On coverings and hyperalgebras of affine algebraic groups

Takeuchi, Mitsuhiro

doi:10.1090/S0002-9947-1975-0429928-3

On coverings and hyperalgebras of affine algebraic groups
HTML articles powered by AMS MathViewer

by Mitsuhiro Takeuchi

Trans. Amer. Math. Soc. 211 (1975), 249-275

DOI: https://doi.org/10.1090/S0002-9947-1975-0429928-3

PDF | Request permission

Abstract:

Over an algebraically closed field of characteristic zero, the universal group covering of a connected affine algebraic group, if such exists, can be constructed canonically from its Lie algebra only. In particular the isomorphism classes of simply connected affine algebraic groups are in 1-1 correspondence with the isomorphism classes of finite dimensional Lie algebras of some sort. We shall consider the counterpart of these results (due to Hochschild) in case of a positive characteristic, replacing the Lie algebra by the “hyperalgebra". We show that the universal group covering of a connected affine algebraic group scheme can be constructed canonically from its hyperalgebra only and hence, in particular, that the category of simply connected affine algebraic group schemes is equivalent to a subcategory of the category of hyperalgebras of finite type which contains all the semisimple hyperalgebras.

References

Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
Mitsuhiro Takeuchi, Tangent coalgebras and hyperalgebras. I, Jpn. J. Math. 42 (1974), 1–143. MR 389896, DOI 10.4099/jjm1924.42.0_{1}

Tangent coalgebras and hyperalgebras

Jean Dieudonné, Lie groups and Lie hyperalgebras over a field of characteristic $p>0$. VI, Amer. J. Math. 79 (1957), 331–388. MR 94413, DOI 10.2307/2372686
G. Hochschild, Algebraic groups and Hopf algebras, Illinois J. Math. 14 (1970), 52–65. MR 257088, DOI 10.1215/ijm/1256053299

The theory of Lie algebras and Chevalley groups

Masayoshi Miyanishi, Une caractérisation d’un groupe algébrique simplement connexe, Illinois J. Math. 16 (1972), 639–650 (French). MR 309949
I. Satake, Classification theory of semi-simple algebraic groups, Lecture Notes in Pure and Applied Mathematics, vol. 3, Marcel Dekker, Inc., New York, 1971. With an appendix by M. Sugiura; Notes prepared by Doris Schattschneider. MR 0316588
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
Mitsuhiro Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251–270. MR 321963, DOI 10.1007/BF01579722