By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This publication incorporates 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 awarded include:
structure and illustration conception of reductive algebraic monoids
monoid schemes and purposes of monoids
monoids with regards to Lie theory
equivariant embeddings of algebraic groups
constructions and houses of monoids from algebraic combinatorics
endomorphism monoids brought on from vector bundles
Hodge–Newton decompositions of reductive monoids
A component to those articles are designed to function a self-contained creation to those themes, whereas the rest contributions are examine articles containing formerly unpublished effects, that are guaranteed to turn into very influential for destiny paintings. between those, for instance, the $64000 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 idea of algebraic team embeddings will take advantage of this exact and huge compilation of a few basic leads to (semi)group idea, 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
Xgyg 1 ; g; g 1 gy/: Now xgyg 1 D xgyeg 1 D xgeg 1 D xe D x and likewise, g 1 xgy D y. This proves the claim. By that claim, Ã is a closed immersion, and a retraction of Ã. Also, we have for any x; x 0 2 X , g; g 0 2 G and y; y 0 2 Y : xgyx 0 g 0 y 0 D xgyex 0 g 0 y 0 D xgex 0 g 0 y 0 D xgeg 0 y 0 : In other words, Ã is a homomorphism of algebraic semigroups, where X given the semigroup structure as in the statement. G Y is 42 M. Brion We next claim that is a homomorphism of algebraic semigroups as well.
Exe/ D A. e/; A. S /; A. //. e; al ; ar ; e/ in the decomposition of Proposition 21. S /. S /. t u Combined with Proposition 16, the above result yields: Corollary 5. Let S be an irreducible algebraic semigroup. (i) All the maximal submonoids of S have the same Albanese variety, and all the maximal subgroups have isogenous Albanese varieties. S /. 36 M. Brion Remark 9. (i) With the notation and assumptions of Proposition 22, the morphism ' ı ˛ W S ! S / is the universal homomorphism to an abelian variety.
G=H the universal homomorphism to an algebraic group, and N the schemetheoretic fiber of ' at 1. Denote by Hred (resp. Nred ) the largest reduced scheme of H (resp. N ). (i) Hred is a closed normal subgroup of G and Nred is a closed submonoid of M , stable under the action of G by conjugation. (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.