Theory and Applications of Categories, Vol. 22, No. 19, 2009, pp. 479–508.

A BRIEF REVIEW OF ABELIAN CATEGORIFICATIONS

MIKHAIL KHOVANOV, VOLODYMYR MAZORCHUK CATHARINA STROPPEL

Abstract. This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra 𝔰𝔩_{n}. The review is intended to give non-experts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles.

1. Introduction

The idea of categorification goes back to Crane and Frenkel [28] and became more and more popular in recent years. In the present review we want to give an idea what abelian categorifications should be and present some examples naturally arising in representation theory. We give references to related categorifications arising in knot theory as well as in symplectic geometry.

In the following we will give a definition of abelian categorifications. To get a feeling for this concept it might be useful to view it as an ”inverse process” of decategorification. Here by decategorification we mean passing from an abelian category C to its (complexified) Grothendieck group, i.e., the complex vector space generated by the isomorphism classes of objects of C, modulo short exact sequences. Decategorification forgets enormous amount of structure, in particular, it does not directly remember about morphisms in the category.

480 MIKHAIL KHOVANOV, VOLODYMYR MAZORCHUK and CATHARINA STROPPEL

Categorification starts with a vector space M and tries to find an (interesting) abelian category C with decategorification M. Assuming that M is in addition a module or an algebra, one would also like to lift this extra structure. The idea of categorification is not completely new in the context of representation theory, but the focus changed in recent years from trying to understand the combinatorics, decomposition numbers and multiplicities of representations, to the opposite—given certain combinatorics one would like to enrich the structure by categorification. This idea was, for instance, successfully applied to constructing functorial link and knot invariants which on the one hand side categorify well-known invariants (such as the Jones-polynomial) ([5], [45], [47], [81], [82]) and on the other side extend to invariants of cobordisms. At this point it becomes clear that one should pass from abelian categories to 2-categories. An axiomatic definition of such a categorification is not yet available, but this review can be viewed as a modest naive step in this direction.

Received by the editors 2008-10-09 and, in revised form, 2009-09-20.

Transmitted by Ieke Moerdijk. Published on 2009-11-02.

2000 Mathematics Subject Classification: 18A22; 18A25; 16D90; 18E10; 18E15; 20C08.

Key words and phrases: category, functor, abelian categorification, braid group, Hecke algebra, Weyl

algebra.

\begin{matrix}c \\ ◯ \end{matrix} Mikhail Khovanov, Volodymyr Mazorchuk and Catharina Stroppel, 2009. Permission to copy for private use granted.

We start by giving the definition of a weak abelian categorification, then proceed with several well-known examples of weak abelian categorifications of modules over the symmetric group and Lie algebras. In Section 2 we give detailed explicit constructions of irreducible representations of the symmetric group. The last section indicates relations to knot theory and combinatorial representation theory.

Chuang and Rouquier, in the recent work [27], introduced the notion of a strong 𝔰𝔩_{2} categorification which is the data of an abelian category together with endofunctors E and F (corresponding to the usual Chevalley generators of 𝔰𝔩_{2}), and elements X and T in the ring of natural transformations of E respectively E^{2} with certain compatibility conditions. Using this machinery of strong 𝔰𝔩_{2}-categorification they proved the so-called Broué conjecture for the representation theory of the symmetric group in positive characteristic: two blocks with the same defect are derived equivalent.

Our list of examples of abelian categorifications is very far from complete. Many great results in the geometric representation theory can be interpreted as categorifications via abelian or triangulated categories. This includes the early foundational work of Beilinson-Bernstein and Brylinsky-Kashiwara on localization [6], [19], [67], the work of Kazhdan and Lusztig on geometric realization of representations of affine Hecke algebras [43], [26], Lusztig’s geometric construction of the Borel subalgebras of quantum groups [60], Nakajima’s realization of irreducible Kac-Moody algebra representations as middle cohomology groups of quiver varieties [68], and various constructions related to Hilbert schemes of surfaces [32], [69], quantum groups at roots of unity [4], geometric Langlands correspondence [30], etc.

Since the first version of the paper appeared, a significant further progress has been made in developing the notion of categorification, especially for categorifications of Lie algebra and quantum groups representations in [17], [20]-[22], [54]-[56], [58], [73] and other papers.

