General linear group over a field as an algebraic group
- The algebraic variety structures arises as follows: it is a Zariski-open subset of , defined by the non-vanishing of the determinant polynomial.
- The group structure is by matrix multiplication. Both the matrix multiplication and the inverse map are given globally by rational functions in the coordinates, and hence are morphisms of algebraic varieties.
General linear group of degree one as an algebraic group
For , the general linear group is the subset of the field . Explicitly:
|Definition as Zariski-open subset of|
|Explicit formula for group multiplication||Group multiplication is field multiplication.|
|Explicit formula for inverse map in group||Group inverse is field inverse.|
General linear group of degree two as an algebraic group
Here, we write a matrix as a bunch of four coordinates . Then:
|Definition as Zariski-open subset of||Nonvanishing determinant condition|
|Explicit formula for group multiplication||Group multiplication is matrix multiplication.|
|Explicit formula for inverse map in group||Group inverse is matrix inverse. Note that denominators do not vanish, and this is precisely because the determinant is nonvanishing on this open subset.|