniquely represented according to thepolar decomposition as the product of aunitary matrix and ahermitian matrix with positiveeigenvalues. Thedeterminant of the unitary matrix is on theunit 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 aspecial unitary matrix (orspecial orthogonal matrix in the real case) and apositive definite hermitian matrix (orsymmetric matrix in the real case) having determinant 1.

Thus the topology of the groupSL(n,C) is theproduct 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 theexponential of atraceless hermitian matrix, and therefore the topology of this is that of(n² − 1)-dimensionalEuclidean space.^[1] Since SU(n) issimply connected,^[2] we conclude thatSL(n,C) is also simply connected, for alln greater than or equal to 2.

The topology ofSL(n,R) is the product of the topology ofSO(n) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. 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. Thus, the groupSL(n,R) has the samefundamental group as SO(n), that is,Z forn = 2 andZ₂ forn > 2.^[3] In particular this means thatSL(n,R), unlikeSL(n,C), is not simply connected, forn greater than 1.

Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are thecommutator subgroup of GL, and the group generated bytransvections. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so [GL, GL] ≤ SL), but in general do not coincide with it.

The group generated by transvections is denotedE(n,A) (forelementary matrices) orTV(n,A). By the secondSteinberg relation, forn ≥ 3, transvections are commutators, so forn ≥ 3,E(n,A) ≤ [GL(n,A), GL(n,A)].

Forn = 2, transvections need not be commutators (of2 × 2 matrices), as seen for example whenA isF₂, the field of two elements, then

where Alt(3) and Sym(3) denote thealternating resp.symmetric group on 3 letters.

However, ifA is a field with more than 2 elements, thenE(2,A) = [GL(2,A), GL(2,A)], and ifA is a field with more than 3 elements,E(2,A) = [SL(2,A), SL(2,A)].^{[dubious –discuss]}

In some circumstances these coincide: the special linear group over a field or aEuclidean domain is generated by transvections, and thestable special linear group over aDedekind domain is generated by transvections. For more general rings the stable difference is measured by thespecial Whitehead groupSK₁(A) := SL(A)/E(A), where SL(A) and E(A) are thestable groups of the special linear group and elementary matrices.

If working over a ring where SL is generated bytransvections (such as afield orEuclidean domain), one can give apresentation of SL using transvections with some relations. Transvections satisfy theSteinberg relations, but these are not sufficient: the resulting group is theSteinberg group, which is not the special linear group, but rather theuniversal central extension of the commutator subgroup of GL.

A sufficient set of relations forSL(n,Z) forn ≥ 3 is given by two of the Steinberg relations, plus a third relation (Conder, Robertson & Williams 1992, p. 19).LetT_ij :=e_ij(1) be the elementary matrix with 1's on the diagonal and in theij position, and 0's elsewhere (andi ≠j). Then

are a complete set of relations for SL(n,Z),n ≥ 3.

Incharacteristic 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. This forms ashort exact sequence of groups:

${\displaystyle 1\to \mathrm{SL}(n,F)\to {\mathrm{SL}}^{\pm }(n,F)\to \{\pm 1\}\to 1.}$

This sequence splits by taking any matrix with determinant−1, for example the diagonal matrix${\displaystyle (-1,1,\dots ,1).}$ If${\displaystyle n=2k+1}$ is odd, the negative identity matrix${\displaystyle -I}$ is inSL^±(n,F) but not inSL(n,F) and thus the group splits as aninternal direct product${\displaystyle {\mathrm{SL}}^{\pm }(2k+1,F)\cong \mathrm{SL}(2k+1,F)\times \{\pm I\}}$. However, if${\displaystyle n=2k}$ is even,${\displaystyle -I}$ is already inSL(n,F) ,SL^± does not split, and in general is a non-trivialgroup extension.

Over the real numbers,SL^±(n,R) has twoconnected components, corresponding toSL(n,R) and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant−1). In odd dimension these are naturally identified by${\displaystyle -I}$, but in even dimension there is no one natural identification.

The groupGL(n,F) splits over its determinant (we useF^× ≅ GL(1,F) → GL(n,F) as themonomorphism fromF^× toGL(n,F), seesemidirect product), and thereforeGL(n,F) can be written as a semidirect product ofSL(n,F) byF^×:

GL(n,F) = SL(n,F) ⋊F^×.

• SL(2,R)
• SL(2,C)
• Modular group (PSL(2,Z))
• Projective linear group
• Conformal map
• Representations of classical Lie groups

This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Special linear group" – news ·newspapers ·books ·scholar ·JSTOR(January 2008) (Learn how and when to remove this message)

• Conder, Marston; Robertson, Edmund; Williams, Peter (1992), "Presentations for 3-dimensional special linear groups over integer rings",Proceedings of the American Mathematical Society,115 (1), American Mathematical Society:19–26,doi:10.2307/2159559,JSTOR 2159559,MR 1079696
• Hall, Brian C. (2015),Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer

• Linear algebra
• Lie groups
• Linear algebraic groups

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