2. A simple framework for categorification

Categorification. The Grothendieck group K(B) of an abelian category B has as generators the symbols [M], where M runs over all the objects of B, and defining relations [M_{2}] = [M_{1}] + [M_{3}], whenever there is a short exact sequence

0 ⟶ M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ 0.

An exact functor F between abelian categories induces a homomorphism [F] between their Grothendieck groups.

Let A be a ring which is free as an abelian group, and a = {a_{i}}_{i∈I} a basis of A such that the multiplication has nonnegative integer coefficients in this basis:

a_{i}a_{j} = ∑_{k} c_{ij}^{k}a_{k}, c_{ij}^{k} ∈ ℤ_{≥0}.  (1)

Let B be a (left) A-module.

2.1. Definition. A (weak) abelian categorification of (A, a, B) consists of an abelian category B, an isomorphism φ : K(B) ⟶^{∼} B and exact endofunctors F_{i} : B ⟶ B such that the following holds:

(C-I) The functor F_{i} lifts the action of a_{i} on the module B, i.e. the action of [F_{i}] on the Grothendieck group of B corresponds to the action of a_{i} on the module B so that the diagram below is commutative.

K(B) ⟶^{[F_{i}]} K(B)
φ↓  ↓^{φ}
B ⟶^{a_{i}} B

(C-II) There are isomorphisms

F_{i}F_{j} ≅ ⊕_{k}F_{k}^{c_{ij}^{k}},

i.e., the composition F_{i}F_{j} decomposes as the direct sum of functors F_{k} with multiplicities c_{ij}^{k} as in (1).

The word weak refers here to the absence of any compatibility constraints on the isomorphisms appearing in (C-II). A stronger version of categorification keeping track of this extra information has to use the notion of 2-categories and is only available so far for special cases (se e.g. [71] for such an example).

If there is a categorification as above we say the action of the functors F_{i} on the category B categorifies the A-module B.

In all our examples, the objects of B will have finite length (finite Jordan-Hölder series). Consequently, if {L_{j}}_{j∈J} is a collection of simple objects of B, one for each isomorphism class, the Grothendieck group K(B) is free abelian with basis elements [L_{j}]. The image of any object M ∈ B in the Grothendieck group is

[M] = ∑_{j} m_{j}(M)[L_{j}]

where m_{j}(M) is the multiplicity of L_{j} in some (and hence in any) composition series of M.

The free group K(B) has therefore a distinguished basis [L_{j}]_{j∈J}, and the action of [F_{i}] in this basis has integer non-negative coefficients:

[F_{i}(L_{j})] = ∑ d_{ij}^{k}[L_{k}],

with d_{ij}^{k} being the multiplicity m_{k}(F_{i}(L_{j})). Via the isomorphism φ we obtain a distinguished basis b = {b_{j}}_{j∈J} of B, and

a_{i}b_{j} = ∑ d_{ij}^{k}b_{k}.  (2)

Conversely, we could fix a basis b of B with a positivity constraint for the action of A. as in (2). Then our definition of a categorification of (A, a, B) can be amended to a similar definition of a categorification of (A, a, B, b), with the additional data being the fixed basis b. Ideally the basis b corresponds then via the isomorphism φ to a basis [M_{j}]_{j∈J} for certain objects M_{j} ∈ B. Varying the choice of basis might give rise to an interesting combinatorial interplay between several, maybe less prominent than [L_{j}]_{j∈J} but more interesting, families {M_{j}}_{j∈J} of objects in B. Typical examples of such an interplay can be found in [13], [31, Section 5].

Of course, any such data (A, a, B, b) admits a rather trivial categorification, via a semisimple category B. Namely, choose a field 𝕜 and denote by 𝕜−vect the category of finite-dimensional 𝕜-vector spaces. Let

B = ⨁_{j∈J} 𝕜−vect

be the direct sum of categories 𝕜−vect, one for each basis vector of B. The category B is semisimple, with simple objects L_{j} enumerated by elements of J, and

Hom_{B}(L_{j}, L_{k}) = \{ \begin{matrix}𝕜 & if j = k, \\ 0 & otherwise. \end{matrix}

We identify K(B) with B by mapping [L_{j}] to b_{j} The functors F_{i} are determined by their action on simple objects, hence, given (2), we can define

F_{i}(L_{j}) = ⊕_{k∈J}L_{k}^{d_{ij}^{k}}

and obtain a categorification of (A, a, B, b). With few exceptions, semisimple categorifications bring little or no new structure into play, and we will ignore them. More interesting instances of categorifications occur for non-semisimple categories B. Here is a sample list.

