Coordinate algebras of algebraic groups: generators and relations

For coordinate algebras of Abelian varieties, the problem of finding a presentation by generators and relations canonically determined by the group structure was considered and solved in the classical works of D. Mumford and G. Kempf. Since every connected algebraic group is an extension of a connected linear algebraic group by an Abelian variety, a similar problem naturally arises for connected affine algebraic groups. In geometric terms, it means finding, for each connected linear algebraic group $G$, an embedding of its group variety into an affine space that is canonically determined by the group structure of $G$. Although some $G$ (e.g., $GL(n)$, $SL(n)$, $SO(n)$, $Sp(n)$, $Spin(n)$) are defined through embeddings into affine spaces, these embeddings, not being defined exclusively in terms of the group structure, in the indicated respect are accidental. This talk aims to describe a solution to the specified problem. It is based on the solution to two problems posed by D. E. Flath and J. Tauber in 1992. From the point of view of this theory, the usual naive presentation of $SL(n)$ as a hypersurface $\det=1$ in $n^2$-dimensional affine space is canonical only for $n=2$: the canonical one defines $SL(3)$ as the intersection of 2 homogeneous and 2 non-homogeneous quadrics in a 12-dimensional affine space, $SL(4)$ as the intersection of 20 homogeneous and 3 non-homogeneous quadrics in a 28-dimensional affine space, etc.