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

现有的宇宙学理论表明，在接近大爆炸或大压缩的宇宙学奇点附近，用空间连续统和基于时空的量子场论来描述宇宙的方法被打破，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，这些变量编码在实分裂Kac-Moody代数的统一形式上的无限维对称空间中。在这种情况下，理解所谓的E10型实分裂Kac-Moody代数的么正形式的表示是特别有趣的。利用生成元的表示和Berman给出的酉型关系，可以构造一个这样的表示，作为它的A9型正则子代数的32维自旋表示的整个酉形的推广.这个项目是在无限维李论和几何群论领域内的纯数学中设置的.它将把特征为0的Kac-Moody代数和Kac-Moody群及其酉型理论中的经典技巧与迅速发展的基于孪生建筑理论的任意域上Kac-Moody群的酉型理论结合起来。它的目的是对Kac-Moody代数和不定型Kac-Moody群的酉型的一致结构理论作出贡献。这个项目的主要重点将分别放在这些酉型的有限维表示和理想和正规子群上，从上述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.