Download Algebraic Monoids, Group Embeddings, and Algebraic by Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang PDF

By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang

This e-book features a number of fifteen articles and is devoted to the 60th birthdays of Lex Renner and Mohan Putcha, the pioneers of the sector of algebraic monoids.

Topics offered include:

structure and illustration conception of reductive algebraic monoids

monoid schemes and purposes of monoids

monoids concerning Lie theory

equivariant embeddings of algebraic groups

constructions and houses of monoids from algebraic combinatorics

endomorphism monoids brought about from vector bundles

Hodge–Newton decompositions of reductive monoids

A part of those articles are designed to function a self-contained advent to those themes, whereas the remainder contributions are study articles containing formerly unpublished effects, that are certain to develop into very influential for destiny paintings. between those, for instance, the real fresh paintings of Michel Brion and Lex Renner displaying that the algebraic semi teams are strongly π-regular.

Graduate scholars in addition to researchers operating within the fields of algebraic (semi)group concept, algebraic combinatorics and the speculation of algebraic crew embeddings will reap the benefits of this particular and vast compilation of a few basic ends up in (semi)group concept, algebraic staff embeddings and algebraic combinatorics merged less than the umbrella of algebraic monoids.

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Additional resources for Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

Sample text

X; to / equals '. (i) There exists a unique morphism « W Y T ! x/; t / on X T . x/ on X T ). Proof. x; t / 7! x; t / 7! OX / D OY in view of Lemma 1. y; to /, where y 2 Y , and the assumption (iii) of that lemma holds with s being the inclusion of Y To in X To . Finally, the assumption (iv)0 of Remark 13 is satisfied, since T is connected. By that remark, we thus have g D g ı s ı f on X T . Hence there 46 M. Brion exists a unique morphism « W Y T ! y; t /. x/ on X T . t u Remark 14. The preceding result has a nice interpretation when X is projective.

In view of Theorem 5, every nontrivial algebraic semigroup law on an irreducible curve S is commutative; by Proposition 17 again, it follows that S has an idempotent F -point whenever S and are defined over F . 4 Rigidity In this subsection, we obtain two rigidity results (both possibly known, but for which we could not locate adequate references) and we apply them to the study of endomorphisms of complete varieties. Our first result is a scheme-theoretic version of a classical rigidity lemma for irreducible varieties (see [8, Lem.

D 0 ; a contradiction. Thus, we must have n D 2, and we obtain a nonconstant morphism D 2 W S ! A1 , where the semigroup law on A1 is the multiplication. The image of contains 0 and a nonempty open subset U of the unit group Gm . Then U U D Gm and hence is surjective. e/ D 1. Then e is the desired nonzero idempotent. t u Remark 10. One may also deduce the above theorem from the description of algebraic semigroup structures on abelian varieties (Proposition 21), when the irreducible curve S is assumed to be nonsingular and nonrational.

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