By using the words of the authors themselves, this book tackles the task of “collecting, unifying and expanding the knowledge about gradings on simple Lie algebras”.
The first chapter contains some general definitions and results on gradings. General gradings are distinguished from group gradings and the latter are the fundamental objects of study in the book. Then, notions as transcendental as universal grading group, fine grading, equivalence and isomorphism are introduced. The connection with the representation theory of diagonalizable group schemes is also treated in this chapter.
In chapter two, the associative case is studied. It is proved that any graded simple associative algebra \(R\) satisfying chain conditions on graded one-sided ideals is isomorphic to a matrix algebra \(M_n(D)\) where \(D\) is a graded division algebra and the grading on \(R\) is determined by the one in \(D\) plus a \(n\)-tuple of elements from the grading group. Then, they reduce the classification problem to that of classifying graded division algebras \(D\) and this goal is achieved in the second section. Roughly speaking \(D\) is a crossed product \(\Delta\ast T\) where \(\Delta\) is a division algebra and \(T\) is a group. In case \(D\) is finite-dimensional and the ground field algebraically closed then \(\Delta\) is the ground field and \(T\) finite. In the last section of this chapter the previous results are joined together to obtain a classification of abelian group gradings on a matrix algebra over an algebraically closed field.
In chapter three we go into the ground of Lie algebras. The main results include a classification of gradings on simple Lie algebras of type \(A_r\), \(B_r\), \(C_r\) or \(D_r\) (ruling out \(D_4\) which is considered in chapter six) over an algebraically closed field of characteristic other than \(2\). For a given abelian group \(G\) all gradings up to isomorphism are described; and all fine gradings up to equivalence. As in previous chapters a pleyade of previously published works on the topic are mentioned in the book. Some of them go beyond the limits of finite-dimensional algebras, and also some of them go out of the algebraically closed case (for instance considering the case of real algebras). The methodology in this chapter includes a realization of the Lie algebra under question as a matrix algebra which allows to apply the results in previous chapters. For the algebras over fields of prime characteristic, their automorphism group schemes have been needed. Thus a number of results concerning this objects (with further application in other chapters) are also provided.
Chapter four is devoted to the description of gradings on a simple Lie algebra of type \(G_2\) over a field of characteristic different from \(2\) and \(3\). For a given abelian group \(G\), all gradings up to isomorphism are described; and all the fine gradings up to equivalence. The key point here is the realization of the Lie algebra under study as the Lie algebra of derivations of a Cayley algebra and the application of transfer theorems from chapter one. Thus, the work of the first author in 1998 [J. Algebra 207, No. 1, 342–354 (1998; Zbl 0915.17022)] classifying the gradings on octonion algebras is a fundamental reference, and also the work by C. Draper and C. Martín in 2006 [Linear Algebra Appl. 418, No. 1, 85–111 (2006; Zbl 1146.17027)] dealing with the gradings on \(G_2\) in the complex case. Three years later (see the work of Yu. A. Bahturin and M. V. Tvalavadze in 2009 [Commun. Algebra 37, No. 3, 885–893 (2009; Zbl 1190.17005)]) the gradings on \(G_2\) over an algebraically closed field of characteristic zero were described. These results were extended by the authors in 2012 to the case of algebras of type \(G_2\) over arbitrary fields of characteristic other than \(2\) and \(3\). The last section of this chapter studies gradings on symmetric composition algebras. This had been classified in 2009 by the first author [J. Algebra 322, No. 10, 3542–3579 (2009; Zbl 1216.17004)].
The aim of chapter five is the classification of gradings on a simple Lie algebra \(L\) of type \(F_4\) over an algebraically closed field of characteristic other than \(2\) (as before gradings by a fixed group \(G\) up to isomorphism and fine gradings up to equivalence). Under suitable condition one can find an isomorphism between the affine group schemes of automorphisms of the Albert algebra and that of \(L\). Taking this into account, the strategy here is to classify gradings on the Albert algebra first and then use automorphism group schemes to transfer the classification of grading on the Albert algebra to gradings on the Lie algebra \(L\). The first classification of fine gradings on both the Albert algebra and the Lie algebra \(F_4\) (over an algebraically closed field of characteristic zero) was obtained in 2009 by C. Draper and C. Martín [Rev. Mat. Iberoam. 25, No. 3, 841–908 (2009; Zbl 1281.17035)]. In this work some computational techniques were used but in 2012 a second work by C. Draper appeared eliminating that computational tools [Rev. Mat. Iberoam. 28, No. 1, 273–296 (2012; Zbl 1257.17039)]. On the other hand the authors published also in 2012 a couple of papers in which they definitively close the classification question for Albert algebras and \(F_4\) type Lie algebras over fields of characteristic other than \(2\).
Chapter six deals with a number of classification problems of gradings and constructions making possible the transference of gradings. For instance the fine gradings (up to equivalence) of \(D_4\) over an algebraically closed field of characteristic zero are described (fine outer gradings on \(D_4\) had been previously studied in 2010 by C. Draper, C. Martín and A. Viruel [Forum Math. 22, No. 5, 863–877 (2010; Zbl 1229.17032)]). The Tits construction relating simple Lie algebras of types \(F_4\), \(E_6\), \(E_7\) and \(E_8\) to Hurwitz algebras and simple Jordan algebras of degree three, is also presented together with another construction which starts from two symmetric composition algebras and produces among others the exceptional algebras mentioned above (with the advantage of including the characteristic three case with minor variations related to \(E_6\) in the exceptional case). The first author published in 2004 the previously mentioned construction based on two symmetric composition algebras which now is fruitfully applied for transferring gradings from symmetric algebras to others.
The classification up to equivalence on gradings on \(E_6\) in characteristic zero, due to C. Draper and A. Viruel, is explained also here. The last section of this chapter presents a table of fine gradings on the simple Lie algebras of types \(E_i\) (\(i=6,7,8\)) which are related to gradings on composition, Jordan and structurable algebras appearing in their construction. This chapter also contains a summary of known fine gradings on the Lie algebras of types \(E_6\), \(E_7\) and \(E_8\). The gradings on \(E_6\) shown in this chapter exhaust the class of fine gradings on \(E_6\), however the classification question of the fine gradings on \(E_7\) and \(E_8\) is still open though a definitive result is expected to appear before long.
Finally the book contains an appendix on affine group schemes which is quite useful since this tool is in the heart itself of the theory of group gradings.