Reconstructing broken symmetry

考虑二维平面。它的对称性是什么？我们可能会想到旋转、反射和平移，以及它们的组合。但可能还有其他的吗？事实上，它们是有的，而且我们可以肯定，通过把这个问题变成代数问题而不是几何问题，我们已经找到了它们。通过这样做，我们发现平面的每一个对称都是线性变换（包括上面提到的所有那些）和进一步的广义（非线性）剪切的组合。转化为代数问题是通过考虑所谓的几何空间的坐标代数来实现的。空间的对称性精确地对应于坐标代数的自同构。代数的自同构是从代数到自身的映射，它保留了代数的加法和乘法结构，并且具有逆-就像对称是从空间到自身的映射，它保留了几何结构（例如角度和长度）并且是“不可做的”。一个代数的自同构的集合在复合作用下形成一个群，因此我们将原来的问题重新表述为描述空间中坐标代数的自同构群的问题。这是一个经典问题，一般来说很难。平面的例子很容易误导人，对于三维及以上的空间，自同构群已被证明包含“野”元素；也就是说，自同构不能像上面那样用初等术语来描述。所以这不是我们要解决的问题。相反，我们的兴趣在于非交换代数几何的世界。（上面提到的坐标代数是交换代数。）这里有一种众所周知但尚未被充分理解的对称性破缺现象。非交换或“量子”空间通常比它们的经典交换对应物更刚性，因为它们的对称性更少。更准确地说，非交换坐标代数通常具有较小的自同构群。这自然引出了以下问题：对称性到哪里去了？这个项目的目的是提供一个答案，表明“隐藏的”对称性可以作为空间不同量化之间的同构而恢复。用专业的语言来说，我们有一个自同构群（“有许多对象的群”），它在经典极限下还原为原始的自同构群。构造这个群样，即使是很小的例子，也需要来自纯数学领域的技术，包括非交换代数、代数几何和上同调理论等。我们的目标是在某些量化下完全理解这个类群。具体来说，我们将考虑平面、高维仿射空间和其他一些精心挑选的例子。在这样做的过程中，我们将发展可以应用于许多其他空间及其量子化的一般理论。

Consider the two-dimensional plane. What are its symmetries? We might think of rotations, reflections and translations, together with combinations of these. But might there be others? In fact, there are and we can be sure we have found them all by turning the question into a problem in algebra rather than geometry. By doing so, we find that every symmetry of the plane is a combinations of linear transformations (which include all of those mentioned above) and a further family of generalised (non-linear) shears.The translation into a problem in algebra is achieved by considering the so-called coordinate algebra of the geometric space. A symmetry of the space then precisely corresponds to an automorphism of the coordinate algebra. An automorphism of an algebra is a map from the algebra to itself that preserves the additive and multiplicative structure of the algebra and that has an inverse - just as a symmetry is a map from the space to itself which preserves geometric structure (e.g. angles and lengths) and is "undo-able". The set of automorphisms of an algebra forms a group under composition, so our original question is reformulated as one of describing the automorphism group of the coordinate algebra of our space.This is a classical problem - and is very hard in general. The example of the plane is rather misleading, as for three dimensions and above, the automorphism group has been proved to contain "wild" elements; that is, automorphisms that cannot be described in elementary terms as above.So this is not the problem we propose to address. Rather, our interests lie in the world of noncommutative algebraic geometry. (The coordinate algebras referred to above are in particular commutative algebras.) Here there is a well-known but not well understood phenomenon of symmetry breaking. Noncommutative or "quantum" spaces are usually more rigid than their classical commutative counterparts, in the sense that they have fewer symmetries. More precisely, noncommutative coordinate algebras typically have smaller automorphism groups.This leads naturally to the following question: where has the symmetry gone? The aim of this project is to provide an answer, showing that the "hidden" symmetries are recoverable as isomorphisms between different quantizations of the space. In technical language, we have an automorphism groupoid ("a group with many objects") that reduces to the original automorphism group in the classical limit. Constructing this groupoid, even for small examples, requires techniques from the spectrum of pure mathematics, includng noncommutative algebra, algebraic geometry and cohomology theory among others.Our goal is to fully understand this groupoid for certain quantizations. Specifically, we shall consider the plane, higher-dimensional affine spaces and some other carefully chosen examples. In doing so we shall develop general theory that can be applied to many further spaces and their quantizations.