# Group and Hopf Algebra Structures for Projective Varieties

Let $V$ be a projective (or affine) variety. Does there exist a bijective correspondence between group structures on $V$ and Hopf algebra structures on the coordinate ring of $V$?

• I'd say affine algebraic groups over $k$ are essentially the same as (fin gen commutative) Hopf $k$-algebras. May 5, 2010 at 22:01
• What do you mean by the coordinate ring of a projective variety? If you mean the global sections of the structure sheaf, this is typically just the constant functions. If you mean "the ring that it's Proj of" then aren't there normally more than one? So in the projective case I can't make sense of the question. May 5, 2010 at 22:13
• I suppose I mean when its coordinate ring when we consider it as a real affine variety - so scrap projective from the question and just consider affine. May 5, 2010 at 22:39
• As far as I can tell, the method of passing from a complex projective variety to an real affine variety (described at mathoverflow.net/questions/6332 ) is rather violent and very far from canonical. May 6, 2010 at 0:23

The starting point is exactly that the homogeneous coordinate ring is not a Hopf algebra, because a line bundle $L$ (defining the embedding) on $X$ does not pull back to $p_1^*L\otimes p_2^*L$ under the group law $X\times X\to X$. (This is what one would need to have induced comultiplication maps $H^0(X,L^n) \to H^0(X\times X, p_1^*L^n\otimes p_2^*L^n)\cong H^0(X, L^n)\otimes H^0(X, L^n)$.)
Instead Mumford finds that one can describe $X\times X\to X\times X$, $(x,y)\mapsto (x+y,x-y)$ explicitly in terms of the homogeneous coordinate ring.