Unitary forms of Kac-Moody algebras and Kac-Moody groups

现有的宇宙学理论表明，在接近宇宙奇点的时候，比如大爆炸或大爆炸，空间连续体和基于时空的量子场论对宇宙的描述被打破，宇宙几何空间变化中编码的信息被转换成空间独立但时间依赖的李代数变量，编码在一个实分裂的Kac-Moody代数的无限维对称空间上。在这种情况下，对所谓E10型的实分裂Kac-Moody代数的酉形式的表示的理解是特别有趣的。一个这样的表示可以被构造为它的A9型正则子代数的32维自旋表示的整个酉形式的扩展，使用伯曼给出的酉形式的生成表示和酉形式的关系。这个项目是在无限维李论和几何群论领域的纯数学中设置的。它将结合经典的Kac-Moody代数理论和特征0上的Kac-Moody群及其幺正形式的技术，以及基于双生建筑理论的快速发展的任意域上的Kac-Moody群的幺正形式理论。其目的是对Kac-Moody代数的酉形式和不定型Kac-Moody群的一致结构理论作出贡献。这个项目的重点将放在有限维表示上，以及这些酉形式的理想子群和正规子群上，从上面提到的在宇宙学中发现的有限维表示开始。

Existing cosmological theories suggest that, close to a cosmological singularity like a big-bang or a big-crunch, the description of the universe in terms of spatial continuum and space-time based quantum field theory breaks down and the information encoded in the spatial variation of the geometryof the universe gets transferred into spatially independent but time-dependent Lie-algebraic variables encoded in the infinite-dimensional symmetric space of a real split Kac-Moody algebra over its unitary form. In this context the understanding of representations of the unitary form of the real split Kac-Moody algebra of so-called type E10 is of particular interest. One such representation can be constructed as an extension to the whole unitary form of the 32-dimensional spin representation of its regular subalgebra of type A9, using a presentation by generators and relations of unitary forms given by Berman.This project is set in pure mathematics within the areas of infinite-dimensional Lie theory and geometric group theory. It will combine classical techniques from the theory of Kac-Moody algebras and Kac-Moody groups in characteristic 0 and their unitary forms with the quickly developing theory of unitary forms of Kac-Moody groups over arbitrary fields based on the theory of twin buildings. Its goal is to contribute to a uniform structure theory of unitary forms of Kac-Moody algebras and of Kac-Moody groups of indefinite type. The main emphasis of this project will be on finite-dimensional representations and on ideals and normal subgroups, respectively, of these unitary forms, starting with the above-mentioned finite-dimensional representation discovered in cosmology.