On conjugacy of Cartan subalgebras in non-fgc Lie tori

We establish the conjugacy of Cartan subalgebras for generic Lie tori "of type A". This is the only conjugacy problem of Lie tori related to Extended Affine Lie Algebras that remained open.Comment: 28 pages, to be published in Transformation Group

On conjugacy of Cartan subalgebras in extended affine Lie algebras

That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate. For symmetrizable Kac-Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac-Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac-Moody Lie algebras

Conjugacy classes of trialitarian automorphisms and symmetric compositions

The trialitarian automorphisms considered in this paper are the outer automorphisms of order 3 of adjoint classical groups of type D_4 over arbitrary fields. A one-to-one correspondence is established between their conjugacy classes and similarity classes of symmetric compositions on 8-dimensional quadratic spaces. Using the known classification of symmetric compositions, we distinguish two conjugacy classes of trialitarian automorphisms over algebraically closed fields. For type I, the group of fixed points is of type G_2, whereas it is of type A_2 for trialitarian automorphisms of type II

Representation varieties of the fundamental groups of compact non-orientable surfaces

Let $\Gamma_g$ be the fundamental group of compact non-orientable surface and K be an algebraically closed field of chracteristic 0. In the paper we describe the structure of representation varieties of the group $\Gamma_g$ into full linear group and into special linear group and their character varieties; namely it is determined a number of their irreducible components, their dimensions and investigated their birational properties, in particular it is proved that all irreducible components of the representation variety into full linear group are Q-rational varieties