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Title Representations of rational Cherednik algebras : Koszulness and localisation
Publication Date
Degree PhD
Degree Level doctoral
University/Publisher University of Edinburgh
Abstract An algebra is a typical object of study in pure mathematics. Take a collection of numbers (for example, all whole numbers or all decimal numbers). Inside, you can add and multiply, but with respect to these operations different collections can behave differently. Here is an example of what I mean by this. The collection of whole numbers is called Z. Starting anywhere in Z you can get to anywhere else by adding other members of the collection: 9 + (-3) + (-6) = 0. This is not true with multiplication; to get from 5 to 1 you would need to multiply by 1/5 and 1/5 doesn’t exist in the restricted universe of Z. Enter R, the collection of all numbers that can be written as decimals. Now, if you start anywhere—apart from 0—you can get to anywhere else by multiplying by members of R—if you start at zero you’re stuck there. By adjusting what you mean by ‘add’ and ‘multiply’, you can add and multiply other things too, like polynomials, transformations or even symmetries. Some of these collections look different, but behave in similar ways and some look the same but are subtly different. By defining an algebra to be any collection of things with a rule to add and multiply in a sensible way, all of these examples (and many more you can’t imagine) can be treated in general. This is the power of abstraction: proving that an arbitrary algebra, A, has some property implies that every conceivable algebra (including Z and R) has that property too. In order to start navigating this universe of algebras it is useful to group them together by their behaviour or by how they are constructed. For example, R belongs to a class called simple algebras. There are mental laboratories full of machinery used to construct new and interesting algebras from old ones. One recipe, invented by Ivan Cherednik in 1993, produces Cherednik algebras. Attached to each algebra is a collection of modules (also called representations). As shadows are to a sculpture, each module is a simplified version of the algebra, with a taste of its internal structure. They are not algebras in their own right: they have no sense of multiplication, only addition. Being individually simple, modules are often much easier to study than the algebra itself. However, everything that is interesting about an algebra is captured by the collective behaviour of its modules. The analogy fails here: for example, shadows encode no information about colour. Sometimes the interplay between its modules leads to subtle and unexpected insights about the algebra itself. Nobody understands what the modules for Cherednik algebras look like. One first step is to simplify the problem by only considering modules which behave ‘nicely’. This is what is referred to as category O. Being Koszul is a rare property of an algebra that greatly helps to describe its behaviour. Also, each Koszul algebra is mysteriously linked with another called its Koszul dual. One of the main results of the thesis is that, in some cases, the modules in category O behave as if they were the modules for some Koszul algebra. It…
Subjects/Keywords 512; algebra ; Cherednik algebras ; Koszul ; geometers ; sheaves
Rights Full text available
Country of Publication uk
Record ID handle:1842/8854
Repository ethos
Date Retrieved
Date Indexed 2019-08-12

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