If \({\mathfrak g}\) is a complex, finite dimensional, semisimple Lie algebra and \({\mathfrak h}\) is a Cartan subalgebra of \({\mathfrak g}\), there is a root space decomposition \({\mathfrak g}={\mathfrak h}\oplus_{\alpha\in\Phi}{\mathfrak g}_{\alpha}\), with set of roots \(\Phi\) and Cartan matrix \(A=(a_{ij})\) for \(a_{ij}=\alpha_j(\alpha_i^{\scriptscriptstyle\vee})\) with \(\alpha_i\in{\mathfrak h}^*,\alpha_j^{\scriptscriptstyle\vee}\in{\mathfrak g}\) simple root and coroot, respectively. The Lie algebra \({\mathfrak g}\) with Cartan matrix \(A\) has a presentation with generators \(e_i\in{\mathfrak g}_{\alpha_i}\), \(f_i\in{\mathfrak g}_{-\alpha_i}\) for each simple root \(\alpha_i\) and their opposites \(-\alpha_i\), together with the coroots \(h_i=\alpha_i^{\scriptscriptstyle\vee}\), and relations \[\begin{equation} [e_i,f_j]=-\delta_{ij}h_i,\quad [h_i,e_j]=a_{ij}e_j,\quad [h_i,f_j]-a_{ij}f_j,\quad [h_i,h_j]=0, \end{equation} \tag{1}\] for all \(1\leq i,j\leq n\), with \(n\) the number of simple roots. This presentation of the Lie algebra \({\mathfrak g}\), which uses information from its Chevalley generators and its Cartan algebra, depends on the type of the Lie algebra. Additional relations of the form \[ \text{ad}(e_i)^{1+|a_{ij}|}e_j= 0\qquad\text{and}\qquad\text{ad}(f_i)^{1+|a_{ij}|}=0 \tag{2} \] for \(i\neq j\) and where \(\text{ad}:{\mathfrak g}\rightarrow \text{End}({\mathfrak g})\) is the adjoint representation \(\text{ad}(x)(y)=[x,y]\), allowed J.-P. Serre to prove that the Lie algebra \({\mathfrak g}\) is characterized by the generators \(e_i, f_i, h_i\) and the relations (1) and (2).

This is the classical background from which, in 1967, two independent developments by R. Moody and V. Kac, considered relaxing the positive definite condition on the Cartan matrix in such a way that the corresponding algebra defined by generators and relations, as in the classical case, is again a Lie algebra which in many cases is infinite dimensional. A good reference for these algebras is V. Kac’s *Infinite dimensional Lie algebras* (Birkhäuser Verlag (1982) first edition, Cambridge University Press (1990), third edition). The generalized Cartan matrices are now square \(n\times n\) matrices \(A=(a_{ij})\) with integer entries such that

- \(a_{ii}=2\) for all \(1\leq i\leq n\).
- \(a_{ij}\leq 0\) for all \(i\neq j\).
- \(a_{ij}=0\) if and only if \(a_{ji}=0\).

Then, if \(A\) has rank \(r\), the *Kac-Moody algebra* \({\mathfrak g}_A\) associated to \(A\) is defined by a complex vector space \({\mathfrak h}\) of dimension \(2n-r\), a set of \(n\) linearly independent elements \(h_i\in {\mathfrak h}\) and a set of generators \(e_i,f_i\), \(1\leq i\leq n\) that satisfy the relations (1) and (2). Here, \({\mathfrak h}\) is the analogue of a Cartan subalgebra of \({\mathfrak g}_A\) and the \(h_i\) are the analogues of the simple coroots of \({\mathfrak g}_A\). There are corresponding analogues of root vectors and spaces and a root space decomposition \[{\mathfrak g}_A={\mathfrak h}\bigoplus_{\alpha\in\Delta}{\mathfrak g}_{\alpha}\] for \(\Delta\subseteq\Phi\) the subset of simple roots.

After a few pages reviewing some basic facts on classical Lie algebras and their classification, the second part of the book under review gives a detailed exposition of Kac-Moody algebras, including properties of the associated Weyl group and the classification of Kac-Moody algebras depending upon properties of its defining generalized Cartan matrix (eg., finite, affine or of indefinite type). There is also a detailed study of the root system of a Kac-Moody algebra, distinguishing between two classes of roots: those whose behavior is as in the classical finite-dimensional case, the so-called *real roots*, and the other ones which are called the *imaginary roots*.

In the classical theory, to a (simple) finite-dimensional (complex) Lie algebra \({\mathfrak g}\) there are associated finitely many connected (complex) Lie groups, ranging from a *minimal* Lie group \(G_{\text{ad}}\), the *adjoint form* of the Lie group, to a *maximal* Lie group \(\widetilde{G}\), the *simply connected form* of the Lie group, while the other groups are finite covers of \(G_{\text{ad}}\) finitely covered by \(\widetilde{G}\). Also, in the classical theory the (simple) and connected finite dimensional (complex) Lie groups are classified by the Dynkin diagrams of finite type, which, in general, also classify the (simple) connected linear algebraic groups over an algebraic closed field of arbitrary characteristic. On the other hand, to any Dynkin diagram of finite type and any field \(k\), there is associated a simple Lie algebra \({\mathfrak g}_k\) over the field \(k\), which in turn gives rise to a finite number of connected algebraic groups over \(k\), the *Chevalley groups* whose Lie algebra is \({\mathfrak g}_k\), ranging from an *adjoint form* \(G_{\text{ad}}\) to a *universal form* \(G_u\).

The third part of the book under review considers the various ways that one can associate a group object to a given Kac-Moody algebra \({\mathfrak g}_A\) and a choice of the underlying category has to be made. The focus is on two particular classes of Kac-Moody groups, the *minimal Kac-Moody groups* associated to real root spaces of the Kac-Moody algebra, and the *maximal Kac-Moody groups* which take into account all root spaces, real and imaginary, of \({\mathfrak g}_A\). The construction in the minimal case is variation of the classical one: choose an integrable linear representation of the algebra \({\mathfrak g}_A\) and using exponentiation maps for the real root spaces \({\mathfrak g}_{\alpha}\) to construct a complex group associated to the given algebra. After proving some properties of these groups, the author lists some shortcomings, e.g., dependency on the representation and validity only over the field of complex numbers. Several other approaches are introduced to overcome these and additional shortcomings, culminating with J. Tits’s functorial and axiomatic approach, which provides a construction of minimal Kac-Moody groups by generators and relations. The exposition is detailed and this helps to clarify and smooth out some parts of the literature that were rather sketchy in the original papers.

The exponentiation construction does not work for imaginary root spaces, so these had to be excluded from the construction of *minimal* Kac-Moody groups. This and other concerns are addressed in the remaining chapter 8 and lead to the construction of maximal Kac-Moody groups, first over the field of complex numbers, and then over an arbitrary field by means of geometric, algebraic or scheme-theoretic completions of minimal Kac-Moody groups. The last chapter ties some loose ends, for example addressing the simplicity or isomorphism problems for Kac-Moody groups.

Two appendices collect some facts that are used in the exposition, the first one on group schemes and the second one on Coxeter groups, buildings and BN-pairs.

Kac-Moody algebras and Kac-Moody groups, being natural extensions of classical Lie algebras and linear algebraic groups, have been of constant interest. The book under review is a welcome accessible introduction to the subject. An interested reader will find the book almost self-contained, assuming only a very basic background on Lie algebras and linear algebraic groups. The book includes many exercises that ask the participation of the reader to prove some facts that are used throughout the exposition. There are also many examples that clarify concepts or results and keep the subject grounded.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is [email protected].