Editing Special linear group

{{Short description|Group of matrices with determinant 1}}
[[File:SL(2,3); Cayley table.svg|thumb|[[Cayley table]] of SL(2,3).]]
{{Group theory sidebar |Topological}}
{{Lie groups |Classical}}

In [[mathematics]], the '''special linear group''' <math>\operatorname{SL}(n,R)</math> of degree <math>n</math> over a [[commutative ring]] <math>R</math> is the set of <math>n\times n</math> [[Matrix (mathematics)|matrices]] with [[determinant]] <math>1</math>, 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 (algebra)|kernel]] of the [[determinant]]

:<math>\det\colon \operatorname{GL}(n, R) \to R^\times.</math>

where <math>R^\times</math> is the [[multiplicative group]] of <math>R</math> (that is, <math>R</math> excluding <math>0</math> when <math>R</math> is a field).

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

When <math>R</math> is the [[finite field]] of order <math>q</math>, the notation <math>\operatorname{SL}(n,q)</math> is sometimes used.

==Geometric interpretation==
The special linear group <math>\operatorname{SL}(n,\R)</math> can be characterized as the group of ''[[volume]] and [[orientation (mathematics)|orientation]] preserving'' linear transformations of <math>\R^n</math>. This corresponds to the interpretation of the determinant as measuring change in volume and orientation.

==Lie subgroup==
{{main|Special linear Lie algebra}}
When <math>F</math> is <math>\R</math> or <math>\C</math>, <math>\operatorname{SL}(n,F)</math> is a [[Lie subgroup]] of <math>\operatorname{GL}(n,F)</math> of dimension <math>n^2-1</math>. The [[Lie algebra]] <math>\mathfrak{sl}(n, F)</math> of <math>\operatorname{SL}(n,F)</math> consists of all <math>n\times n</math> matrices over <math>F</math> with vanishing [[trace (matrix)|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 [[eigenvalue]]s. The [[determinant]] of the unitary matrix is on the [[unit circle]], while that of the Hermitian matrix is real and positive. 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 matrix|positive definite]] Hermitian matrix (or [[symmetric matrix]] in the real case) having determinant 1.

It follows that the topology of the group <math>\operatorname{SL}(n,\C)</math> is the [[product topology|product]] of the topology of <math>\operatorname{SU}(n)</math> 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 [[matrix exponential|exponential]] of a [[traceless]] Hermitian matrix, and therefore the topology of this is that of <math>(n^2-1)</math>-dimensional [[Euclidean space]].<ref>{{harvnb|Hall|2015}} Section 2.5</ref> Since <math>\operatorname{SU}(n)</math> is [[simply connected]],<ref>{{harvnb|Hall|2015}} Proposition 13.11</ref> then <math>\operatorname{SL}(n,\C)</math> is also simply connected, for all <math>n\geq 2</math>.

The topology of <math>\operatorname{SL}(n,\R)</math> is the product of the topology of [[special orthogonal matrix|SO]](''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 {{nowrap|(''n'' + 2)(''n'' − 1)/2}}-dimensional Euclidean space. Thus, the group <math>\operatorname{SL}(n,\R)</math> has the same [[fundamental group]] as <math>\operatorname{SO}(n)</math>; that is, <math>\Z</math> for <math>n=2</math> and <math>\Z_2</math> for <math>n>2</math>.<ref>{{harvnb|Hall|2015}} Sections 13.2 and 13.3</ref> In particular this means that <math>\operatorname{SL}(n,\R)</math>, unlike <math>\operatorname{SL}(n,\C)</math>, is not simply connected, for <math>n>1</math>.

==Relations to other subgroups of GL(''n'', ''A'')==
{{see also|Whitehead's lemma}}

Two related subgroups, which in some cases coincide with <math>\operatorname{SL}</math>, and in other cases are accidentally conflated with <math>\operatorname{SL}</math>, are the [[commutator subgroup]] of <math>\operatorname{GL}</math>, and the group generated by [[Shear mapping|transvections]]. These are both subgroups of <math>\operatorname{SL}</math> (transvections have determinant 1, and det is a map to an abelian group, so <math>[\operatorname{GL},\operatorname{GL}]<\operatorname{SL}</math>), but in general do not coincide with it.

The group generated by transvections is denoted <math>\operatorname{E}(n,A)</math> (for [[elementary matrices]]) or <math>\operatorname{TV}(n,A)</math>. By the second [[Steinberg relations|Steinberg relation]], for <math>n\geq 3</math>, transvections are commutators, so for <math>n\geq 3</math>, <math>\operatorname{E}(n,A)<[\operatorname{GL}(n,A),\operatorname{GL}(n,A)]</math>.

<!-- Does equality hold? Dunno. -->
For <math>n=2</math>, transvections need not be commutators (of <math>2\times 2</math> matrices), as seen for example when <math>A</math> is <math>\mathbb{F}_2</math>, the field of two elements. In that case 
:<math>A_3 \cong [\operatorname{GL}(2, \mathbb{F}_2),\operatorname{GL}(2, \mathbb{F}_2)] < \operatorname{E}(2, \mathbb{F}_2) = \operatorname{SL}(2, \mathbb{F}_2) = \operatorname{GL}(2, \mathbb{F}_2) \cong S_3,</math>
where <math>A_3</math> and <math>S_3</math> respectively denote the [[alternating group|alternating]] and [[symmetric group]] on 3 letters.

