Exceptional Lie algebras from number theory

09 Nov 2015 NUS mathematicians describe one powerful method called the local theta correspondence or Howe correspondence.

Abstract:

Let g be a simple Lie algebra of type F₄or E_ndefined over a local or global

field k of characteristic zero. We show that g can be obtained by the Tits’ construction from an octonion algebra O and a cubic Jordan algebra J.

In mathematics and physics, a Lie group is a smooth differentiable object which exhibits highly symmetric features called the group structure. Lie groups could be studied both from the point of views of algebra and analysis. It is these close inter-connections between algebra and analysis that give Lie groups very rich mathematical structures. Closely related to a Lie group is its Lie algebra, which is defined as the vector space of left invariant tangent fields on the Lie group. Lie algebras in general are complicated objects so mathematicians restrict their attention a smaller class of Lie algebras called reductive Lie algebras. These Lie algebras are built up from simple Lie algebras.

The classification of simple Lie algebras was obtained by Elie Cartan in his PhD thesis in 1894. He divided them into types A to G. Types A to D are well known simple Lie algebras like the trace zero n by n matrices, the skew symmetric n by n matrices etc which are taught in an undergraduate linear algebra class. These are known as the classical Lie algebras. The modern proof of the classification theorem uses the theory of roots which eventually reduces the classification problem to some graphs called the Dynkin diagrams. There are three Dynkin diagrams of type E which we illustrate below_:

Lie algebras of types E, F and G are called the exceptional Lie algebras and they are more mysterious. Jacques Tits later gave a uniform construction of the exceptional Lie algebras of type E and F. He constructed such a Lie algebra g using an octonion k-algebra O and a central simple cubic Jordan k-algebra J. This construction is now known as Tits’ construction. Up to this point, the base field k is either the real numbers or the complex numbers. On the other hand, the construction above works fine if we replace the base field by any other field. In particular replacing it with the p-adic numbers or the rational numbers reveals many deep and interesting results in number theory.

A team led by Prof LOKE Hung Yean from the Department of Mathematics in NUS together with Prof Gordan SAVIN from the University of Utah, they concentrated on the Lie algebras of types E and F. If the base field k is algebraically closed, then each Dynkin diagram corresponds to exactly one Lie algebra. However over a non-algebraically closed field, there are usually more than one Lie algebra which they call the rational forms of the Lie algebra. One question is whether Tits’ construction gives all rational forms of the exceptional Lie algebras of types E and F.

In a joint paper with Prof Savin, they show that Tits’ construction indeed gives all the rational forms of type E and F if the field k is a local field or global field of characteristic 0. In order to further explain our results, we need to get a little technical. In the case of type E and the field k is a non-archimedean local field, different pairs of O and J give different rational forms of the Lie algebra g. In the case of type E and the field k is a global field, different pairs of O and J do not necessarily give different forms of the Lie algebra g, but the fiber is reflected at the achimedean places.

The result quoted above also gives a classification of dual pairs d x h in the exceptional Lie algebra g where d and h are the derivation Lie algebras of the octonion algebra O and the Jordan algebra J respectively. In the research on exceptional dual pair correspondences, one restricts a representation of an exceptional Lie group to a dual pair. It is useful to know what are the possible exceptional dual pairs that one encounters.

References

1. E Cartan. “Sur la structure des groupes de transformations finis et continus.” Thèse Paris, 1894; 2d éd., Paris, Vuibert, 1933.

2. Loke HY, G Savin. “Rational forms of exceptional dual pairs.” Journal of Algebra, 422 (2015).arXiv:1212.1957.

3. J Tits. “Algebres alternatives, algebres de Jordan, et algebres de Lie exceptionelle.” Nederl. Akad. Weten. Proc. Ser. A, 69 (1966) 223.