By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This booklet incorporates a selection of fifteen articles and is devoted to the 60th birthdays of Lex Renner and Mohan Putcha, the pioneers of the sphere of algebraic monoids.
Topics awarded include:
structure and illustration conception of reductive algebraic monoids
monoid schemes and functions of monoids
monoids on the topic of 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 element of those articles are designed to function a self-contained advent to those issues, whereas the rest contributions are examine articles containing formerly unpublished effects, that are guaranteed to develop into very influential for destiny paintings. between those, for instance, the real contemporary 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 conception, algebraic combinatorics and the idea of algebraic crew embeddings will reap the benefits of this particular and vast compilation of a few primary leads to (semi)group thought, algebraic staff embeddings and algebraic combinatorics merged below the umbrella of algebraic monoids.
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Extra info for Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics
Keep the notation and assumptions of Proposition 11. k/ D 0), then the scheme-theoretic fibers of ' are reduced. (ii) If M is normal, then H is a connected algebraic group; moreover, the schemetheoretic fibers of ' are reduced and irreducible. Proof. (i) Denote by cartesian square W G ! G=H the quotient homomorphism and form the X ? 0? y M '0 ' ! G ? y ! G=H: Since and ' are equivariant for the actions of G by left multiplication, X is equipped with a G-action such that 0 and ' 0 are equivariant.
Ii) G Hred Nred is an algebraic monoid, and the natural map WG Hred Nred ! M is a finite bijective homomorphism of algebraic monoids. (iii) Nred is unit dense and its unit group is Hred . (iv) is birational. Proof. (i) The assertion on Hred is well-known. That on Nred follows readily from the fact that N is a closed submonoid scheme of M , stable under the G-action by conjugation. (ii) The natural map G=Hred ! G=H is a purely inseparable homomorphism of algebraic groups, and hence is finite and bijective.
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.