Let G be an affine connected algebraic group. When a subvariety of G with codimension one is a subgroup. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:31:54Z http://mathoverflow.net/feeds/question/92940 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92940/let-g-be-an-affine-connected-algebraic-group-when-a-subvariety-of-g-with-codi Let G be an affine connected algebraic group. When a subvariety of G with codimension one is a subgroup. gauss 2012-04-02T23:36:05Z 2012-04-03T09:50:14Z <p>Let G be an affine connected algebraic group, and K[G] be its coordinate ring. Let Y be a subvariety of G defined as a zero set for some f in K[G]. For which f, Y is a closed subgroup of G</p> http://mathoverflow.net/questions/92940/let-g-be-an-affine-connected-algebraic-group-when-a-subvariety-of-g-with-codi/92981#92981 Answer by Tom De Medts for Let G be an affine connected algebraic group. When a subvariety of G with codimension one is a subgroup. Tom De Medts 2012-04-03T09:46:55Z 2012-04-03T09:46:55Z <p>It is perhaps easiest to express this in terms of Hopf algebras. The coordinate ring $K[G]$ has the structure of a Hopf algebra; the subvariety $Y$ is a closed subgroup of $G$ if and only if the ideal $(f)$ is a Hopf ideal, i.e. \begin{align*} \Delta(f) &amp;\in (f) \otimes K[G] + K[G] \otimes (f); \\ \epsilon(f) &amp;= 0; \\ S(f) &amp;\in (f), \end{align*} where $\Delta$, $\epsilon$ and $S$ are the comultiplication, counit and antipode of the Hopf algebra $K[G]$, respectively.</p> <p>(See, for instance, Milne's freely available <a href="http://www.jmilne.org/math/CourseNotes/ala.html" rel="nofollow">course notes</a> on linear algebraic groups.)</p>