The goal of this seminar is to understand the recent approaches to categorical actions of (and categorifications of) Kac-Moody lie algebras.
When given a categorical setup, any statements about its decategorification are really statements about objects in the category. Starting with Chuang and Rouquier's 2005 paper on derived equivalences, people began to realize the additional power one gains over a categorification by understanding the natural transformations between functors. This led to the simultaneous discovery by Rouquier and Khovanov-Lauda of a collection of algebras whose module categories categorify the positive half of the quantum group of any Kac-Moody lie algebra. These algebras are now called quiver Hecke algebras, or KLR algebras after their discoverers. In addition, both Khovanov-Lauda and Rouquier came up with independent and subtly different methods of discussing categorification of the entire quantum group U_q(g).
Calendar of talks:
| September | |||
| 13 |
Harvard |
Kazhdan |
Introduction |
| 20 |
MIT |
Elias | sl_2 categorification and the NilHecke algebra |
| 27 |
Harvard |
Elias | sl_2 categorification and the NilHecke algebra redux |
| October | |||
| November | |||
| 1 |
MIT |
Elias | The "cellular" filtration and isotypic categorification |
| 8 |
Harvard |
Ben Webster | Categorifications of tensor products (for sl_2) |
| 22 |
Harvard |
Ben Webster | TBA |
| December | |||
| 6 |
Harvard |
Ben Webster | TBA |
The authors look at some of the standard categorifications of sl_2 representations, and notice that cyclotomic (degenerate affine) Hecke algebras act as endotransformations of iterated functors E^n. They then examine the theory of categorical setups (abelian categories with raising and lowering functors which are biadjoint) where such endotransformations are present, and prove structural results. A categorified representation has a filtration whose subquotients categorify isotypic components, and categorifications of isotypic components have a nice form (i.e. there is a canonical categorification of an irreducible). The action of the Weyl reflection lifts to an equivalence of homotopy categories. This particular statement is then applied to representations of symmetric groups in characteristic p in order to prove the Broue conjecture. It is also applied to category O in finite characteristic, proving a conjecture of Rickard. This paper is heavily algebraic. Where linear algebra is used to prove statements about sl_2 representations, homological algebra proves statements about categorical sl_2 representations.
There are 3 1-hour lectures by Chuang as part of this conference. The lecture series by Kleshchev and Rouquier are also relevant to different aspects of this seminar.
Instead of using cyclotomic degenerate affine Hecke algebras, one can use cyclotomic NilHecke algebras to categorify sl_2 instead. If one merely uses NilHecke algebras without taking the cyclotomic quotient, one will categorify the positive half of U_q(sl_2). This is done in the first paper above, which also provides the generalization of NilHecke algebras to other simply-laced types. In the second paper, Lauda gives a larger 2-category which categorifies the entirety of U_q(sl_2), and therefore understands the natural transformations between combinations of functors E and F, not just between powers of E. The third paper describes the canonical categorification of the irreducible modules for sl_2, using the action on grassmanians of the cyclotomic NilHecke algebra. These papers use mostly diagrammatic methods.
An expository paper. Very readable.
Now we generalize from sl_2 to an arbitrary Cartan datum. The first and third papers provide roughly identical algebras which categorify the positive half of the quantum group U^+_q(g); these algebras are now called quiver Hecke algebras, or KLR algebras. The second and third paper give separate notions of what it means to have a categorical action of the whole quantum group. The second paper generalizes the previous algebra of Lauda to give a 2-category which categorifies U_q(g) as an algebra (in type A, conjecturally in other types), while the third paper only assumes the existence of a biadjunction without specifying the morphisms between E and F precisely. As of now, it is not known that these notions are equivalent. The third paper explores more deeply the structure of categorifications, and provides many results similar to Chuang and Rouquier.
This paper shows that quiver Hecke algebras describe precisely the morphisms between certain perverse sheaves on quiver varieties, and thus the indecomposables in the Khovanov-Lauda category descend to the canonical basis of U^+_q.
This paper contains the proof that the cyclotomic quiver Hecke quotients do categorify the appropriate irreducible modules. In addition, the crystal structure is given as follows: the vertices correspond to simple modules, and the raising operators come from cosocles of inductions, while the lowering operators come from socles of restrictions.
