The Unapologetic Mathematician

Mathematics for the interested outsider

The Special Linear Group (and others)

We’ve got down the notion of the general linear group \mathrm{GL}(V) of a vector space V, including the particular case of the matrix group \mathrm{GL}(n,\mathbb{F}) of the space \mathbb{F}^n. We also have defined the orthogonal group \mathbb{O}(n,\mathbb{F}) of n\times n matrices over \mathbb{F} whose transpose and inverse are the same, which is related to the orthogonal group \mathrm{O}(V,B) of orthogonal transformations of the real vector space V preserving a specified bilinear form B. Lastly, we’ve defined the group \mathrm{U}(n) of unitary transformations on \mathbb{C}^nn\times n complex matrices whose conjugate transpose and inverse are the same.

For all of these matrix groups — which are all subgroups of some appropriate \mathrm{GL}(n,\mathbb{F}) — we have a homomorphism to the multiplicative group of \mathbb{F} given by the determinant. We originally defined the determinant on \mathrm{GL}(n\mathbb{F}) itself, but we can easily restrict it to any subgroup. We actually know that for unitary and orthogonal transformations the image of this homomorphism must lie in a particular subgroup of \mathbb{F}^\times. But in any case, the homomorphism must have a kernel, and this kernel turns out to be important.

In the case of the general linear group \mathrm{GL}(V), the kernel of the determinant homomorphism consists of the automorphisms of V with determinant {1}. We call this subgroup of \mathrm{GL}(V) the “special linear group” \mathrm{SL}(V), and transformations in this subgroup are sometimes called “special linear transformations”. Of course, we also have the particular special linear group \mathrm{SL}(n,\mathbb{F})\subseteq\mathrm{GL}(n,\mathbb{F}). When we take the kernel of any of the other groups, we prepend the adjective “special” and an \mathrm{S} to the notation. Thus we have the special orthogonal groups \mathrm{SO}(V,B) and \mathrm{SO}(n,\mathbb{F}) and the special unitary group \mathrm{SU}(n).

In a sense, all the interesting part of the general linear group is contained in the special linear subgroup. Outside of that, what remains is “just” a scaling. It’s a little more complicated than it seems on the surface, but not much.

September 8, 2009 Posted by | Algebra, Linear Algebra | 6 Comments