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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

- $detcolon\; operatorname\{GL\}(n,\; F)\; to\; F^times.$

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

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

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^2-1 dimensional Euclidean space.

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. 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.

The group SL(n, C), like SU(n), is simply connected while SL(n, R), like SO(n), is not. SL(n, R) has the same fundamental group as GL^{+}(n, R) or SO(n), that is, Z for n = 2 and Z_{2} for n > 2.

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 $[operatorname\{GL\},operatorname\{GL\}]leqoperatorname\{SL\}$), 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 $ngeq\; 3$, transvections are commutators, so for $ngeq3$, $operatorname\{E\}\_n(A)\; leq\; [operatorname\{GL\}\_n(A),operatorname\{GL\}\_n(A)]$. 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 $operatorname\{Alt\}(3)\; cong\; [operatorname\{GL\}\_2(mathbb\{Z\}/2mathbb\{Z\}),operatorname\{GL\}\_2(mathbb\{Z\}/2mathbb\{Z\})]\; <\; operatorname\{E\}\_2(mathbb\{Z\}/2mathbb\{Z\})\; =\; operatorname\{SL\}\_2(mathbb\{Z\}/2mathbb\{Z\})\; =\; operatorname\{GL\}\_2(mathbb\{Z\}/2mathbb\{Z\})\; cong\; operatorname\{Sym\}(3).$

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.

A sufficient set of relations for $operatorname\{SL\}(n,mathbf\{Z\})$ for $ngeq\; 3$ is given by two of the Steinberg relations, plus a third relation . 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 $ineq\; j$). Then

- $begin\{align\}$

- GL(n, F) = SL(n, F) ⋊ F
^{×}.

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Last updated on Thursday October 02, 2008 at 04:40:05 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday October 02, 2008 at 04:40:05 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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