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algebraic group

Given a field $k$, an algebraic $k$-group is a group object in the category of $k$-varieties. An algebraic $k$-group is linear if it is a Zariski-closed subgroup of the general linear group $\mathrm{GL}(n,k)$ for some $n$. Another important class are commutative algebraic $k$-groups whose underlying variety is projective, namely the abelian varieties; in dimension $1$ these are precisely the elliptic curves. If $k$ is a perfect field and $G$ an algebraic $k$-group, the theorem of Chevalley says that there is a unique linear subgroup $H\subset G$ such that $G/H$ is an abelian variety.

The group objects in the category of algebraic schemes and formal schemes are called (algebraic) group schemes and formal groups, respectively. One should mention among group schemes ‘the infinite-dimensional algebraic groups’ of Shafarevich. An algebraic group scheme is affine if the underlying scheme is affine. Algebraic analogues of loop group?s are in the category of ind-schemes. All linear algebraic $k$-groups are affine. The category of affine group schemes is the opposite of the category of commutative Hopf algebras.

References

The standard references are

• M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970

• M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, Schemas en groupes, i.e. SGA III-1, III-2, III-3

• A. Borel, Linear algebraic groups, Springer (2nd edition much expanded)

• W. Waterhouse, Introduction to affine group schemes, GTM 66, Springer 1979.

• S. Lang, Abelian varieties, Springer 1983.

• D. Mumford, Abelian varieties, 1970, 1985.

• J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

• T. Springer, Linear algebraic groups, Progress in Mathematics 9, Birkhäuser Boston (2nd ed. 1998, reprinted 2008)