However, if <math>A</math> is a field with more than 2 elements, then {{nowrap|1=E(2, ''A'') = [GL(2, ''A''), GL(2, ''A'')]}}, and if <math>A</math> is a field with more than 3 elements, {{nowrap|1=E(2, ''A'') = [SL(2, ''A''), SL(2, ''A'')]}}. {{Dubious - discuss|date=March 2019}}

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. For more general rings the stable difference is measured by the [[special Whitehead group]] <math>SK_1(A)=\operatorname{SL}(A)/\operatorname{E}(A)</math>, where <math>\operatorname{SL}(A)</math> and <math>\operatorname{E}(A)</math> are the [[direct limit of groups|stable group]]s of the special linear group and elementary matrices.

==Generators and relations==
If working over a ring where <math>\operatorname{SL}</math> is generated by [[Shear mapping|transvections]] (such as a [[Field (mathematics)|field]] or [[Euclidean domain]]), one can give a [[presentation of a group|presentation]] of <math>\operatorname{SL}</math> using transvections with some relations. Transvections satisfy the [[Steinberg relations]], but these are not sufficient: the resulting group is the [[Steinberg group (K-theory)|Steinberg group]], which is not the special linear group, but rather the [[universal central extension]] of the commutator subgroup of <math>\operatorname{GL}</math>.

A sufficient set of relations for {{nowrap|SL(''n'', '''Z''')}} for {{nowrap|''n'' ≥ 3}} is given by two of the Steinberg relations, plus a third relation {{harv|Conder|Robertson|Williams|1992|p=19}}.
Let {{nowrap|1=''T<sub>ij</sub>'' := ''e<sub>ij</sub>''}}(1) be the elementary matrix with 1's on the diagonal and in the ''ij'' position, and 0's elsewhere (and ''i'' ≠ ''j''). Then

:<math>\begin{align}
            \left[ T_{ij},T_{jk} \right] &= T_{ik}     && \text{for } i \neq k \\[4pt]
         \left[ T_{ij},T_{k\ell} \right] &= \mathbf{1} && \text{for } i \neq \ell, j \neq k \\[4pt]
  \left(T_{12}T_{21}^{-1}T_{12}\right)^4 &= \mathbf{1}
\end{align}</math>

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

==SL<sup>±</sup>(''n'',''F'')==
In [[Characteristic (algebra)|characteristic]] other than 2, the set of matrices with determinant {{math|±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 a [[short exact sequence]] of groups:
:<math>1\to\operatorname{SL}(n, F) \to \operatorname{SL}^{\pm}(n, F) \to \{\pm 1\}\to1.</math>

This sequence splits by taking any matrix with determinant {{math|−1}}, for example the diagonal matrix <math>(-1, 1, \dots, 1).</math> If <math>n = 2k + 1</math> is odd, the negative identity matrix <math>-I</math> is in {{math|SL<sup>±</sup>(''n'',''F'')}} but not in {{math|SL(''n'',''F'')}} and thus the group splits as an [[internal direct product]] <math>\operatorname{SL}^\pm(2k + 1, F) \cong \operatorname{SL}(2k + 1, F) \times \{\pm I\}</math>. However, if <math>n = 2k</math> is even, <math>-I</math> is already in {{math|SL(''n'',''F'')}} , {{math|SL<sup>±</sup>}} does not split, and in general is a non-trivial [[group extension]].

Over the real numbers, {{math|SL<sup>±</sup>(''n'', ''R'')}} has two [[connected component (topology)|connected components]], corresponding to {{math|SL(''n'', ''R'')}} and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant {{math|−1}}). In odd dimension these are naturally identified by <math>-I</math>, but in even dimension there is no one natural identification.

==Structure of GL(''n'',''F'')==
The group <math>\operatorname{GL}(n,F)</math> splits over its determinant (we use <math>F^\times = \operatorname{GL}(1,F)\to \operatorname{GL}(n,F)</math> as the [[monomorphism]] from <math>F^\times</math> to <math>\operatorname{GL}(n,F)</math>, see [[semidirect product]]), and therefore <math>\operatorname{GL}(n,F)</math> can be written as a semidirect product of <math>\operatorname{SL}(n,F)</math> by <math>F^\times</math>:

:<math>\operatorname{GL}(n,F)=\operatorname{SL}(n,F)\rtimes F^\times</math>.

==See also==
* [[SL2(R)|SL(2, '''R''')]]
* [[SL2(C)|SL(2, '''C''')]]
* [[Modular group]] (PSL(2, '''Z'''))
* [[Projective linear group]]
* [[Conformal map]]
* [[Representations of classical Lie groups]]

==References==
{{Reflist}}
{{more citations needed|date=January 2008}}
*{{Citation | last1=Conder | first1=Marston|author1-link=Marston Conder | last2=Robertson | first2=Edmund | last3=Williams | first3=Peter | title=Presentations for 3-dimensional special linear groups over integer rings | mr=1079696  | year=1992 | journal=Proceedings of the American Mathematical Society | volume=115 | issue=1 | pages=19–26 |doi=10.2307/2159559 | publisher=American Mathematical Society | jstor=2159559 }}
*{{Citation| last=Hall|first=Brian C.|title=Lie groups, Lie algebras, and representations: An elementary introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015}}

[[Category:Linear algebra]]
[[Category:Lie groups]]
[[Category:Linear algebraic groups]]