The first two papers give a geometric categorification of tensor products of irreducible modules, first for sl_2 and then for the symmetric Kac-Moody case. The main tool is a localization of a certain category of perverse sheaves on quiver varieties. This is inspired by constructions of Lusztig and Nakajima. Finding the simple perverse sheaves in this context would be difficult, but Webster realized that one can construct a number of natural perverse sheaves which have these simples as direct summands. The third paper realizes the morphisms between these sheaves diagrammatically. The third and fourth papers apply this categorification of tensor products to categorifying knot invariants.
This paper gives a combinatorial realization of the crystal structure from the Lauda-Vazirani paper above. There are numerous other papers by these authors on the crystal structure. For more on the crystal structure, also see the papers of Lauda-Vazirani and Webster above.
The authors define the notion of a geometric sl_2 categorification: instead of specifying certain morphisms between functors, they specify that these functors roughly admit a one-parameter "deformation." This will imply (and explain) the existence of certain morphisms, giving a Chuang-Rouquier style categorification. For more general g, the functors should admit a deformation with base space given by the cartan subalgebra h. The first paper gives definitions, and the latter two check that the standard actions on Grassmanians and quiver varieties meet this description. The authors have several other papers too.
A formula for the nilpotency of the dot on a particular strand, using combinatorics and "antigravity." Some good fun computational practice.
To categorify a central element of an algebra, one typically desires a central object in a category - which is significantly more data than just an object, consisting of a collection of natural isomorphisms. The authors categorify the casimir using a complex of functors in Khovanov and Lauda's 2-category, and provide the needed maps to show that this complex of functors is central.
When dealing with diagrammatic categories one often needs to take the Karoubi envelope, which sweeps many calculations under the rug. The authors here give an explicit description of that Karoubi envelope, which involves numerous tricky formulae involving Schur polynomials.
The title says it all.
I think this is the paper which states that representations of the symmetric group in characteristic p give you an ^sl_p categorification. This is the categorification mainly studied by Chuang-Rouquier. The natural transformations within give the degenerate affine hecke algebra.
I think this is the same story except for modular representations of general linear groups (also studied in Chuang-Rouquier).
Here it is shown explicitly that cyclotomic Hecke algebras and cyclotomic quiver Hecke algebras are isomorphic, linking the two styles of categorification. In particular, this puts a secret Z-grading on cyclotomic Hecke algebras and even symmetric groups. The authors have numerous other papers on the topic. See also Kleshchev's talks at the INI.
The most important tricks here, for both equivariant cohomology and equivariant derived categories, are the induction and restriction principles. Learn those and you can compute the cohomology for most of the interesting spaces in categorification theory. One also needs to understand chern classes and the cohomology of flag varieties. References to come once I find them.
The classic text on the equivariant derived category.
This talk will give an introduction to the work of Chuang and Rouquier.
Abstract: We give an introduction to sl_2 categorification. Roughly
speaking, the action of the raising and lowering operators e and f
on a representation V are lifted to raising and lowering functors E
and F on a category V. We start by describing the two most
fundamental examples in the literature: Grassmanian chains, and
p-induction for Symmetric groups. Examining these representations,
Chuang and Rouquier made the fundamental and paradigm-changing
observation that specifying the structure of the morphisms between
functors will imbue the sl_2 categorification with richer
structure. This led to the discovery of quiver Hecke algebras, and
categorifications of quantum groups themselves (in addition to various
representations).
There are two distinct and important features to a categorical sl_2
action: the categorified action of U^+(sl_2), and the biadjoint
operators which give the action of U^-(sl_2). For the former, we give
the easier algebraic version using the NilHecke algebras (from the
Grassmanian example) instead of degenerate affine Hecke algebras (from
the p-induction example). These algebras allow one to define functors
E^(n) which categorify the divided powers e^(n). Adding biadjoint
operators into the mix, we can construct complexes of functors
E^(n)F^(m) (where differentials come from counits of adjunction) which
categorify the action of the Weyl group reflection.
Etc.
Sep 20, Ben Elias: sl_2 categorification and the NilHecke algebra
Sep 27, Ben Elias: sl_2 categorification and the NilHecke algebra redux