The ring of regular functions of an algebraic monoid

Renner, Lex; Rittatore, Alvaro

doi:10.1090/S0002-9947-2011-05335-3

The ring of regular functions of an algebraic monoid
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Trans. Amer. Math. Soc. 363 (2011), 6671-6683 Request permission

Abstract:

Let $M$ be an irreducible normal algebraic monoid with unit group $G$. It is known that $G$ admits a Rosenlicht decomposition, $G=G_{\operatorname {ant}}G_{\operatorname {aff}} \cong (G_{\operatorname {ant}} \times G_{\operatorname {aff}})/G_{\operatorname {aff}}\cap G_{\operatorname {ant}}$, where $G_{\operatorname {ant}}$ is the maximal anti-affine subgroup of $G$, and $G_{\operatorname {aff}}$ the maximal normal connected affine subgroup of $G$. In this paper we show that this decomposition extends to a decomposition $M=G_{\operatorname {ant}}M_{\operatorname {aff}} \cong G_{\operatorname {ant}}*_{G_{\operatorname {aff}}\cap G_{\operatorname {ant}}}M_{\operatorname {aff}}$, where $M_{\operatorname {aff}}$ is the affine submonoid $M_{\operatorname {aff}}=\overline {G_{\operatorname {aff}}}$. We then use this decomposition to calculate $\mathcal {O}(M)$ in terms of $\mathcal {O}(M_{\operatorname {aff}})$ and $G_{\operatorname {aff}}, G_{\operatorname {ant}}\subset G$. In particular, we determine when $M$ is an anti-affine monoid, that is $\mathcal {O}(M)=\Bbbk$.

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