Kernels, regularity and unipotent radicals in linear algebraic monoids

Abstract

An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid with G (⊊ M) its unit group, ker(M) its semigroup kernel, E(ker(M)) the set of minimal idempotents of M. Algebraically, ker(M) is the minimum regular 𝒥-class in M; geometrically, ker(M) is a retract of M but also the unique closed orbit under the natural G × G-action on M. We study regularity conditions for M and the relationships between ker(M) and Ru(G). We also study an algebraic group embedding problem. The principal results are: (i) dim Ru(G) = dim E(ker(M)) + dim Ru(H) + dim Ru(G(e)), where e ∈ E(ker(M)), H is a maximal subgroup of ker(M) and G(e) ≔ {x ∈ G | xe = ex = e}. (ii) A connected linear algebraic group with nontrivial characters can be realized as a proper unit group of some irreducible normal regular algebraic monoid. (iii) When char (K) = 0, if M is regular, dim Ru(G) = dim ker(M) implies Ru(G) ≅ ker(M) as algebraic varieties.