Special linear group

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

where R^× is the multiplicative group of R (that is, R excluding 0).

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[edit]

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.

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Lie subgroup[edit]

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

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Topology[edit]

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 (n² − 1)-dimensional Euclidean space.^[1] Since SU(n) is simply connected,^[2] 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.^[3] In particular this means that SL(n, R), unlike SL(n, C), is not simply connected, for n greater than 1.

SL(2, R)

SL(2, C)

(PSL(2, Z))

Modular group

Projective linear group

Conformal map

Representations of classical Lie groups

; 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

Conder, Marston

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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