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Inmathematics, thespecial linear group${\displaystyle \mathrm{SL}(n,R)}$ of degree${\displaystyle n}$ over acommutative ring${\displaystyle R}$ is the set of${\displaystyle n\times n}$matrices withdeterminant${\displaystyle 1}$, with the group operations of ordinarymatrix multiplication andmatrix inversion. This is thenormal subgroup of thegeneral linear group given by thekernel of thedeterminant

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

where${\displaystyle R^{\times }}$ is themultiplicative group of${\displaystyle R}$ (that is,${\displaystyle R}$ excluding${\displaystyle 0}$ when${\displaystyle R}$ 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).

When${\displaystyle R}$ is thefinite field of order${\displaystyle q}$, the notation${\displaystyle \mathrm{SL}(n,q)}$ is sometimes used.

The special linear group${\displaystyle \mathrm{SL}(n,\mathbb{R})}$ can be characterized as the group ofvolume andorientation preserving linear transformations of${\displaystyle {\mathbb{R}}^n}$. This corresponds to the interpretation of the determinant as measuring change in volume and orientation.

When${\displaystyle F}$ is${\displaystyle \mathbb{R}}$ or${\displaystyle \mathbb{C}}$,${\displaystyle \mathrm{SL}(n,F)}$ is aLie subgroup of${\displaystyle \mathrm{GL}(n,F)}$ of dimension${\displaystyle n^2-1}$. TheLie algebra${\displaystyle \mathfrak{s}\mathfrak{l}(n,F)}$ of${\displaystyle \mathrm{SL}(n,F)}$ consists of all${\displaystyle n\times n}$ matrices over${\displaystyle F}$ 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. 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.

It follows that the topology of the group${\displaystyle \mathrm{SL}(n,\mathbb{C})}$ is theproduct of the topology of${\displaystyle \mathrm{SU}(n)}$ 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${\displaystyle (n^2-1)}$-dimensionalEuclidean space.^[1] Since${\displaystyle \mathrm{SU}(n)}$ issimply connected,^[2] then${\displaystyle \mathrm{SL}(n,\mathbb{C})}$ is also simply connected, for all${\displaystyle n\ge 2}$.

The topology of${\displaystyle \mathrm{SL}(n,\mathbb{R})}$ 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 group${\displaystyle \mathrm{SL}(n,\mathbb{R})}$ has the samefundamental group as${\displaystyle \mathrm{SO}(n)}$; that is,${\displaystyle \mathbb{Z}}$ for${\displaystyle n=2}$ and${\displaystyle {\mathbb{Z}}_2}$ for${\displaystyle n>2}$.^[3] In particular this means that${\displaystyle \mathrm{SL}(n,\mathbb{R})}$, unlike${\displaystyle \mathrm{SL}(n,\mathbb{C})}$, is not simply connected, for${\displaystyle n>1}$.

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

The group generated by transvections is denoted${\displaystyle \mathrm{E}(n,A)}$ (forelementary matrices) or${\displaystyle \mathrm{TV}(n,A)}$. By the secondSteinberg relation, for${\displaystyle n\ge 3}$, transvections are commutators, so for${\displaystyle n\ge 3}$,.

For${\displaystyle n=2}$, transvections need not be commutators (of${\displaystyle 2\times 2}$ matrices), as seen for example when${\displaystyle A}$ is${\displaystyle {\mathbb{F}}_2}$, the field of two elements. In that case

where${\displaystyle A_3}$ and${\displaystyle S_3}$ respectively denote thealternating andsymmetric group on 3 letters.

However, if${\displaystyle A}$ is a field with more than 2 elements, thenE(2,A) = [GL(2,A), GL(2,A)], and if${\displaystyle A}$ 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 group${\displaystyle S{K}_1(A)=\mathrm{SL}(A)/\mathrm{E}(A)}$, where${\displaystyle \mathrm{SL}(A)}$ and${\displaystyle \mathrm{E}(A)}$ are thestable groups of the special linear group and elementary matrices.

If working over a ring where${\displaystyle \mathrm{SL}}$ is generated bytransvections (such as afield orEuclidean domain), one can give apresentation of${\displaystyle \mathrm{SL}}$ 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${\displaystyle \mathrm{GL}}$.

A sufficient set of relations for${\displaystyle \text{SL}(n,\mathbb{Z})}$ for${\displaystyle n\ge 3}$ is given by two of the Steinberg relations, plus a third relation (Conder, Robertson & Williams 1992, p. 19).Let${\displaystyle T_{ij}:=e_{ij}(1)}$ be the elementary matrix with${\displaystyle 1}$'s on the diagonal and in the${\displaystyle ij}$ position, and${\displaystyle 0}$'s elsewhere (and${\displaystyle i\ne j}$). Then

are a complete set of relations for${\displaystyle \text{SL}(n,\mathbb{Z}),n\ge 3}$.

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 group${\displaystyle \mathrm{GL}(n,F)}$ splits over its determinant (we use${\displaystyle F^{\times }=\mathrm{GL}(1,F)\to \mathrm{GL}(n,F)}$ as themonomorphism from${\displaystyle F^{\times }}$ to${\displaystyle \mathrm{GL}(n,F)}$, seesemidirect product), and therefore${\displaystyle \mathrm{GL}(n,F)}$ can be written as a semidirect product of${\displaystyle \mathrm{SL}(n,F)}$ by${\displaystyle F^{\times }}$:

${\displaystyle \mathrm{GL}(n,F)=\mathrm{SL}(n,F)⋊F^{\times }}$.

• 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

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• Linear algebraic groups

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