The Weyl algebra. 1 Let A_{1} be the first Weyl algebra (the algebra of polynomial differential operators in one variable) with integer coefficients,

A_{1} = ℤ⟨x, ∂⟩/(∂x − x∂ − 1).

One takes {x^{i}∂^{j}}_{i,j≥0} as the basis a of A_{1}. The ℤ-lattice B ⊂ ℚ[x] with the basis b = \{\frac{x^{n}}{n!} \}_{n≥0} is an A_{1}-module.

To categorify this data we consider the category B = ⊕_{n≥0}R_{n}−mod, i.e. the direct

sum of the categories of finite-dimensional modules over the nilCoxeter 𝕜-algebra R_{n}. The latter has generators Y_{1}, …, Y_{n−1} subject to relations

Y_{i}^{2} = 0,
Y_{i}Y_{j} = Y_{j}Y_{i} if |i − j| > 1,
Y_{i}Y_{i+1}Y_{i} = Y_{i+1}Y_{i}Y_{i+1}.

The algebra R_{n} has a unique, up to isomorphism, finite dimensional simple module L_{n}, and K(R_{n}−mod) ≅ ℤ. The Grothendieck group K(B) is naturally isomorphic to the A_{1}- module B, via the isomorphism φ which maps [L_{n}] to \frac{x^{n}}{n!} . The endofunctors X, D in B that lift the action of x and ∂ on B are the induction and restriction functors for the inclusion of algebras R_{n} ⊂ R_{n+1}. Basis elements lift to functors X^{i}D^{j}, and the isomorphisms (C-II) of definition 2.1 are induced by an isomorphism of functors DX ≅ XD ⊕ Id which lifts the defining relation ∂x = x∂ + 1 in A_{1}. A detailed analysis of this categorification can be found in [46].

The symmetric group. 2. The regular representation of the group ring ℤ[S_{n}] of the symmetric group S_{n} has a categorification via projective functors acting on a regular block of the highest weight BGG category O from [12] for 𝔰𝔩_{n}. (For an introduction to the representation theory of semisimple Lie algebras we refer to [36]).

To define category O, start with the standard triangular decomposition 𝔰𝔩_{n} = 𝔫_{+} ⊕ 𝔥 ⊕ 𝔫_{−}, where the first and the last terms are the Lie algebras of strictly upper-triangular (resp. lower-triangular) matrices, while 𝔥 is the algebra of traceless diagonal matrices. The highest weight category O of 𝔰𝔩_{n} is the full subcategory of the category of finitelygenerated 𝔰𝔩_{n}-modules consisting of 𝔥-diagonalize (possibly infinite dimensional) modules on which U(𝔫_{+}) acts locally-nilpotently. Thus, any M ∈ O decomposes as

M = ⊕_{λ∈𝔥^{∗}}M_{λ},

where hx = λ(h)x for any h ∈ 𝔥 and x ∈ M_{λ}. Here 𝔥^{∗} is the dual vector space of 𝔥, its elements are called weights.

The one-dimensional modules \underline{ℂ}_{λ} = ℂv_{λ} over the positive Borel subalgebra 𝔟_{+} = 𝔫_{+} ⊕ 𝔥 are classified by elements λ of 𝔥^{∗}. The subalgebra 𝔫_{+} acts trivially on v_{λ}, while hv_{λ} = λ(h)v_{λ} for h ∈ 𝔥.

The Verma module M(λ) is the 𝔰𝔩_{n}-module induced from the 𝔟_{+}-module \underline{ℂ}_{λ},

M(λ) = U(𝔰𝔩_{n}) ⊗_{U(𝔟_{+})} \underline{ℂ}_{λ}.

The Verma module M(λ) has a unique simple quotient, denoted L(λ), and any simple object of O is isomorphic to L(λ) for some λ ∈ 𝔥^{∗}.

We call a weight λ positive integral if ⟨λ, α⟩ ∈ ℤ_{≥0} for any positive simple root α ∈ 𝔥^{∗}. The representation L(λ) is finite-dimensional if and only if λ is a positive integral weight.

Although most of the objects in O are infinite dimensional vector spaces, every object M of O has finite length, i.e. there is an increasing filtration by subobjects 0 = M^{0} ⊂ M^{1} ⊂ ⋯ ⊂ M^{m} = M such that the subsequent quotients M^{i+1}/M^{i} are isomorphic to simple objects, hence have the form L(λ) (where λ may vary). The Grothendieck group of O is thus a free abelian group with generators [L(λ)] for λ ∈ 𝔥^{∗}.

It turns out that O has enough projective objects: given M there exists a surjection P ↠ M with a projective P ∈ O. Moreover, isomorphism classes of indecomposable projective objects are enumerated by elements of 𝔥^{∗}. The indecomposable projective object P (λ) is determined by the property of being projective and

Hom_{O}(P (λ), L(μ)) = \begin{cases}ℂ & if λ = μ, \\ 0 & otherwise. \end{cases}

We should warn the reader that the P (λ)’s are not projective when viewed as objects of the category of all 𝔰𝔩_{n}-modules, while the L(λ)’s remain simple in the latter category.

The symmetric group S_{n}, the Weyl group of 𝔰𝔩_{n}, acts naturally on 𝔥 by permuting the diagonal entries and then also on 𝔥^{∗}. Let ρ ∈ 𝔥^{∗} be the half-sum of positive roots. In the study of the category O an important role is played by the shifted (dot) action of S_{n},

w · λ = w(λ + ρ) − ρ.

Two simple modules L(λ), L(λ′) have the same central character (i.e. are annihilated by the same maximal ideal of the center of the universal enveloping algebra) if and only if λ and λ′ belong to the same S_{n}-orbit under the shifted action. Consequently, O decomposes into a direct sum of categories

O = ⊕_{ν∈𝔥^{∗}/S_{n}}O_{ν}  (3)

indexed by orbits ν of the shifted action of S_{n} on 𝔥^{∗}. Here, O_{ν} consists of all modules with composition series having only simple subquotients isomorphic to L(λ) for λ ∈ ν. There is no interaction between O_{ν} and O_{ν′} for different orbits ν, ν′. More accurately, if ν ≠ ν′ then Ext_{O}^{i}(M, M′) = 0 for any i ≥ 0, M ∈ O_{ν} and M′ ∈ O_{ν′}.

Furthermore, each O_{ν} is equivalent to the category of finite-dimensional modules over some finite-dimensional ℂ-algebra A_{ν}. Here’s the catch, though: explicitly describing A_{ν} for n > 3 and interesting ν is very hard, see [79]. For an implicit description we just form P = ⊕_{λ∈ν}P (λ), the direct sum of all indecomposable projectives over λ ∈ ν. Then

A_{ν} ≅ Hom_{O}(P, P)^{op}.

An orbit ν (for the shifted action) is called generic if w · λ − λ is never integral, for λ ∈ ν and w ∈ S_{n}, w ≠ 1. For a generic orbit ν, the category O_{ν} is boring and equivalent to the direct sum of n! copies of the category of finite-dimensional ℂ-vector spaces, one for each λ ∈ ν. For such λ we have P (λ) = M(λ) = L(λ), i.e. the Verma module with the highest weight λ is simple as well as projective in O.

We call an orbit integral if it is a subset of the integral weight lattice in 𝔥^{∗}. In [76] it is shown that O_{ν} for non-integral ν reduces to those for integral weights. From now on we therefore assume that ν is integral. Then the category O_{ν} is indecomposable (unlike in the generic case). Moreover, the complexity of O_{ν} only depends on the type of the orbit. If two orbits ν and ν′ contain points λ ∈ ν, λ′ ∈ ν′ with identical stabilizers, then the categories O_{ν} and O_{ν′} are equivalent, see [11], [76]. If the stabilizer of ν under the shifted action is trivial, the category O_{ν} is called a regular block. Regular blocks are the most complicated indecomposable direct summands of O, for instance in the sense of having the maximal number of isomorphism classes of simple modules.

