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| CATEGORIES ABOUT ALGEBRAIC GROUP | |
| algebraic groups | |
| properties of groups | |
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Several important classes of groups are algebraic groups, including:
Two important classes of algebraic groups arise, that for the most part are studied separately: '' Abelian Varieties '' (the 'projective' theory) and '' Linear Algebraic Group s'' (the 'affine' theory). There are certainly examples that are neither one nor the other — these occur for example in the modern theory of Integrals Of The Second And Third Kinds such as the Weierstrass Zeta Function , or the theory of Generalized Jacobian s. But according to a basic theorem the general algebraic group is an extension of an Abelian Variety by a linear algebraic group. According to another basic theorem, any group in the category of of the 2×2 special linear group that are Lie groups, but have no faithful linear representation. A more obvious difference between the two concepts arises because the Identity Component of an affine algebraic group ''G'' is necessarily of finite index in ''G''. When one wants to work over a base ring ''R'' (commutative), there is the in the category of Scheme s over ''R''. ''Affine group scheme'' is the concept dual to a type of Hopf Algebra . There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties. |
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