Maximal abelian diagonalizable groups and fine gradings on simple Lie algebras

Abstract

The oldest and best known grading on a (semisimple) Lie algebra is the root space decomposition with respect to a maximal torus. This is a grading by a free abelian group (the root lattice) and it is \emph{fine} in the sense that it cannot be refined. In general, there is a bijective correspondence between fine gradings by abelian groups on any finite-dimensional algebra over an algebraically closed field $\mathbb{F}$ of characteristic zero and maximal abelian diagonalizable subgroups (known as \emph{MAD-subgroups}) of the automorphism group of the algebra. This has been one of the approaches to finding all fine gradings on simple (finite-dimensional) Lie algebras over $\mathbb{F}$, which is an important problem concerning Lie algebras. The computation of the MAD-subgroups of simple Lie (or algebraic) groups is also of independent interest in the theory of Lie groups. The work \cite{LGII} gives a complete list of MAD-subgroups of the group $\mathrm{Aut}( \mathfrak{gl}(n,\mathbb{C}))\cong \mathrm{PGL}(n,\mathbb{C})\rtimes \langle \tau\rangle$, as well as precise indications about how to obtain the MAD-subgroups for the Lie algebras of all classical types (except $\mathfrak{d}_4$). However, it is not easy to obtain the corresponding fine gradings from this list and to decide which of them are equivalent to each other (in other words, which of the MAD-subgroups are conjugate). More conceptual approaches to the study of fine gradings were proposed by several authors, including Y. Bahturin and M. Kotchetov \cite{BahKotch}, A. Elduque \cite{Eld}, and, for some exceptional cases, C. Mart\'\i n, A. Viruel and myself \cite{f4spin,e6}. Most of this material is presented in the monograph \cite{Book}, published two years ago, including the solution of the classification problem for fine gradings on simple Lie algebras other than the exceptional Lie algebras $\mathfrak{e}_7$ and $\mathfrak{e}_8$. For the latter, conjectural lists of fine gradings were given. These lists turned out to be correct, in view of the recent results announced by Jun Yu \cite{Yu} who classified the conjugacy classes of a certain set of abelian subgroups (containing the MAD-subgroups) of compact simple Lie groups and taking into account that the classification problem over $\mathbb{F}$ is equivalent to the one over complex numbers \cite{bastaC}. Thus, the classification of fine gradings (up to equivalence) on finite-dimensional simple Lie algebras is now complete. The various viewpoints taken in the above works have their own merits, but it is not easy to recognize that the classification results agree with each other. Thus, it is useful to have a mental picture of MAD-groups and fine gradings at the same time. In particular, this helps us to determine which of the representations of a simple Lie algebra are compatible with a given grading. In this talk, we will first review the general concepts, examples and motivation of gradings on Lie algebras, and then we will focus on the Lie algebras $\mathfrak{g}_2$, $\mathfrak{f}_4$ and $\mathfrak{e}_6$ to illustrate the different approaches, emphasizing the correspondence between MAD-groups and fine gradings.