By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This e-book features a choice 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 provided 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 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a self-contained advent to those issues, whereas the remainder contributions are learn articles containing formerly unpublished effects, that are bound to turn into very influential for destiny paintings.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 thought, algebraic combinatorics and the speculation of algebraic team embeddings will reap the benefits of this distinct and extensive 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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10]. M / is a commutative algebraic semigroup, defined over F by [33, Thm. 13]. Now applying Proposition 17 to N yields the desired idempotent. t u Remark 8. The above observations leave open all the rationality questions for an algebraic semigroup S over a field F , not necessarily perfect. In fact, S has an idempotent F -point if it has an F -point, as follows from the main result of . But some results do not extend to this setting: for example, the kernel of an algebraic F -monoid may not be defined over F , as shown by a variant of the standard example 34 M.
G=H induced by the inclusion of Gx in H , we obtain a morphism ' W M ! G=H . 1/ is the neutral element of G=H . Thus, the restriction 'jG is the quotient homomorphism . By density, ' is a homomorphism of monoids, and D ı '. So ' is the desired homomorphism. t u Remark 5. M /. M /, are all conjugate in G (Proposition 8). By that proposition, we may take for x any minimal idempotent of M . (ii) As another consequence, any irreducible semigroup S has a universal homomorphism to an algebraic group (in the sense of the above proposition).
V) Theorem 6 extends readily to those irreducible algebraic semigroups that are defined over a perfect subfield F of k, and that have an F -point; indeed, this implies the existence of an idempotent F -point by Proposition 17. 44 M. Brion Likewise, the results of Sects. 2 extend readily to the setting of perfect fields. In view of Theorem 5, every nontrivial algebraic semigroup law on an irreducible curve S is commutative; by Proposition 17 again, it follows that S has an idempotent F -point whenever S and are defined over F .
Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics by Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang