Let $V$ be a real algebraic variety and let ${\cal O}(V)$ denote its algebra of regular functions. If we put a group structure on $V$ (not necessarily an algebraic group structure) it will induce a Hopf algebra structure on ${\cal O}(V)$ in the usual manner. My question is, is there a bijective correspondence between the possible group structures on $V$ and the possible Hopf algebra structures on ${\cal O}(V)$?
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$\begingroup$ What definition of "real algebraic variety" are you using here? If it is an integral separated scheme of finite type over $\mathbb{R}$, then how are you defining a group structure? $\endgroup$– S. Carnahan ♦Oct 31, 2010 at 9:06
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$\begingroup$ Just the plain ordinary zero set of an ideal of polynomials definition of an algebraic variety. $\endgroup$– Abtan MassiniOct 31, 2010 at 18:23
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1 Answer
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If $V$ is an affine algebraic variety over any field $k$, then there is a bijection between the algebraic group structures on $V$ and the Hopf algebra structures on $O(V)$. The reason is that the category of affine varieties over $k$ is the just the contravariant category (the category with arrows reversed) of algebras over $k$. So a group law $m:V \times V \to V$ becomes a coproduct $m:O(V) \to O(V) \otimes O(V)$, and this correspondence is a bijection. (Because also, the tensor product of algebras corresponds to the Cartesian product of varieties.)
If you are interested in group structures on $V$ that are not necessarily algebraic group structures, then there isn't necessarily a good relation to $O(V)$ either.
answered Oct 30, 2010 at 20:50