Abstract

The special orthogonal group SO(n,R) is a maximal compact subgroup of SL(n,R). Its Lie algebra therefore is called a maximal compact subalgebra of sl(n,R), and it can be characterized as the fixed point set of the Cartan-Chevalley involution sending a matrix to minus its transpose. Kac-Moody algebras were introduced in the 1960’s to generalise complex semisimple Lie algebras and have since then found applications in theoretical physics. For a Kac-Moody algebra one can similarly define its maximal compact subalgebra as the fixed points of the involution. In the case of E(10), theoretical physicists have discovered that the spin representation of so(10) can be extended to a representation of the maximal compact subalgebra of E(10). In this talk, we discuss this representation and introduce a general framework which encompasses it. With the help of these so-called generalized spin representations, we derive some algebraic properties of maximal compact subalgebras of simply-laced Kac-Moody algebras. This is joint work with Ralf Gramlich.