Affine Group Schemes Examples

Another group naturally occurring is the set of all invertible matrices commuting with a given matrix, say with But as it stands this is 

The crucial additional property of our functors is that the elements in G(R) are given by finding the solutions in R of some family of polynomial equations (with coefficients in k). In most of the examples this is obvious the elements in SL2(A), for instance, are given by quadruples a, b, c, d in R satisfying the equation ad —be = 1. Invertibility can be expressed in this manner because an element uniquely determines its inverse if it has one. That is, the elements x in G (A) correspond precisely to the solutions in R of the equation xy = 1. 

Affine group schemes are exactly the group functors constructed by solution of equations. But such a definition would be technically awkward, since quite different collections of equations can have essentially the same solutions. For this reason the official definition is postponed to the next section, where we translate the condition into something less familiar but more manageable. 

Suppose we have some family of polynomial equations over k. We can then form a most general possible solution of the equations as follows. Take a polynomial ring over k, with one indeterminate for each variable in the equations. Divide by the ideal generated by the relations which the equations express. Call the quotient algebra A. From the equation for SL2, for instance, we get A — X12, X2u X22 I[X X22 12 21 ) The 

Every fc-algebra A arises in this way from some family of equations. To see this, take any set of generators x for A, and map the polynomial ring k[ X ] onto A by sending Xa to xa. Choose polynomials /, generating the kernel. (If we have finitely many generators and k noetherian, only finitely many/, are needed (A.5).) Clearly then x is the most general possible solution of the equations f = 0. In summary  

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A homomorphism of affine group schemes is a natural map G -+ H for which each G(R) - H(R) is a homomorphism. We have already seen the example det GL - Gm. The Yoneda lemma shows as expected that such maps correspond to Hopf algebra homomorphisms. But since any map between groups preserving multiplication also preserves units and inverses, we need to check only that A is preserved. An algebra homomorphism between Hopf algebras which preserves A must automatically preserve S and e. 

The ease of formation of the carbene depends on the nucleophilicity of the anion associated with the imidazolium. For example, when Pd(OAc)2 is heated in the presence of [BMIM][Br], the formation of a mixture of Pd imidazolylidene complexes occurs. Palladium complexes have been shown to be active and stable catalysts for Heck and other C-C coupling reactions [34]. The highest activity and stability of palladium is observed in the ionic liquid [BMIM][Brj. Carbene complexes can be formed not only by deprotonation of the imidazolium cation but also by direct oxidative addition to metal(O) (Scheme 5.3-3). These heterocyclic carbene ligands can be functionalized with polar groups in order to increase their affinity for ionic liquids. While their donor properties can be compared to those of donor phosphines, they have the advantage over phosphines of being stable toward oxidation. 

The common characteristics of the above mentioned heterocycles are electron withdrawing and a site of unsaturation that can stabilize the negative charge developed by the displacement reaction through resonance. For example, the thiazole activated halo displacement is similar to that of a conventional activating group as shown in Scheme 1. The activation is derived from the electron affinity and the stabilization of the negative... 

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