Special linear group
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
where we write F× for the multiplicative group of F (that is, F 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).
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.
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