论文标题
广义量子包裹代数，彩色的Kac-Moody代数和Langlands插值
Generalised Quantum Enveloping Algebras, Coloured Kac-Moody Algebras, and Langlands Interpolation
论文作者
论文摘要
我们在本文中提出了Kac-Moody代数及其表示的新变形过程。变形方向由称为着色的数字集合给出。自然数将导致经典代数导致，而量子数则导致相关的量子代数。我们首先在着色上建立足够和必要的条件，以允许该过程在多项式上取决于形式参数，并提供广义的量子包络（GQE）代数。然后，我们提高限制，并表明该过程仍然通过彩色的KAC Moody代数存在。我们制定了GQE猜想，该猜想预测Kac-Moody代数类别中的每个表示形式都可以变形为相关的GQE代数的表示。我们给出了这种猜想的各种证据，并通过证明没有塞雷关系的Kac-Moody代数可以将其变形为GQE代数而没有塞雷关系，从而朝着其解决方案迈出了第一步。如果猜想成立，我们为彩色的kac-moody代数建立了一个模拟结果，我们证明了变形的表示理论与经典的代数平行，我们向GQE代数的变形serre表示，我们提出的后者是后者是kac-mood ys fin fin and fin and fin的代表。作为一种应用，我们用插入式lie代数的代表之间的插值来解释，并且我们提出了一种新方法，旨在证明Frenkel-Hernandez的猜想。通常，我们证明，可以通过第三个代表来插值两个等源性的kac-moody代数的表示。观察到标准量子代数满足GQE的猜想，我们给出了前面提到的经典兰兰兹二元性的新证明。
We propose in this thesis a new deformation process of Kac-Moody algebras and their representations. The direction of deformation is given by a collection of numbers, called a colouring. The natural numbers lead for example to the classical algebras, while the quantum numbers lead to the associated quantum algebras. We first establish sufficient and necessary conditions on colourings to allow the process depend polynomially on a formal parameter and to provide the generalised quantum enveloping (GQE) algebras. We then lift the restrictions and show that the process still exists via the coloured Kac Moody algebras. We formulate the GQE conjecture which predicts that every representation in the category Oint of a Kac-Moody algebra can be deformed into a representation of an associated GQE algebra. We give various evidences for this conjecture and make a first step towards its resolution by proving that Kac-Moody algebras without Serre relations can be deformed into GQE algebras without Serre relations. In case the conjecture holds, we establish an analog result for coloured Kac-Moody algebras, we prove that the deformed representation theories are parallel to the classical one, we explicit a deformed Serre presentation for GQE algebras, we prove that the latter are the representatives of a natural class of formal deformations of Kac-Moody algebras and are h-trivial in finite type. As an application, we explain in terms of interpolation both classical and quantum Langlands dualities between representations of Lie algebras, and we propose a new approach which aims at proving a conjecture of Frenkel-Hernandez. In general, we prove that representations of two isogenic coloured Kac-Moody algebras can be interpolated by representations of a third one. Observing that standard quantum algebras satisfy the GQE conjecture, we give a new proof of the previously mentioned classical Langlands duality.