There is a natural bijection between the following three sets: positive integral weights, isomorphism classes of irreducible finite-dimensional representations of 𝔰𝔩_{n}, and regular blocks of O for 𝔰𝔩_{n}. A positive integral weight λ is the highest weight of an irreducible finitedimensional representation L(λ), determined by the weight uniquely up to isomorphism. In turn, L(λ) belongs to the regular block O_{ν}, where ν = S_{n} · λ is the orbit of λ.

Any two regular blocks of O are equivalent as categories, as shown in [40]. For this reason, we can restrict our discussion to the uniquely defined regular block which contains the one-dimensional trivial representation L(0) of 𝔰𝔩_{n}. We denote this block by O_{0}. It has n! simple modules L(w) = L(w·0), enumerated by all permutations w ∈ S_{n} (with the identity element e of S_{n} corresponding to L(0) which is the only finite dimensional simple module in O_{0}). Thus, K(O_{0}) is free abelian of rank n! with basis {[L(w)]}_{w∈S_{n}}. Other notable objects in O_{0} are the Verma modules M(w) = M(w·0) and the indecomposable projective modules P (w) = P (w · 0), over all w ∈ S_{n}. The sets {[M(w)]}_{w∈S_{n}} and {[P (w)]}_{w∈S_{n}} form two other prominent bases in K(O_{0}). For the set {[M(w)]}_{w∈S_{n}} this is easy to see, because the transformation matrix between Verma modules and simple modules is upper triangular with ones on the diagonal. For the set {[P (w)]}_{w∈S_{n}} this claim is not obvious and relates to the fact that O_{0} has finite homological dimension, see [12].

Equivalences between regular blocks are established by means of translation functors. First note that we can tensor two U(𝔰𝔩_{n})-modules over the ground field. If V is a finitedimensional 𝔰𝔩_{n}-module it follows from the definitions that V ⊗ M lies in O whenever M is in O. Hence, tensoring with V defines an endofunctor V ⊗_{−} of the category O. Taking direct summands of the functors V ⊗ _{−} provides a bewildering collection of different functors and allows one to analyze O quite deeply. By definition, a projective functor is any endofunctor of O isomorphic to a direct summand of V ⊗_{−} for some finite-dimensional 𝔰𝔩_{n}-module V. Projective functors were classified by J. Bernstein and S. Gelfand [11]. Translation functors are special cases of projective functors—they are direct summands of projective functors obtained by first restricting to a certain block and afterwards also projecting onto a fixed block.

Let us restrict our discussion to projective endofunctors in the regular block O_{0}. Each projective endofunctor O_{0} ⟶ O_{0} decomposes into a finite direct sum of indecomposable functors θ_{w}, enumerated by permutations w and determined by the property θ_{w}(M(e)) ≅ P (w). We have P (e) = M(e) and the functor θ_{e} is the identity functor. The composition or the direct sum of two projective functors are again projective functors. With respect to these two operations, projective endofunctors on O_{0} are (up to isomorphism) generated by the projective functors θ_{i} := θ_{s_{i}} corresponding to the simple transpositions/reflections s_{i} = (i, i + 1). The functor θ_{i} is called the translation through the i-th wall. The functor θ_{w} is a direct summand of θ_{i_{k}} … θ_{i_{1}}, for any reduced decomposition w = s_{i_{1}} … s_{i_{k}}. The induced endomorphism [θ_{i}] of the Grothendieck group acts (in the basis given by Verma modules) by

[θ_{i}][(M(w))] = [θ_{i}(M(w))] = [M(w)] + [M(ws_{i})].

Now we are prepared to explain the categorification. We first fix the unique isomorphism φ of groups

φ : K(O_{0}) ⟶ ℤ[S_{n}]
[M(w)] ⟼ w

and define C_{w} := φ([P (w)]). Then the C_{w}, w ∈ W, form a basis a of ℤ[S_{n}].

The action of [θ_{i}] corresponds under φ to the endomorphism of ℤ[S_{n}] given by right multiplication with C_{s_{i}} := 1 + s_{i}.

The defining relations of the generators 1 + s_{i} in ℤ[S_{n}] lift to isomorphisms of functors as follows

θ_{i}^{2} ≅ θ_{i} ⊕ θ_{i},
θ_{i}θ_{j} ≅ θ_{j}θ_{i} if |i − j| > 1,
θ_{i}θ_{i+1}θ_{i} ⊕ θ_{i+1} ≅ θ_{i+1}θ_{i}θ_{i+1} ⊕ θ_{i}.

