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| [[File:SL(2,3); Cayley table.svg|thumb|[[Cayley table]] of SL(2,3).]]
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| {{Group theory sidebar |Topological}}
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| {{Lie groups |Classical}}
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| In [[mathematics]], the '''special linear group''' of degree ''n'' over a [[Field (mathematics)|field]] ''F'' is the set of {{nowrap|''n'' × ''n''}} [[Matrix (mathematics)|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 (algebra)|kernel]] of the determinant
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| :<math>\det\colon \operatorname{GL}(n, F) \to F^\times.</math>
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| where we write ''F''<sup>×</sup> for the multiplicative group of ''F'' (that is, excluding 0). | |
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| These elements are "special" in that they fall on a [[Algebraic variety|subvariety]] of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).
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| ==Geometric interpretation==
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| The special linear group {{nowrap|SL(''n'', '''R''')}} can be characterized as the group of ''[[volume]] and [[orientation (mathematics)|orientation]] preserving'' linear transformations of '''R'''<sup>''n''</sup>; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.
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| ==Lie subgroup==
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| When ''F'' is '''R''' or '''C''', {{nowrap|SL(''n'', ''F'')}} is a [[Lie subgroup]] of {{nowrap|GL(''n'', ''F'')}} of dimension {{nowrap|''n''<sup>2</sup> − 1}}. The [[Lie algebra]] <math>\mathfrak{sl}(n, F)</math> of SL(''n'', ''F'') consists of all {{nowrap|''n'' × ''n''}} matrices over ''F'' with vanishing [[trace (matrix)|trace]]. The Lie bracket is given by the [[commutator]].
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| ==Topology==
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| 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 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 matrix|positive definite]] hermitian matrix (or [[symmetric matrix]] in the real case) having determinant 1.
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| Thus the topology of the group {{nowrap|SL(''n'', '''C''')}} is the [[product topology|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 [[matrix exponential|exponential]] of a [[traceless]] hermitian matrix, and therefore the topology of this is that of {{nowrap|(''n''<sup>2</sup> − 1)}}-dimensional [[Euclidean space]].
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| The topology of {{nowrap|SL(''n'', '''R''')}} 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.
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| The group {{nowrap|SL(''n'', '''C''')}}, like SU(''n''), is simply connected while {{nowrap|SL(''n'', '''R''')}}, like SO(''n''), is not. {{nowrap|SL(''n'', '''R''')}} has the same fundamental group as {{nowrap|GL<sup>+</sup>(''n'', '''R''')}} or SO(''n''), that is, '''Z''' for {{nowrap|1=''n'' = 2}} and '''Z'''<sub>2</sub> for {{nowrap|''n'' > 2}}.
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| ==Relations to other subgroups of GL(''n'',''A'')==
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| {{see also|Whitehead's lemma}}
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| 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 [[Shear mapping|transvections]]. 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.
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| The group generated by transvections is denoted {{nowrap|E(''n'', ''A'')}} (for [[elementary matrices]]) or {{nowrap|TV(''n'', ''A'')}}. By the second [[Steinberg relations|Steinberg relation]], for {{nowrap|''n'' ≥ 3}}, transvections are commutators, so for {{nowrap|''n'' ≥ 3}}, {{nowrap|E(''n'', ''A'') ≤ [GL(''n'', ''A''), GL(''n'', ''A'')]}}.
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| <!-- Does equality hold? Dunno. -->
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| For {{nowrap|1=''n'' = 2}}, transvections need not be commutators (of {{nowrap|2 × 2}} matrices), as seen for example when ''A'' is '''F'''<sub>2</sub>, the field of two elements, then
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| :<math>\operatorname{Alt}(3) \cong [\operatorname{GL}(2, \mathbf{F}_2),\operatorname{GL}(2, \mathbf{F}_2)] < \operatorname{E}(2, \mathbf{F}_2) = \operatorname{SL}(2, \mathbf{F}_2) = \operatorname{GL}(2, \mathbf{F}_2) \cong \operatorname{Sym}(3),</math>
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| where Alt(3) and Sym(3) denote the [[alternating group|alternating]] resp. [[symmetric group]]<!--- I suppose this is meant; that article does not mention "Sym(n)" notation---> on 3 letters.
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| However, if ''A'' is a field with more than 2 elements, then {{nowrap|1=E(2, ''A'') = [GL(2, ''A''), GL(2, ''A'')]}}, and if ''A'' is a field with more than 3 elements, {{nowrap|1=E(2, ''A'') = [SL(2, ''A''), SL(2, ''A'')]}}.
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| 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]] {{nowrap|1=SK<sub>1</sub>(''A'') := SL(''A'')/E(''A'')}}, where SL(''A'') and E(''A'') are the [[direct limit of groups|stable group]]s of the special linear group and elementary matrices.
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| ==Generators and relations==
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| If working over a ring where SL 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 SL 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 GL.
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| 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}}.
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| 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
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| :<math>\begin{align}
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| \left[ T_{ij},T_{jk} \right] &= T_{ik} && \mbox{for } i \neq k\\
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| \left[ T_{ij},T_{kl} \right] &= \mathbf{1} && \mbox{for } i \neq l, j \neq k\\
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| (T_{12}T_{21}^{-1}T_{12})^4 &= \mathbf{1}\\
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| \end{align}</math>
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| are a complete set of relations for SL(''n'', '''Z'''), ''n'' ≥ 3.
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| ==Structure of GL(''n'',''F'')==
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| The group {{nowrap|GL(''n'', ''F'')}} splits over its determinant (we use {{nowrap|''F''<sup>×</sup> ≅ GL(1, ''F'') → GL(''n'', ''F'')}} as the [[monomorphism]] from ''F''<sup>×</sup> to {{nowrap|GL(''n'', ''F'')}}, see [[semidirect product]]), and therefore {{nowrap|GL(''n'', ''F'')}} can be written as a [[semidirect product]] of {{nowrap|SL(''n'', ''F'')}} by ''F''<sup>×</sup>:
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| :GL(''n'', ''F'') = SL(''n'', ''F'') ⋊ ''F''<sup>×</sup>.
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| ==See also==
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| * [[SL2(R)|SL(2, '''R''')]]
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| * [[SL2(C)|SL(2, '''C''')]]
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| * [[Modular group]]
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| * [[Projective linear group]]
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| ==References==
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| {{refimprove|date=January 2008}}
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| *{{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 }}
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| [[Category:Linear algebra]]
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| [[Category:Lie groups]]
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| [[pl:Pełna grupa liniowa#Specjalna grupa liniowa]]
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