Coordinate algebras of algebraic groups: generators and relations

The problem of finding a “canonical” presentation by generators and relations of coordinate algebras of abelian varieties has been explored in the classical papers by Mumford and Kempf. Since every connected algebraic group is an extension of a connected linear algebraic group by an abelian variety, the analogous problem for connected linear algebraic group naturally arises. I shall describe its solution. From the standpoint of this theory, the usual naive presentation of $SL(n)$ as a hypersurface $\det=1$ in a $n^2$-dimensional affine space is adequate only for $n=2$: the “canonical” presentation defines $SL(3)$ as the intersection of 2 homogeneous and 2 nonhomogeneous quadrics in a 12-dimensional affine space, $SL(4)$ as an intersection of 20 homogeneous and 3 nonhomogeneous quadrics in a 28-dimensional space etc.