By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This ebook 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 conception of reductive algebraic monoids
monoid schemes and purposes 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 on from vector bundles
Hodge–Newton decompositions of reductive monoids
A component of those articles are designed to function a self-contained creation to those subject matters, whereas the rest contributions are examine articles containing formerly unpublished effects, that are absolute to turn into very influential for destiny paintings. between those, for instance, the real 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 thought, algebraic combinatorics and the speculation of algebraic team embeddings will take advantage of this targeted and extensive compilation of a few basic ends up in (semi)group concept, algebraic staff 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
Exe/ D A. e/; A. S /; A. //. e; al ; ar ; e/ in the decomposition of Proposition 21. S /. S /. t u Combined with Proposition 16, the above result yields: Corollary 5. Let S be an irreducible algebraic semigroup. (i) All the maximal submonoids of S have the same Albanese variety, and all the maximal subgroups have isogenous Albanese varieties. S /. 36 M. Brion Remark 9. (i) With the notation and assumptions of Proposition 22, the morphism ' ı ˛ W S ! S / is the universal homomorphism to an abelian variety.
G. [7, Cor. 9]). The image of Gx under the homomorphism ˛ W G ! A is affine (as the image of an affine group scheme by a homomorphism of group schemes) and proper (as a subgroup scheme of the abelian variety G=Gaff ), hence finite. H / is finite. Also, the kernel of the homomorphism ˛jH is a subgroup scheme of Gaff , and hence is affine. Thus, the reduced scheme Hred is an extension of a finite group by an affine algebraic group, and hence is affine. Thus, so is Nred in view of Theorem 2 and of Proposition 13.
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.
Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics by Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang