Misplaced Pages

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
v t e

In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R 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

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

where R is the multiplicative group of R (that is, R excluding 0 when R is a field).

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

When R is the finite field of order q, the notation SL(n, q) is sometimes used.

Geometric interpretation

The special linear group SL(n, R) can be characterized as the group of volume and orientation preserving linear transformations of R; 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, F) is a Lie subgroup of GL(n, F) of dimension n − 1. The Lie algebra ${\displaystyle {\mathfrak {sl}}(n,F)}$ of SL(n, F) 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 − 1)-dimensional Euclidean space. Since SU(n) is simply connected, we conclude that SL(n, C) is also simply connected, for all n greater than or equal to 2.

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 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 SL(n, R) has the same fundamental group as SO(n), that is, Z for n = 2 and Z₂ for n > 2. In particular this means that SL(n, R), unlike SL(n, C), is not simply connected, for n greater than 1.

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 ≤ SL), but in general do not coincide with it.

The group generated by transvections is denoted E(n, A) (for elementary matrices) or TV(n, A). By the second Steinberg relation, for n ≥ 3, transvections are commutators, so for n ≥ 3, E(n, A) ≤ .

For n = 2, transvections need not be commutators (of 2 × 2 matrices), as seen for example when A is F₂, the field of two elements, then

${\displaystyle \operatorname {Alt} (3)\cong <\operatorname {E} (2,\mathbf {F} _{2})=\operatorname {SL} (2,\mathbf {F} _{2})=\operatorname {GL} (2,\mathbf {F} _{2})\cong \operatorname {Sym} (3),}$

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

However, if A is a field with more than 2 elements, then E(2, A) = , and if A is a field with more than 3 elements, E(2, A) = .

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 SK₁(A) := SL(A)/E(A), where SL(A) and 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 field or Euclidean domain), 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 SL(n, Z) for n ≥ 3 is given by two of the Steinberg relations, plus a third relation (Conder, Robertson & Williams 1992, p. 19). 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 i ≠ j). Then

${\displaystyle {\begin{aligned}\left&=T_{ik}&&{\text{for }}i\neq k\\\left&=\mathbf {1} &&{\text{for }}i\neq \ell ,j\neq k\\\left(T_{12}T_{21}^{-1}T_{12}\right)^{4}&=\mathbf {1} \end{aligned}}}$

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

SL(n,F)

In characteristic 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 a short exact sequence of groups:

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

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 in SL(n,F) but not in SL(n,F) and thus the group splits as an internal direct product ${\displaystyle \operatorname {SL} ^{\pm }(2k+1,F)\cong \operatorname {SL} (2k+1,F)\times \{\pm I\}}$. However, if ${\displaystyle n=2k}$ is even, ${\displaystyle -I}$ is already in SL(n,F) , SL does not split, and in general is a non-trivial group extension.

Over the real numbers, SL(n, R) has two connected components, corresponding to SL(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.

Structure of GL(n,F)

The group GL(n, F) splits over its determinant (we use F ≅ GL(1, F) → GL(n, F) as the monomorphism from F to 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

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

References

1. Hall 2015 Section 2.5
2. Hall 2015 Proposition 13.11
3. Hall 2015 Sections 13.2 and 13.3

This article needs additional citations for verification. Please help improve this article by adding 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