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

This e-book includes 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 offered include:

structure and illustration concept of reductive algebraic monoids

monoid schemes and functions of monoids

monoids relating to Lie theory

equivariant embeddings of algebraic groups

constructions and houses of monoids from algebraic combinatorics

endomorphism monoids triggered from vector bundles

Hodge–Newton decompositions of reductive monoids

A section of 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creation to those subject matters, whereas the remainder contributions are examine articles containing formerly unpublished effects, that are guaranteed to turn into very influential for destiny paintings.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 workforce embeddings will reap the benefits of this distinct and vast compilation of a few primary ends up in (semi)group concept, algebraic workforce embeddings and algebraic combinatorics merged lower than the umbrella of algebraic monoids.

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Additional info for Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

Example text

S / D feg Al Ar B. Proof. (i) By [19, Chap. II, §4, Cor. x/ C WA A ! y/ C x0 ; where x0 2 A and ', are endomorphisms of the algebraic group A. y/. '/ \ Ker. '/ \ Ker. '/ \ Im. '/ \ Im. /, so that ' (resp. ) is the projection of A to Al B (resp. Ar B). The uniqueness of this decomposition follows from that of ' and . A; / ! G be a homomorphism to an algebraic group. Then the image of is a complete irreducible variety, and hence generates an abelian subvariety of G . Thus, we may assume that G is an abelian variety, with group law also denoted additively.

A1 ; / in the construction of this example). Also, note that Chevalley’s structure theorem fails over any imperfect field F (see [32, Exp. XVII, App. ], and [35] for recent developments). Thus, Gaff may not be defined over F with the notation and assumptions of Proposition 19. Yet the Albanese morphism still exists for any F -variety equipped with an F -point (see [36, App. A]) and hence for any algebraic F -semigroup equipped with an F -idempotent. 1 Abelian Varieties In this subsection, we begin by describing all the algebraic semigroup laws on an abelian variety.

Xo satisfy the assumptions of Lemma 4. , g D h ı f on Xo . In particular, g maps the scheme-theoretic fiber of f at any point of Yo to a single point. Next, let Y1 be an irreducible component of Y intersecting Yo . Y1 / is an irreducible component of X ; moreover, the restrictions f1 W X1 ! Y1 , g1 W X1 ! Z and s1 W Y1 ! X1 satisfy the assumptions of the above lemma, for any point y1 of Yo \ Y1 . Thus, g D h ı f on Xo [ X1 . Iterating this argument completes the proof in view of the connectedness of X .

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