Right distributive quasigroups on algebraic varieties
TL;DR: A structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes has been developed in this article, where the algebraic Fischer groups are generated such that the mapping is an automorphism.
Abstract: In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes
$$\mathfrak{D}$$
generating algebraic Fischer groups (in the sense of [6]) such that the mappingx ↦x
−1
ax, fora e
$$\mathfrak{D}$$
, is an automorphism of
$$\mathfrak{D}$$
(as variety) We also give examples of algebraic Fischer groups where this does not happen It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups
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TL;DR: In this paper, it was shown that the elliptic curve withj-invariant 0 is the only abelian variety with fixed point free automorphism, and this result was extended to abelians of higher dimensions and some connected commutative algebraic groups.
Abstract: An automorphismf of an abelian varietyX is called fixed point free if it admits no fixed points other than the origin and this is of multiplicity one. It is well known that the elliptic curve withj-invariant 0 is the only elliptic curve admitting a fixed point free automorphism. In this note, this result is extended to abelian varieties of higher dimensions and some connected commutative algebraic groups.
17 citations
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TL;DR: In this article, the authors determine all algebraic transformation groups G, defined over an algebraically closed field k, which operate transitively, but not primitively, on a variety W, sub-ject to the following conditions.
Abstract: In this paper we determine all algebraic transformation groups G, defined over an algebraically closed field k, which operate transitively, but not primitively, on a variety W, sub- ject to the following conditions. We require that the (non-eective) action of G on the variety of blocks is sharply 2-transitive, as well as the action on a block D of the normalizer GD. Also we require sharp transitivity on pairs ðX ; Y Þ of independent points of W, i.e. points contained in dierent blocks.
3 citations
TL;DR: In this article, a generalization of the geometric quasigroup construction on the complex plane has been proposed, which can be expressed by a semidirect product G of the translation group (which is sharply transitive on the points of the plane and hence may be identified with the plane) by a finite cyclic group of rotations.
Abstract: In terms of regular n-gons a left distributive quasigroup operation is defined on the complex plane. This operation can be expressed by means of a semidirect product G of the translation group (which is sharply transitive on the points of the plane and hence may be identified with the plane) by a finite cyclic group of rotations of order n. That observation makes possible a wide generalization of this geometric quasigroup construction. The connection in general between algebraic properties of the quasigroup and various properties of the group G is discussed, in particular it is studied what the consequences for the quasigroup Q are if G is interpreted as a topological group or an algebraic group.
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Book•
01 Jan 1991
TL;DR: Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions are discussed in this paper.
Abstract: Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
2,919 citations
Book•
01 Jan 1975
TL;DR: A survey of rationality properties of semisimple groups can be found in this paper, where a survey of rational properties of algebraic groups is also presented, as well as a classification of reductive groups representations.
Abstract: Algebraic geometry affine algebraic groups lie algebras homogeneous spaces chracteristic 0 theory semisimple and unipoten elements solvable groups Borel subgroups centralizers of Tori structure of reductive groups representations and classification of semisimple groups survey of rationality properties.
2,070 citations
Book•
01 Jan 1981
TL;DR: A linear algebraic group over an algebraically closed field k is a subgroup of a group GL n (k) of invertible n × n-matrices with entries in k, whose elements are precisely the solutions of a set of polynomial equations in the matrix coordinates as mentioned in this paper.
Abstract: A linear algebraic group over an algebraically closed field k is a subgroup of a group GL n (k) of invertible n × n-matrices with entries in k, whose elements are precisely the solutions of a set of polynomial equations in the matrix coordinates. The present article contains a review of the theory of linear algebraic groups.
1,202 citations
TL;DR: The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic group of matrices, or actually of neither of these types as discussed by the authors.
Abstract: The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.
528 citations
"Right distributive quasigroups on a..." refers background in this paper
...It is well known that G o is the product of its maximal connected affine subgroup L by its smallest subgroup A (7~ 1) giving rise to a linear factor group ([ 18 ])....
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