Linear algebraic monoids.
The theory of algebraic monoids was founded in the 1980s and has been well developed mainly by Putcha and the author. It culminates in a natural blend of algebraic groups, torus embeddings and semigroups. In studying algebraic monoids, Putcha mainly takes the algebraic point of view, while the author mainly takes the geometric point of view, and they meet at algebraic combinatorics. The monograph by \textit{M. S. Putcha} [Linear algebraic monoids, Cambridge Univ. Press, Cambridge (1988; Zbl 0647.20066)] is an algebraic version of the theory; this one by the author is essentially a geometric version of algebraic monoids. As far how Putcha's algebraic and the author's geometric approaches meet in algebraic combinatorics, one is advised to read \textit{L. Solomon} [``An introduction to reductive monoids'', in NATO ASI Ser., Ser. C, Math. Phys. Sci. 466, 295-352 (1995; Zbl 0870.20047)], which is equipped with many illustrative examples. The book under review is a survey based monograph (or monograph style survey) of algebraic monoids, with focus on the geometry, classifications and representations of normal reductive algebraic monoids, plus a survey of their extensions to monoids of Lie type and spherical embeddings. The book consists of 15 chapters. Chapter 1 is an introduction to the history of algebraic monoids; Chapter 2 is for introducing tools of algebraic geometry, linear algebraic groups and semigroups; Chapter 3 is for introducing relative algebraic monoids. Chapters 4-14 form the main body, surveying the geometry of algebraic groups or algebraic monoids, classifications and representations of normal reductive algebraic monoids, the monoids of Lie type. Chapter 15 is a brief survey of closely related developments in other branches.
scientific article; zbMATH DE number 2155967
scientific article