Idempotent lattices, Renner monoids and cross section lattices of the special orthogonal algebraic monoids. (Q1421782)

In the theory of reductive algebraic monoids, the idempotent lattice, the cross-section lattice and the Renner monoid play central roles. Let \(M=\overline{K^*\text{SO}_n(K)}\subset M_n(K)\), where \(K\) is an algebraically closed field. \(M\) is referred to as a special orthogonal algebraic monoid. \(M\) is a reductive monoid. The article explicitly determines the idempotent lattice, the cross-section lattice and the Renner monoid of \(M\) by using admissible subsets and the Weyl group. It turns out that there is a one-to-one relationship between \(E(\overline T)\) and the admissible subsets, where \(T\) is a maximal torus of the unit group of \(M\), that the cross-section lattice is a sublattice of the cross-section lattice of \(M_n(K)\), and that the Renner monoid of \(M\) is a submonoid of the Renner monoid of \(M_n(K)\). For closely related work see the author [Semigroup Forum 66, No. 2, 273-287 (2003; Zbl 1018.20054)].