Categorical representation theory

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 42

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• 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.

مفاهيم خاطئة شائعة

• 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.

اكتشافات اختيارية

• ← Quiver representations

قراءات إضافية

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories (2015)
• Volodymyr Mazorchuk, Lectures on Algebraic Categorification (2012)