By Ibrahim Assem

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It is trivial to check in general that when X is dense in Y, then X is irreducible itT Y is. 2 Connected Components of Algebraic Matrix Groups Theorem. Let S be an algebraic matrix group. Let SO be the connected component containing the unit e. Then SO is a normal subgroup of finite index; it is irreducible, and the other irreducible components are its cosets. Let S = X t U ... U Xm be the decomposition into irreducible components. We know X 1 is not contained in anyone other Xi and hence by irreducibility is not contained in their union.

Our set S has" enough" points in the sense that a nonzero element of the ring cannot vanish on them all, but for more delicate questions we can see it might be better to expand our space and include all the maximal ideals. If the base ring k is not a field, then even maximal ideals tum out to be not quite all we want. The kernel of a homomorphism iE[X] -+ iE, for instance, is not maximal. The next natural generalization is to prime ideals, and these do indeed give a satisfactory theory. 4 Spec A The spectrum Spec A of a ring A is the collection of its prime ideals.

U®V®A is a comodule structure corresponding to the action g. (u ® v) = g. u ® g. v. A submodule Wof V is a subcomodule if p(W) ~ W ® A, which is equivalent (Ex. 3) to saying that G(R) always maps W ® R to itself. g. ) If W is a subcomodule, then V ~ V ® A ~ (V/W) ® A factors through V/Wand makes V/Wa quotient comodule; it of course corresponds to the representation induced on the quotient space. 3 Finiteness Theorems The last theorem shows that all linear representations are given by formulas. Over fields this now implies that both they and the Hopf algebras have important finiteness properties.