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

This booklet includes 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 offered include:

structure and illustration thought 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 component to those articles are designed to function a self-contained creation to those subject matters, whereas the remainder contributions are examine articles containing formerly unpublished effects, that are guaranteed to develop into very influential for destiny paintings. between those, for instance, the $64000 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 thought, algebraic combinatorics and the idea of algebraic staff embeddings will reap the benefits of this distinctive and wide compilation of a few primary ends up in (semi)group conception, algebraic crew 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**

**Sample text**

Iii) The above map ' is the universal homomorphism to an algebraic group. On Algebraic Semigroups and Monoids 41 Proof. (i) By Proposition 3 (iii), it suffices to show that e is the unique idempotent of eSe. eSe/, then xy D xf y for all x; y 2 S , by Lemma 2. Taking x D y D e yields e D ef e D f . eye/ for all x; y 2 S . (iii) Let G be an algebraic group and let W S ! G be a homomorphism of algebraic semigroups. exe/ for all x 2 S . Thus, factors uniquely as the homomorphism ' followed by some homomorphism of algebraic groups eSe !

Then X has a structure of algebraic monoid with unit group G if and only if the Albanese morphism ˛ W X ! X / is affine. Proof. In view of Proposition 16, it suffices to show that X is an algebraic monoid if ˛ is affine. X /. X / Š G=K equivariantly for the left (or right) action of G. G/ Y ! G/; for the left (or right) G-action, this yields the fiber bundle G K Y ! G=K. Since ˛ is affine, so is Y . G/ by conjugation. Thus, the group scheme K is quasi-affine, and hence is affine. We now show that the group law K W K K !

On Algebraic Semigroups and Monoids 37 Step 1: we show that every idempotent of S is either a neutral or a zero element. S /. Since Se is a closed irreducible subvariety of S , it is either the whole S or a single point; in the latter case, Se D feg. Thus, one of the following cases occurs: (i) Se D eS D S . , e is the neutral element. (ii) Se D feg and eS D S . Then for any x; y 2 S , we have xe D e and ey D y. Thus, xy D xey D ey D y. So D r in the notation of Example 1 (i), a contradiction since is assumed to be nontrivial.