Orthogonal decompositions and integral lattices.

*(English)*Zbl 0855.11033
De Gruyter Expositions in Mathematics. 15. Berlin: De Gruyter. x, 535 p. (1994).

This book presents an important area of recent research in integral lattices and Lie algebras. Almost all the material was obtained by the authors and their collaborators at Moscow University during the last fifteen years. Up to now it could only be found in journal articles, so this publication will be highly welcomed.

The book is divided into two parts. Part I deals with the existence and classification of decompositions of a complex simple Lie algebra into a direct sum of Cartan subalgebras which are pairwise orthogonal with respect to the Killing form. The question of existence has been solved in the positive except for type \(A_n\) (where a negative answer is conjectured when \(n+1\) is not a prime power) and type \(C_n\) (where a negative answer is conjectured when \(n\) is not a 2-power). The first three chapters give the basic constructions, making use of concepts from finite geometry (such as symplectic spreads) and graph theory. The next three chapters are devoted to orthogonal decompositions whose automorphism groups act irreducibly on the Lie algebra. This more advanced subject involves, among other things, some applications of the classification of finite simple groups. The isolated final chapter of Part I deals with certain orthogonal decompositions of semisimple associative algebras.

Part II studies Euclidean lattices on a complex simple Lie algebra \({\mathcal L}\). The automorphism group \(G\) of an irreducible orthogonal decomposition (in the sense explained above) acts rationally on \({\mathcal L}\), and the authors consider integral lattices invariant under some subgroup of \(G\) that is still irreducible on \({\mathcal L}\). The ambitious goals are to describe and classify these invariant lattices and to determine their full automorphism groups. In Chapters 9 and 10 this difficult project is carried through for type \(A_{p-1}\), \(p\) prime. In particular, one meets an apparently new series of even unimodular lattices of dimension \(p^2- 1\), but also the well-known Craig lattices occur (under the name \(\Gamma_k\)) as projections of invariant lattices onto one Cartan component. Subsequent chapters deal with the other Lie types. Here the results are less complete, although deep and complicated; for part of the cumbersome details the authors refer to the original papers. As regards to type \(E_8\), which was the original point of departure (by Thompson) in Lie lattice realizations of finite simple groups, the authors describe the automorphism groups of all invariant lattices; one of them is the notorious Smith-Thompson lattice (of course, also in this book not really visible). The final Chapter 14 gives an overview of other lattices that have occured in connection with group representations (Barnes-Wall, Gow, Gross-Elkies), directing the interested reader to the most recent work by the second author.

The book is divided into two parts. Part I deals with the existence and classification of decompositions of a complex simple Lie algebra into a direct sum of Cartan subalgebras which are pairwise orthogonal with respect to the Killing form. The question of existence has been solved in the positive except for type \(A_n\) (where a negative answer is conjectured when \(n+1\) is not a prime power) and type \(C_n\) (where a negative answer is conjectured when \(n\) is not a 2-power). The first three chapters give the basic constructions, making use of concepts from finite geometry (such as symplectic spreads) and graph theory. The next three chapters are devoted to orthogonal decompositions whose automorphism groups act irreducibly on the Lie algebra. This more advanced subject involves, among other things, some applications of the classification of finite simple groups. The isolated final chapter of Part I deals with certain orthogonal decompositions of semisimple associative algebras.

Part II studies Euclidean lattices on a complex simple Lie algebra \({\mathcal L}\). The automorphism group \(G\) of an irreducible orthogonal decomposition (in the sense explained above) acts rationally on \({\mathcal L}\), and the authors consider integral lattices invariant under some subgroup of \(G\) that is still irreducible on \({\mathcal L}\). The ambitious goals are to describe and classify these invariant lattices and to determine their full automorphism groups. In Chapters 9 and 10 this difficult project is carried through for type \(A_{p-1}\), \(p\) prime. In particular, one meets an apparently new series of even unimodular lattices of dimension \(p^2- 1\), but also the well-known Craig lattices occur (under the name \(\Gamma_k\)) as projections of invariant lattices onto one Cartan component. Subsequent chapters deal with the other Lie types. Here the results are less complete, although deep and complicated; for part of the cumbersome details the authors refer to the original papers. As regards to type \(E_8\), which was the original point of departure (by Thompson) in Lie lattice realizations of finite simple groups, the authors describe the automorphism groups of all invariant lattices; one of them is the notorious Smith-Thompson lattice (of course, also in this book not really visible). The final Chapter 14 gives an overview of other lattices that have occured in connection with group representations (Barnes-Wall, Gow, Gross-Elkies), directing the interested reader to the most recent work by the second author.

Reviewer: H.G.Quebbemann (Oldenburg)

##### MSC:

11H56 | Automorphism groups of lattices |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

17B20 | Simple, semisimple, reductive (super)algebras |

20C10 | Integral representations of finite groups |