On the equations defining affine algebraic groups

For the coordinate algebras of abelian varieties, the problem of finding a presentation by generators and relations canonically determined by the group structure has been explored and solved by D. Mumford. Since every connected algebraic group is an extension of a connected linear algebraic group by an abelian variety, the analogous problem for connected affine algebraic group naturally arises. The talk is intended to describing its solution based on solving two problems posed by D. E. Flath and J. Towber in 1992. From the standpoint of this theory, the usual naive presentation of $SL(n)$ as a hypersurface $\det=1$ in an $n^2$-dimensional affine space is adequate only for $n=2$: the canonical presentation defines $SL(3)$ as the intersection of 2 homogeneous and 2 inhomogeneous quadrics in a 12-dimensional affine space, $SL(4)$ as the intersection of 20 homogeneous and 3 inhomogeneous quadrics in a 28-dimensional affine space, etc.