nLab
general linear group

Definition

Given a field $k$, the general linear group $\mathrm{GL}(n,k)$ (or ${\mathrm{GL}}_n(k)$) is the group of invertible linear transformations of the vector space $k^n$. It can be canonically identified with the group of $n\times n$ matrices with entries in $k$ having nonzero determinant.

This group can be considered as a (quasi-affine) subvariety of the affine space $M_{n\times n}(k)$ of square matrices of size $n$ defined by the condition that the determinant of a matrix is nonzero. It can be also presented as an affine subvariety of the affine space $M_{n\times n}(k)\times k$ defined by the equation $\det (M)t=1$ (where $M$ varies over the factor $M_{n\times n}(k)$ and $t$ over the factor $k$).

This variety is an algebraic $k$-group, and if $k$ is the field of real or complex numbers it is a Lie group over $k$.

One can in fact consider the set of invertible matrices over an arbitrary unital ring, not necessarily commutative. Thus ${\mathrm{GL}}_n:R\mapsto {\mathrm{GL}}_n(R)$ becomes a presheaf of groups on $\mathrm{Aff}={\mathrm{Ring}}^{\mathrm{op}}$ where one can take rings either in commutative or in noncommutative sense. In the commutative case, this functor defines a group scheme; it is in fact an affine group scheme represented by the commutative ring $R=\mathbb{Z}[x_{11},\dots ,x_{nn},t]/(\det (X)t-1)$.

Coordinate rings of general linear groups and of special general linear groups have quantum deformations called quantum linear group?s.

Stable and unstable versions:

The above is sometimes referred to as the unstable general linear group, whilst the result if one lets $n$ go to infinity is called the stable general linear group of $R$, and is then written $\mathrm{GL}(R)$ with no suffix.

Related concepts

• Gauss decomposition,

• special linear group, orthogonal group, unitary group,

• Borel subgroup, flag variety.

• group of units

References

• O.T. O’Meara, Lectures on Linear Groups, Amer. Math. Soc., Providence, RI, 1974.

• B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.