1
$\begingroup$

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

$\endgroup$
1

1 Answer 1

2
$\begingroup$

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) &\in (f) \otimes K[G] + K[G] \otimes (f); \\\\ \epsilon(f) &= 0; \\\\ S(f) &\in (f), \end{align*} where $\Delta$, $\epsilon$ and $S$ are the comultiplication, counit and antipode of the Hopf algebra $K[G]$, respectively.

(See, for instance, Milne's freely available course notes on linear algebraic groups.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.