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The Lumenary
Njia za utafitiCategory TheoryImepangwa

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

Onyesha kila kinachohitajika kufikia mada hii

Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.

Inajengwa juu ya

Higher categoriesAbelian categoriesRepresentation theory of finite groups

Masharti ya awali ya lazima 42 (hadi chini kabisa)

Utajifunza

  • 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.

Fasili

  • 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.

Nadharia

  • 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.

Mawazo potofu ya kawaida

  • 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.

Ugunduzi wa hiari

Usomaji zaidi

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