研究路径Category Theory计划中
Categorical representation theory
Optional research direction: representations on categories rather than vector spaces, including tensor and fusion categories, module categories, categorical actions and categorification, such as Chuang–Rouquier sl₂-categorification and Khovanov homology.
math.cat.categorical-representation-theory
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前置知识
42 项必需的前置知识（一直追溯到底）
你将学会
- Explain categorification through examples: from natural numbers to finite sets, from Euler characteristic to homology, and from the Jones polynomial to Khovanov homology.
- Define tensor and fusion categories and module categories over them, with Rep(G) as an example.
- Read a survey on categorical actions and separate established results from open problems.
定义
- Categorification replaces a set-based structure by a category-based one from which the original can be recovered, often by passing to a Grothendieck group.
- A fusion category is a rigid semisimple k-linear monoidal category with finitely many simple objects and a simple unit object.
- The Grothendieck group K₀(A) of an abelian category is generated by classes [X] of objects, with [B] = [A] + [C] for every short exact sequence 0 → A → B → C → 0.
定理
- For a finite group G, Rep(G) over ℂ is a fusion category whose Grothendieck ring is the representation ring of G.
- Khovanov homology is a link invariant whose graded Euler characteristic is the (unnormalised) Jones polynomial.
- Chuang and Rouquier used sl₂-categorification to prove Broué's abelian defect group conjecture for symmetric groups.
常见误解
- Thinking categorification is a single construction; it is a family of methods, and a structure may have many categorifications or none known.
- Assuming the Grothendieck group remembers everything about a category; much information is lost.
可选发现
延伸阅读
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories (2015)
- Volodymyr Mazorchuk, Lectures on Algebraic Categorification (2012)