Affinity groups

HarM64 Harrison, M. A. On the classification of Boolean functions by the general linear and affine groups. SIAM J. 12 (1964) 285-299. 

An interesting imaging probe Id that can selectively target bacteria was recently reported by Smith et al. [31] also based on a heptamethine chromophore. The probe is composed of a bacterial affinity group, which is a synthetic zinc (II) coordination complex that targets the anionic surfaces of bacterial cells and a near infrared dye. The probe allowed detection of Staphylococcus aureus in a mouse leg infection model using whole animal near-infrared fluorescence imaging. 

Figure 7.22 Fluorescent dyes such as an amine-reactive Cy5 derivative can be coupled to amine-dendrimers at relatively high substitution levels to create intensely fluorescent detection agents. If the dendrimer also is deriva-tized to contain an affinity group or a targeting group then specific fluorescent detection at high sensitivity can be realized.

More detailed discussions of avidin-biotin systems as well as the process of adding a biotin affinity group to proteins, nucleic acids, and other biomolecules can be found in Chapters 11 and 23. 

I swallowed hard. He didn t look like the sort of person who would appreciate my stories of fighting the police at the Berkeley barricades shoulder-to-shoulder with affinity groups like the Persian Fuckers and the Acid Anarchists. Nor did my participation in the Human Be-In or the rolling orgies of the Summer of Love in... 

Billy M. Williams For my other point, you mentioned your affinity groups and employee networks. Within Dow, we have had in place the past several years companywide affinity networks. We now have a women s innovation network, an Asian development network, an African American network, a Hispanic network, and a gay and lesbian network across the company, which we have found to be effective in providing the type of environment that you talked about and that has been discussed in other venues here. 

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. 

Such F are called representable, and one says that A represents F. We can now officially define an affine group scheme over k as a representable functor from k-algebras to groups. 

Theorem. Affine group schemes over k correspond to Hopf algebras over k. 

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. 

Theorem. Characters of an affine group scheme G represented by A correspond to group-like elements in A. 

If G and H are any abelian group functors over k, we can always get another group functor Hom(G, H) by attaching to JR the group Hom(GR, Hr). This is the functorial version of Horn, and for H = Gm it is a functorial character group for finite G it is GD. In general it will not be an affine group scheme even when G and H are Cartier duality is one case where it is representable. 

Let F, G, and H be commutative affine group schemes over k. Show that homomorphisms F - Hom(G, H) correspond to natural biadditive maps F x G- H. 

Theorem. Let G be an affine group scheme represented by A. Then linear representations of G on V correspond to k-linear maps p V - V A such that... 

We call an affine group scheme G algebraic if its representing algebra is finitely generated. 

Corollary. Every affine group scheme G over a field is an inverse limit of algebraic affine group schemes. 

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