Here, the last isomorphism follows from the existence of decompositions of functors

θ_{i}θ_{i+1}θ_{i} ≅ θ_{w_{1}} ⊕ θ_{i},
θ_{i+1}θ_{i}θ_{i+1} ≅ θ_{w_{1}} ⊕ θ_{i+1},

where w_{1} = s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1}. In particular, [θ_{w_{1}}] corresponds under φ to the right multiplication with 1 + s_{i} + s_{i+1} + s_{i}s_{i+1} + s_{i+1}s_{i} + s_{i}s_{i+1}s_{i}.

By the classification theorem of projective functors, the endomorphism [θ_{w}], w ∈ W, corresponds then to right multiplication with the element C_{w}. From this one can then actually deduce that the multiplication in the basis a has non-negative integral coefficients

C_{w}C_{w′} = ∑_{w′′} c_{w,w′}^{w′′}C_{w′′}, c_{w,w′}^{w′′} ∈ ℤ_{≥0}.  (4)

Hence we are in the situation of (1) and are looking for an abelian categorification of (ℤ[S_{n}], a, ℤ[S_{n}]). We already have the isomorphism φ and the exact endofunctor θ_{w} corresponding to the generator C_{w} satisfying condition (C-I).

The composition of two projective functors decomposed as a direct sum of indecomposable functors [θ_{w}], w ∈ W, has nonnegative integral coefficients, and the equations (4) turn into isomorphisms of functors

θ_{w}θ_{w′} ≅ ⨁_{w′′} (θ_{w′′})^{c_{ww′}^{w′′}}, c_{ww′}^{w′′} ∈ ℤ_{≥0},  (5)

It turns out that each [θ_{w}] acts by a multiplication with a linear combination of y’s for y ≤ w. Moreover, all coefficients are nonnegative integers. For instance, if w ∈ S_{4} then [θ_{w}] ≐ ∑_{y≤w} y, with two exceptions:

\begin{matrix}[θ_{w}] & ≐ & ∑_{y≤w} y + 1 + s_{2}, & w = s_{2}s_{1}s_{3}s_{2}, \\ [θ_{w}] & ≐ & ∑_{y≤w} y + 1 + s_{1} + s_{3} + s_{1}s_{3}, & w = s_{1}s_{3}s_{2}s_{1}s_{3}. \end{matrix}

We can summarize the above results into a theorem.

2.2. Theorem. The action of the indecomposable projective functors θ_{w}, w ∈ S_{n}, on the block O_{0} for 𝔰𝔩_{n} categorifies the right regular representation of the integral group ring of the symmetric group S_{n} (in the basis a of the elements C_{w}, w ∈ S_{n}).

This theorem is due to Bernstein and Gelfand, see [11], where it was stated in different terms, since the word “categorification” was not in the mathematician’s vocabulary back then. In fact, Bernstein and Gelfand obtained a more general result by considering any simple Lie algebra 𝔤 instead of 𝔰𝔩_{n} and its Weyl group W in place of S_{n}.

In the explanation to the theorem we did not give a very explicit description of the basis a due to the fact that there is no explicit (closed) formula for the elements C_{w} available. However, the elements C_{w} can be obtained by induction (on the length of w) using the Kazhdan-Lusztig theory [41], [42]. The Kazhdan-Lusztig theory explains precisely the complicated interplay between the basis a and the standard basis of ℤ[S_{n}].

3. Parabolic blocks of O categorify representations of the symmetric group S_{n} induced from the sign representation of parabolic subgroups.

Let μ = (μ_{1}, …, μ_{k}), μ_{1} + ⋯ + μ_{k} = n, be a composition of n and λ = (λ_{1}, …, λ_{k}) the corresponding partition. In other words, λ is a permutation of the sequence μ with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k}. Denote by p_{μ} the subalgebra of 𝔰𝔩_{n} consisting of μ-block uppertriangular matrices. Consider the full subcategory O^{μ} of O which consists of all modules M on which the action of U(p_{μ}) is locally finite. The category O^{μ} is an example of a parabolic subcategory of O, introduced in [70]. A simple object L(λ) of O belongs to O^{μ} if and only if the weight λ is positive integral with respect to all roots of the Lie algebra p_{μ}. The two extreme cases are μ = (1, 1, …, 1), in which case O^{μ} is all of O, and μ = (n), for O^{(n)} is the semisimple category consisting exactly of all finite-dimensional 𝔰𝔩_{n}-modules.

