Authors

Citation

Wx. Huang, Reductive and semisimple algebraic monoids, FORUM MATH, 13(4), 2001, pp. 495-504

Citations number

26

Language

INGLESE

art.tipo

Article

Categorie Soggetti

Mathematics

Journal title

FORUM MATHEMATICUM

ISSN journal

0933-7741
→ ACNP

Volume

13

Issue

4

Year of publication

2001

Pages

495 - 504

Database

ISI

SICI code

0933-7741(2001)13:4<495:RASAM>2.0.ZU;2-M

Abstract

Let Mbe an irreducible linear algebraic monoid over an algebraically closed
field. M is reductive if its unit group G is a reductive (algebraic) group
; M is semisimple if M not equal G and G is reductive with a one dimensiona
l center. The following theorems are proved:
(1) M is reductive iff M is regular with its kernel (in the sense of semigr
oup) a reductive group.
(2) M is semisimple iff M is regular with exactly two central idempotents a
nd with its kernel, ker(M), a semisimple (algebraic) group iff M-e, the irr
educible component of the identity in {a epsilon M \ ae = ea = e}, is a sem
isimple monoid and ker(M) is a semisimple group, where e is a minimal idemp
otent of M.
(3) If dim M = 4 and M not equal G is nonsolvable, then M is semisimple.
(4) If dim M = 5 and M not equal G is nonsolvable, then M is reductive iff
rank(G) not equal 2.
Five dimensional reductive monoids with zero are further analyzed in terms
of semisimple monoids.