The equations defining algebraic groups

The problem of finding presentations by generators and relations of the coordinate algebras of abelian varieties that are canonically determined by their group structures, has been explored and solved in 1966 by D. Mumford. Since every connected algebraic group is an extension of a connected affine algebraic groups by means of an abelian variety, it is natural to consider the analogous problem for affine algebraic groups. The talk is intended to describe its solution. The latter is based on solving two problems posed in 1992 by D. Flath and J. Towber. From the viewpoint of this theory, the naive presentation of $\mathrm{SL}(n)$ as a hypersurface $\det=1$ in the $n^2$-dimensional affine space is adequate for $n < 3$ only: the canonical presentation of the coordinate algebra of $\mathrm{SL}(n)$ by generators and relations embodies $\mathrm{SL}(3)$ as the intersection of 2 homogeneous and 2 inhomogeneous quadrics in the 12-dimensional affine space, $\mathrm{SL}(4)$ as the intersection of 20 homogeneous and 3 inhomogeneous quadrics in the 28-dimensional affine space, etc.