By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This e-book includes a number 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 awarded include:
structure and illustration concept of reductive algebraic monoids
monoid schemes and functions of monoids
monoids with regards 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 component to those articles are designed to function a self-contained creation to those subject matters, whereas the remainder contributions are learn articles containing formerly unpublished effects, that are certain to turn into very influential for destiny paintings. between those, for instance, the real fresh 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 idea of algebraic team embeddings will reap the benefits of this distinct and huge compilation of a few basic leads to (semi)group thought, algebraic team embeddings and algebraic combinatorics merged below the umbrella of algebraic monoids.
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Extra resources for Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics
3]. But the morphism ˛ is generally nonaffine, and the finiteness of ˇ is an open question in this setting. We now characterize algebraic monoids among equivariant embeddings: Theorem 4. Let X be an equivariant embedding of a connected algebraic group G. 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.
S; / (over k) is defined over F , or an algebraic F -semigroup, if S is an F -variety and the morphism is defined over F . F /. Note that an algebraic F -semigroup may well have no F -point; for example, an F -variety without F -point equipped with the trivial semigroup law l or r . But this is the only obstruction to the existence of F -idempotents, as shown by the following: Proposition 17. S; / be an algebraic F -semigroup. S / (viewed as closed subsets of S ) are defined over F . (ii) If S is commutative, then its smallest idempotent is defined over F .
C tNn 1 , where c 2 k. xy/ D 0 for all x; y 2 S . Thus, xy belongs to the fiber of n at 0, a finite set containing 0. , D 0 ; a contradiction. Thus, we must have n D 2, and we obtain a nonconstant morphism D 2 W S ! A1 , where the semigroup law on A1 is the multiplication. The image of contains 0 and a nonempty open subset U of the unit group Gm . Then U U D Gm and hence is surjective. e/ D 1. Then e is the desired nonzero idempotent. t u Remark 10. One may also deduce the above theorem from the description of algebraic semigroup structures on abelian varieties (Proposition 21), when the irreducible curve S is assumed to be nonsingular and nonrational.
Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics by Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang