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.Part of the properties of an EALA (E, H) is a root space decomposition: E = α∈Ψ E α with E 0 = H. The "root system" Ψ is an example of an extended affine root system. The main question, of course, is whether Ψ is an invariant of E. In other words, if H ′ is a subalgebra of E for which the pair (E, H ′ ) is given an EALA structure, is the resulting root system Ψ ′ isomorphic (in the sense of [extended affine] root systems) to Ψ? That this is true follows immediately from the main result of our paper. 0.1. Theorem (Theorem 7.6). Let (E, H) be an extended affine Lie algebra of fgc type. Assume E admits the second structure (E, H ′ ) of an extended affine Lie algebra. Then H and H ′ are conjugate, i.e., there exists a k-linear automorphism f of the Lie algebra E such that f (H) = H ′ .

Brain lesions, including stroke and tumours, have a high degree of variability in terms of location, size, intensity and form, making automatic segmentation difficult. We propose an improvement to existing segmentation methods by exploiting the bilateral quasi-symmetry of healthy brains, which breaks down when lesions are present. Specifically, we use nonlinear registration of a neuroimage to a reflected version of itself ("reflective registration") to determine for each voxel its homologous (corresponding) voxel in the other hemisphere. A patch around the homologous voxel is added as a set of new features to the segmentation algorithm. To evaluate this method, we implemented two different CNN-based multimodal MRI stroke lesion segmentation algorithms, and then augmented them by adding extra symmetry features using the reflective registration method described above. For each architecture, we compared the performance with and without symmetry augmentation, on the SISS Training dataset of the Ischemic Stroke Lesion Segmentation Challenge (ISLES) 2015 challenge. Using affine reflective registration improves performance over baseline, but nonlinear reflective registration gives significantly better results: an improvement in Dice coefficient of 13 percentage points over baseline for one architecture and 9 points for the other. We argue for the broad applicability of adding symmetric features to existing segmentation algorithms, specifically using nonlinear, template-free methods.

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings.

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We construct a counterexample to the conjugacy of maximal abelian diagonalizable subalgebras in extended affine Lie algebras.