Topics in lie theory: tensor categories

Lectures: 2 sessions / week, 1.5 hours / session

An important feature of category theory is that it allows one to imitate, at a higher level, many classical notions of elementary algebra. "Higher level" means that one allows maps (morphisms) not only between algebraic structures, but also between their elements. Such imitation is called categorification. The very notion of a category is a categorification, in this sense, of the notion of a set. Further, monoidal categories categorify monoids, and tensor categories (i.e., monoidal categories with a compatible additive structure), categorify associative rings.

Tensor categories are ubiquitous in mathematics. They arise in representation theory (representation categories of classical and quantum groups), algebraic geometry (categories of coherent sheaves on algebraic varieties, categories of local systems, categories of motives), topology (topological quantum field theory, invariants of knots, links, and 3-manifolds) the theory of operator algebras (bimodule categories for subfactors), 2-dimensional conformal field theory (fusion categories of modules over a vertex operator algebra), quantum statistical mechanics (nonabelian anyons in the quantum Hall effect), etc.

This course will be an attempt to give a detailed introduction to the theory of tensor categories and to review some of its connections to other subjects, time permitting (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.

We will develop from scratch the theory of monoidal and tensor categories and module categories, covering the following topics:

Before beginning this course, you are expected to know basic category theory, basic algebra, and fundamentals of group representations. Hopf algebras will often appear in the course, but no previous knowledge of them is expected.

There will be five homework problem sets assigned during the course. It is ok to collaborate on homework if you creatively participate in solving it and understand what you write.

Monoidal functors

MacLane's strictness theorem

MacLane coherence theorem

Rigid monoidal categories

Invertible objects

Tensor and multitensor categories

Tensor product and tensor functors

Unit object

Grothendieck rings

Groupoids

Finite abelian categories

Fiber functors

Coalgebras

Quantum groups

Skew-primitive elements

Pointed tensor categories

Coradical filtration

Chevalley's theorem and Chevalley property

Andruskeiwitsch-Schneider conjecture

Cartier-Kostant theorem

Quasi-bialgebras and quasi-Hopf algebras

Quantum traces

Pivotal categories and dimensions

Spherical categories

Multitensor cateogries

Multifusion rings

Frobenius-Perron theorem

Tensor categories

Deligne's tensor product

Finite (multi)tensor categories

Categorical freeness

Distinguished invertible object

Integrals in quasi-Hopf algebras

Cartan matrix

Basics of Module categories

Exact module categories

Algebras in categories

Internal Hom

Main Theorem

Categories of module functors

Dual categories

Don't show me this again

This is one of over 2,200 courses on OCW. Find materials for this course in the pages linked along the left.

MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.

No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.

Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.

Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse (just remember to cite OCW as the source.)

Learn more at Get Started with MIT OpenCourseWare