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（如大爆炸或大坍缩）时，用空间连续性和基于时空的量子场论来描述宇宙的方法被打破了，编码在宇宙几何空间变化中的信息被转换成空间独立但依赖于时间的李代数变量，编码在无穷大中-酉型上的真实的分裂Kac-Moody代数的一维对称空间。在这方面的理解表示的酉形式的真实的分裂卡茨-穆迪代数的所谓类型E10是特别感兴趣的。一个这样的表示可以构造为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Berman给出的生成元和酉形式关系的表示。这个项目是在无限维Lie理论和几何群论领域内的纯数学中设置的。它将结合联合收割机的经典技术，从理论的Kac-Moody代数和Kac-Moody群在特征0和他们的酉形式与快速发展的理论的酉形式的Kac-Moody群在任意领域的基础上的理论的孪生建筑物。它的目标是有助于一个统一的结构理论的酉形式的卡茨-穆迪代数和卡茨-穆迪群的不定型。这个项目的主要重点将是有限维表示和理想和正规子群，分别对这些酉形式，从上述有限维表示宇宙学发现。

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.