Lie algebras and algebraic groups /

A strong undergraduate mathematics collection should feature a certain number of books that offer students a glimpse beyond the horizon. Pride of place should go to detailed, systematic, and self-contained treatises on advanced but foundational subjects that also happen to clearly build on undergraduate mathematics. Such a collection would have several works on Lie theory, a subject one may approach from analysis (see the books of A. Knapp), from topology (see F. Warner's book or Mimura and Toda's), or even from differential equations (see P.J. Olver's Applications of Lie Groups to Differential Equations, 1986; 2nd ed., 1993.) As Tauvel and Yu (both, Universite de Poitiers, France) focus on algebraic groups, they approach Lie theory via algebraic geometry and even develop that subject from scratch, taking half their book to do so! (Extracting just the necessary background from a general introduction to algebraic geometry would make a daunting task.) For the purpose at hand, Tauvel and Yu's work compares favorably with two otherwise excellent books (which Tauvel and Yu inexplicably fail to cite!), both with the title Introduction to Algebraic Geometry and Algebraic Groups, one the classic (1980) by M. Demazure and P. Gabriel, the other a recent entry (2004) by M. Geck. ^BSumming Up: Highly recommended. Upper-division undergraduates through professionals. D. V. Feldman University of New Hampshire