By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This publication encompasses a selection 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 idea of reductive algebraic monoids
monoid schemes and functions of monoids
monoids regarding Lie theory
equivariant embeddings of algebraic groups
constructions and houses of monoids from algebraic combinatorics
endomorphism monoids caused from vector bundles
Hodge–Newton decompositions of reductive monoids
A section of those articles are designed to function a self-contained advent to those subject matters, whereas the remainder contributions are examine articles containing formerly unpublished effects, that are certain to develop into very influential for destiny paintings. between those, for instance, the $64000 contemporary paintings of Michel Brion and Lex Renner exhibiting 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 staff embeddings will take advantage of this detailed and vast compilation of a few primary leads to (semi)group conception, algebraic workforce embeddings and algebraic combinatorics merged less than the umbrella of algebraic monoids.
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Additional info for Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics
G1 g2 ; g2 1 y1 g2 y2 / 24 M. 1G ; 1N / (this defines the semi-direct product of G with N ). gh 1 ; hy/: In other words, ' W M ! G=H identifies M to the fiber bundle G H N ! G=H associated to the principal H -bundle G ! G=H and to the variety N on which H acts by left multiplication. We say that the algebraic monoid M is induced from N . If we no longer assume that H is an algebraic group, then N is just a submonoid scheme of M , and the above properties hold in the setting of monoid schemes. We now obtain slightly weaker versions of these properties in the setting of algebraic monoids.
It follows that X is reduced. But X Š G N and hence N is reduced. If in addition H is connected, then the fibers of are irreducible; hence the same holds for 0 , and X is irreducible. As above, it follows that N is irreducible. o o (ii) Consider the reduced neutral component Hred Â H ; then Hred is a closed o normal subgroup of G. Moreover, the natural map ı W G=Hred ! G=H is a finite morphism and sits in a commutative square G ? y o G=Hred Ã ı ! M ? 'y ! G=H; On Algebraic Semigroups and Monoids 23 o where Ã denotes the inclusion.
M is a finite birational homomorphism of algebraic monoids. Maff /. 28 M. Brion (v) M is normal if and only if Maff is normal and Ä is an isomorphism. Then the assignment I 7! I \ Maff defines a bijection between the two-sided ideals of M and those of Maff ; the inverse bijection is given by J 7! GJ . Maff /. Proof. Maff / contains Gaff as an open subgroup. Maff / D Gaff . Hence Maff is affine by Theorem 2. (ii) The assertion follows readily from the Rosenlicht decomposition: since Gaff \ Gant is contained in the center of Gaff , its action on Maff by conjugation is trivial.
Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics by Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang