Special linear group

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Groups

Basic notions
Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product direct sum

Discrete groups
Classification of finite simple groups Cyclic group Z_n Alternating group A_n Sporadic groups Mathieu group M_11..12,M_22..24 Conway group Co_1..3 Janko group J_1..4 Fischer group F_22..24 Baby Monster group B Monster group M

Continuous groups
Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) E₆ E₇ E₈ F₄ G₂ SL₂ Lorentz group Poincaré group

Infinite dimensional groups
conformal group diffeomorphism group Quantum group O(∞) SU(∞) Sp(∞)

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In mathematics, the special linear group of degree n over a field F 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

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

where we write F^× for the multiplicative group of F (that is, excluding 0).

1 Geometric interpretation
2 Lie subgroup
3 Topology
4 Relations to other subgroups of GL(n,A)
5 Generators and relations
6 Structure of GL(n,F)
7 References

[edit] Geometric interpretation

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

[edit] Lie subgroup

When F is R or C, SL(n) is a Lie subgroup of GL(n) of dimension n² − 1. The Lie algebra of SL(n) consists of all n×n matrices over F with vanishing trace. The Lie bracket is given by the commutator.

[edit] Topology

The group SL(n, C) is simply connected while SL(n, R) is not. SL(n, R) has the same fundamental group as GL⁺(n, R), that is, Z for n = 2 and Z₂ for n > 2.

[edit] Relations to other subgroups of GL(n,A)

See also: Whitehead's lemma

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 $[\operatorname{GL},\operatorname{GL}]\leq\operatorname{SL}$ ), but in general do not coincide with it.

The group generated by transvections is denoted $\operatorname{E}_n(A)$ (for elementary matrices) or $\operatorname{TV}_n(A)$ . By the second Steinberg relation, for $n\geq 3$ , transvections are commutators, so for $n\geq3$ , $\operatorname{E}_n(A) \leq [\operatorname{GL}_n(A),\operatorname{GL}_n(A)]$ . For $n = 2$ , transvections need not be commutators (of 2×2 matrices), as seen for example when $A$ is the field of two elements, then $\operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] < \operatorname{E}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}) \cong \operatorname{Sym}(3)$ .

In some circumstances these coincide: the special linear group over a field or the integers 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 $SK_1(A) := \operatorname{SL}(A)/\operatorname{E}(A)$ , where $\operatorname{SL}(A)$ and $\operatorname{E}(A)$ are the stable groups of the special linear group and elementary matrices.

[edit] Generators and relations

If working over a ring where SL is generated by transvections (such as a ring or the integers), 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 $\operatorname{SL}(n,\mathbf{Z})$ for $n\geq 3$ is given by two of the Steinberg relations, plus a third relation (Conder, Robertson & Williams 1992, p. 19). Let $T i j : = e i j (1)$ be the elementary matrix with 1's on the diagonal and in the $i j$ position, and 0's elsewhere (and $i\neq j$ ). Then

$\begin{align} \left[ T_{ij},T_{jk} \right] &= T_{ik} && \mbox{for } i \neq k\\ \left[ T_{ij},T_{kl} \right] &= \mathbf{1} && \mbox{for } i \neq l, j \neq k\\ (T_{12}T_{21}^{-1}T_{12})^4 &= \mathbf{1}\\ \end{align}$

are a complete set of relations for $\operatorname{SL}(n,\mathbf{Z})$ , $n\geq 3$ .

[edit] Structure of GL(n,F)

The group $\det\colon \operatorname{GL}(n, F)$ splits over its determinant (we use $F^\times \stackrel{\sim}{\to} \operatorname{GL}(1,F) \hookrightarrow \operatorname{GL}(n,F)$ as the group homomorphism needed from $F^\times$ to $\operatorname{GL}(n,F)$ , 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^×

[edit] References

This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (January 2008)

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): 19–26, MR 1079696, ISSN 0002-9939 , available at JSTOR.

Categories: Abstract algebra | Linear algebra | Lie groups

Hidden category: Articles needing additional references from January 2008

Special linear group

From Wikipedia, the free encyclopedia

Contents

[edit] Geometric interpretation

[edit] Lie subgroup

[edit] Topology

[edit] Relations to other subgroups of GL(n,A)

[edit] Generators and relations

[edit] Structure of GL(n,F)

[edit] References

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