By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This booklet features a choice 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 thought of reductive algebraic monoids
monoid schemes and functions of monoids
monoids concerning Lie theory
equivariant embeddings of algebraic groups
constructions and homes of monoids from algebraic combinatorics
endomorphism monoids triggered from vector bundles
Hodge–Newton decompositions of reductive monoids
A element of those articles are designed to function a self-contained advent to those themes, whereas the remainder contributions are examine articles containing formerly unpublished effects, that are bound to turn into very influential for destiny paintings.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 idea, algebraic combinatorics and the idea of algebraic workforce embeddings will reap the benefits of this specified and large compilation of a few basic leads to (semi)group conception, algebraic workforce embeddings and algebraic combinatorics merged below the umbrella of algebraic monoids.
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This booklet includes a choice of fifteen articles and is devoted to the 60th birthdays of Lex Renner and Mohan Putcha, the pioneers of the sector of algebraic monoids. subject matters awarded include:structure and illustration idea of reductive algebraic monoidsmonoid schemes and purposes of monoidsmonoids with regards to Lie theoryequivariant embeddings of algebraic groupsconstructions and houses of monoids from algebraic combinatoricsendomorphism monoids triggered from vector bundlesHodge–Newton decompositions of reductive monoidsA component of those articles are designed to function a self-contained creation to those issues, whereas the remainder contributions are study articles containing formerly unpublished effects, that are certain to develop into very influential for destiny paintings.
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Additional resources for Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics
This proves the first assertion. The second assertion is proved by the argument of Proposition 14 (ii); note that any two-sided ideal of Maff is stable under conjugation by G, in view of (ii) above. t u Example 5. Let n be a positive integer, n the group scheme of nth roots of unity, and A an abelian variety containing n as a subgroup scheme (any ordinary elliptic curve will do). A2 ; /, and H the unit subgroup scheme of N ; then H Š n Gm . Next, let G WD A Gm ; this is a connected commutative algebraic group containing H as a subgroup scheme.
K/. V ˝ W / D W ˝ W as subspaces of V ˝ V . K/, this yields the desired equality. 32 M. Brion Since Y is associative on the dense subscheme K, it is associative everywhere; likewise, Y admits 1K as a neutral element. Thus, Y is an algebraic monoid law on Y . We may now form the induced monoid G K Y as in Sect. 2, to get the desired structure on X . 5 Algebraic Semigroups and Monoids over Perfect Fields In this subsection, we extend most of the above results to the setting of algebraic semigroups and monoids defined over a perfect field.
Moreover, by Lemma 2, the image of the morphism is exactly the kernel of S ; this is a simple algebraic semigroup in view of Proposition 5. One may thus deduce part of Theorem 6 from the structure of simple algebraic semigroups presented in Remark 3 (i). Yet we will provide a direct, selfcontained proof by adapting the arguments of Proposition 5. Proof of Theorem 6. One readily checks that the map (resp. ) as in the statement yields an algebraic semigroup structure on X G Y (resp. on S ); compare with Example 1 (ii).