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BOOK REVIEWS 365, groups in arbitrary characteristic It stated that a connected solvable linear. algebraic group defined over an algebraically closed field was conjugate to a. group of upper triangular matrices See Kolchin 13, By the 1950s however the systematic use of developments in algebraic. geometry began to supplant the traditional Lie theoretic approach to algebraic. groups This infusion of new techniques made possible not only the construc. tion of a theory which was valid in arbitrary characteristic but also the. discovery of new results which had previously been unknown even in char. acteristic zero The celebrated Borel fixed point theorem provided a prime. example of this phenomenon Borel showed 1 that a connected solvable linear. algebraic group over an algebraically closed field which acted on a complete. variety in the appropriate sense had a fixed point When applied to the flag. variety of a vector space on which the group acted linearly this result yielded a. new algebraic geometric proof of the above mentioned Lie Kolchin theorem. Making use of this new approach Chevalley succeeded in completely. classifying the simple algebraic groups over an algebraically closed field k His. attack was two pronged He showed 5 that two simple groups over k with. isomorphic root systems and weight lattices were isomorphic More generally. he determined the isogenics between two simple or semisimple groups In his. great T hoku paper 6 however he actually carried out a construction of. the simple adjoint groups This process which began with the corresponding. complex simple Lie algebras also gave rise to the finite Chevalley groups and. a little later to the important finite groups of Lie type See Carter 4 for a. treatment of this aspect Amazingly enough in spite of the substantial. recasting and generalization the new classification perfectly mirrored its. classical characteristic zero analogue, After Chevalley s discovery of the classification theorems algebraic group. theory grew and prospered both as an area of mathematics in its own right and. as a useful tool in such fields as finite groups algebraic geometry and. automorphic forms See Borel s article 3 for a discussion of these matters. As a testimony to the subject s coming of age introductory accounts began to. appear in the late 1960s and 1970s most notably Borel s Linear algebraic. groups 2 1969 and Humphreys book 12 1975 of the same title Most. recently G P Hochschild and T A Springer have each made important. contributions to this growing stock of introductory literature. Hochschild sets the tone of his book in the preface when he states that the. emphasis is on developing the major general mathematical tools used for. gaining control over algebraic groups rather than on securing final definitive. results p v His two most powerful tools algebraic geometry and the. theory of Lie algebras demand quite a bit of attention In fact Hochschild. develops these topics to the extent that his treatments could serve as introduc. tions for the uninitiated novice, The discussion of algebraic geometry he gives here closely follows that found. in Humphreys 12 but Hochschild departs from Humphreys sketch by. meticulously including every proof Primarily he spares no effort in dealing. with the pertinent areas of commutative algebra For example he thoroughly. develops both the dimension theory of local rings and the structure theory of. 366 BOOK REVIEWS, regular local rings In discussing Lie algebras the author reproduces the.

polished development he gave in 11 to obtain such classical results as. Cartan s solvability criterion and Weyl s complete reducibility theorem He. finishes up with the canonical presentation of the basic structure theory as well. as the representation theory of semisimple Lie algebras. With these results and techniques at his disposal Hochschild describes the. two main slants on the theory of linear algebraic groups He gives expositions. both of the general structure theory over arbitrary algebraically closed fields. and of the more classical characteristic zero theory His handling of the. arbitrary field case proceeds along traditional lines and concludes with the. theory of Borel subgroups He makes full use of algebraic geometry in. obtaining such results as the existence of quotient varieties and the Borel. fixed point theorem As for the Lie algebra theory he calls this into service to. prove more incisive results about the groups in characteristic zero Some of this. comes from Chevalley 7 of course but Hochschild includes much more We. find for example a proof of Mostow s semidirect product theorem 14 not to. mention the masterful closing chapter where Hochschild presents an elegant. construction based on 10 of the simply connected groups having Lie. algebras with nilpotent radicals This is definitely the high point of the book. The author s treatment of the characteristic zero theory is a tour de force. In spite of these strengths this book will not appeal to everyone While the. author succeeds in achieving his announced purpose of developing the funda. mental tools the lack of direction and the rarity of clarifying examples. represent potential stumbling blocks for the beginner A glance at the table of. contents reveals a medley of topics from Representative Functions and Hopf. Algebras to Derivations and Lie Algebras then from Algebraic Varieties. to Semisimple Lie Algebras In this last chapter there is neither an example. of a semisimple Lie algebra nor one of a specific irreducible representation. although these are the objects under investigation. The more experienced reader however will appreciate Hochschild s book. for its concise and authoritative style as well as for its treatment of various. topics not usually found in introductory texts We point out in addition to the. final chapter mentioned above his discussion of automorphism groups and. observable subgroups In summary this book is a rich and potentially useful. treatment of algebraic groups The same may also be said of T A Springer s. new text on the same subject, Based on lectures given at the University of Notre Dame in 1968 Springer s. book provides a complete and essentially self contained account of the classifi. cation of reductive and hence simple algebraic groups over a general algebrai. cally closed field k In contrast to the traditional approach Springer presents a. new construction of these reductive groups which makes no use of the classical. Lie algebra theory Thus he limits his preliminary material to basic algebraic. geometry In about fifty pages and with a minimal amount of fuss the reader. acquires all of the tools needed for the task ahead. First Springer tackles the general structure theory Here he follows the usual. outline but his proofs are often new and improved The development culminates. in Chapters 9 and 10 where he associates to each reductive group its root. BOOK REVIEWS 367, datum R X Rv Xv in the sense of 8 and then proves such fundamental. results as the Bruhat decomposition lemma, Next in Chapter 11 he carries out half of Chevalley s classification program. by showing that two reductive groups over k with isomorphic root data are. isomorphic The novel argument comes in several parts Given a reductive. group G with root system R Springer suitably normalizes the usual one. parameter unipotent subgroups xa a R of G This gives rise to structure. constants for G via Chevalley s commutator formula By giving a generators. and relations presentation of G he shows that these constants are determined. up to a natural equivalence by the root datum of G The desired result then. follows straightaway with the existence of graph automorphisms as an easy. corollary Springer concludes this train of thought with a brief sketch of. Chevalley s isogeny theorem, The coup de gr ce comes in Chapter 12 when the author proves the existence. of a reductive group having a prescribed root datum Reducing to the special. case of a semisimple group G of adjoint type he presents G as an automor. phism group of a Lie algebra provided R is simply laced His slick argument in. constructing the Lie algebra hinges on a device first used by Frenkel and Kac. 9 Finally in case R is not simply laced he obtains G from the fixed points of. a suitable graph automorphism, This sharp treatment of algebraic groups should become the standard.

