Special linear group

In mathematics, the special linear group SL(n, F) 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 where F× is the multiplicative group of F (that is, F 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 F is a finite field of order q, the notation SL(n, q) is sometimes used. 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. When F is R or C, SL(n, F) is a Lie subgroup of GL(n, F) of dimension n2 − 1. The Lie algebra of SL(n, F) consists of all n × n matrices over F with vanishing trace. The Lie bracket is given by the commutator. 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.