reference for classification theory Informal and engaging in style the work. makes for pleasant reading It is further enhanced by the presence of numerous. examples and timely exercises on such topics as Schubert varieties and Bruhat. orderings in Weyl groups The inordinate number of typographical errors. however definitely detracts from its overall appeal although it presents no real. Despite their common ground these two books are remarkably different. For those interested primarily in the characteristic zero theory Hochschild s. work should prove invaluable whereas both books should benefit the en. thusiast of the general theory Each in his own way Springer and Hochschild. have filled gaps in the literature,REFERENCES, 1 A Borel Groupes lin aires alg briques Ann of Math 2 64 1956 20 80. 2 Linear algebraic groups Benjamin New York 1969, 3 On the development of Lie group theory Math Intelligencer 2 1980 67 72. 4 R Carter Simple groups of Lie type Wiley London and New York 1972. 5 C Chevalley S minaire sur la classification des groupes de Lie alg briques cole Norm Sup. Paris 1956 1958, 6 Sur certains groupes simples T hoku Math J 7 1955 14 66. 7 Th orie des groupes de Lie Tome II Groupes alg briques 1951 Tome III. Th or mes g n raux sur les alg bres de Lie 1955 Hermann Paris. 8 M Demazure and A Grothendieck Sch mas en groupes Lecture Notes in Math vols 151. 152 155 Springer Verlag Berlin and New York 1970, 9 I Frenkel and V Kac Basic representations of affine Lie algebras and dual resonance models. Invent Math 62 1980 23 66, 10 G Hochschild Algebraic Lie algebras and representative functions Illinois J Math 3 1959.

499 523 supplement ibid 4 1960 609 618,368 BOOK REVIEWS. 11 The structure of Lie groups Holden Day San Francisco 1965. 12 J Humphreys Linear algebraic groups Springer Verlag Berlin and New York 1975. 13 E Kolchin Algebraic matrix groups and the Picard Vessiot theory of homogeneous linear. ordinary differential equations Ann of Math 2 49 1948 1 42. 14 G Mostow Fully reducible subgroups of algebraic groups Amer J Math 78 1956 200 221. BRIAN PARSHALL,BULLETIN New Series OF THE,AMERICAN MATHEMATICAL SOCIETY. Volume 9 Number 3 November 1983,1983 American Mathematical Society. 0273 0979 83 1 00 25 per page, Computational methods f or fluid flow by Roger Peyret and Thomas D Taylor. Springer Series in Computational Physics Springer Verlag New York 1983. x 358 pp 42 50 ISBN 0 3871 1147 6, I read this book by Peyret and Taylor with pleasure At last there is an.

introductory book on computational fluid dynamics that is intelligent fairly. well informed and reasonably up to date Its perspective is not however fully. consistent with a mathematician s point of view, Computational fluid dynamics CFD is the process of solving problems in. fluid dynamics numerically on a computer It was recognized long ago that the. equations of fluid dynamics are particularly amenable to numerical solution. and as early as the 1920 s L Richardson tried to describe how this could be. done In 18 he imagined a gigantic concert hall full of hundreds of human. computers passing pieces of calculations to each other under the majestic. guidance of a conductor s baton The invention of the nonhuman computer. greatly streamlined the logistics and since the forties there has arisen an. enormous CFD enterprise There are thousands of researchers innumerable. applications and an enormous literature billions of dollars are spent every. year on CFD calculations These calculations have affected an amazing range. of sciences from astrophysics geophysics meteorology biology and chem. istry to most branches of engineering Airplanes space shuttles weather. forecasts bombs and nuclear power plants are brought to you in part by CFD. Artificial hearts and energy producing devices are being added to this partial. list CFD is useful in all these fields because it supplements or even replaces. experiments that are expensive uninformative or as in the case of astrophysics. and meteorology uncontrollable, The equations of fluid dynamics are well known and their validity is not in. doubt one might think that at least in principle CFD is merely an elaborate. exercise in numerical analysis and approximation theory However the phe. nomena of fluid mechanics turbulence combustion multidimensional shock. patterns boundary layers etc are so complicated that with present methods. many of them could not be fully analyzed on any finite computer As a result. CFD has acquired a life of its own as a major research area one in which. mathematics physical intuition and computer science interact in original and. unexpected ways Those who are not familiar with fluid mechanics can see. 364 BOOK REVIEWS REFERENCES 1 A Basic theory of algebraic groups and Lie algebras complex simple Lie algebras

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