The direct sum decomposition (3) induces a similar decomposition of the parabolic category:

O^{μ} ≅ ⊕_{ν∈𝔥^{∗}/S_{n}}O_{ν}^{μ}.

Each category O_{ν}^{μ} is either trivial (i.e. contains only the zero module) or equivalent to the category of finite-dimensional modules over some finite-dimensional ℂ-algebra (but describing this algebra explicitly for interesting μ and ν is a hard problem, see [18]). Unless μ = (1^{n}), for generic ν the summand O_{ν}^{μ} is trivial. Again, the most complicated summands are the O_{ν}^{μ} where the orbit ν contains a dominant regular integral weight. Translation functors establish equivalences between such summands for various such ν, and allow us to restrict our consideration to the block O_{0}^{μ} corresponding to the (shifted) orbit through 0. The inclusion

O_{0}^{μ} ⊂ O_{0}

is an exact functor and induces an inclusion of Grothendieck groups

K(O_{0}^{μ}) ⊂ K(O_{0}).  (6)

Indeed, the Grothendieck group of O_{0} is free abelian with generators [L(w)], w ∈ S_{n}. A simple module L(w) lies in O_{0}^{μ} if and only if w is a minimal left coset representative for the subgroup S_{μ} of S_{n} (we informally write w ∈ (S_{μ}\S_{n})_{short}). The Grothendieck group of O_{0}^{μ} is then the subgroup of K(O_{0}) generated by such L(w).

The analogues of the Verma modules in the parabolic case are the so-called parabolic Verma modules

M(p_{μ}, V) = U(𝔰𝔩_{n}) ⊗_{U(p_{μ})} V,

where V is a finite-dimensional simple p_{μ}-module. The module M(p_{μ}, V) is a homomorphic image of some ordinary Verma module from O, in particular, it has a unique simple quotient isomorphic to some L(w) for some unique w ∈ S_{n}. In this way we get a canonical bijection between parabolic Verma modules in O_{0}^{μ} and the set (S_{μ}\S_{n})_{short} of shortest coset representatives. Hence it is convenient to denote the parabolic Verma module with simple quotient L(w), w ∈ (S_{μ}\S_{n})_{short}, simply by M^{μ}(w).

Generalized Verma modules provide a basis for the Grothendieck group of O_{0}^{μ}. Under the inclusion (6) of Grothendieck groups the image of the generalized Verma module M^{μ}(w) is the alternating sum of Verma modules, see [70] and [59]:

[M^{μ}(w)] = ∑_{u∈S_{μ}}(−1)^{l(u)}[M(uw)].  (7)

Since the projective endofunctors θ_{w} preserve O_{0}^{μ}, the inclusion (6) is actually an inclusion of S_{n}-modules, and, in view of the formula (7), we can identify K(O_{0}^{μ}) with the submodule I_{μ}^{−} of the regular representation of S_{n} isomorphic to the representation induced from the sign representation of S_{μ},

I_{μ}^{−} ≅ Ind_{S_{μ}}^{S_{n}}ℤv,

where we denoted by ℤv the sign representation, so that wv = (−1)^{l(w)}v for w ∈ S_{μ}. To summaries, we have:

2.3. Theorem. The action of the projective functors θ_{w}, w ∈ W, on the parabolic subcategory O_{0}^{μ} of O categorifies the induced representation I_{μ}^{−} of the integral group ring of the symmetric group S_{n} (with basis a = {C_{w}}_{w∈S_{n}}).

As in the previous example the Grothendieck group K(O_{0}^{μ}) has three distinguished basis, given by simple objects, projective objects, and parabolic Verma modules respectively.

Remark: If we choose a pair μ, μ′ of decompositions giving rise to the same partition λ of n, then the modules I_{μ}^{−} and I_{μ′}^{−} are isomorphic, and will be denoted I_{λ}^{−}. However, the categories O^{μ} and O^{μ′} are not equivalent in general, which means the two categorifications of the induced representation I_{λ}^{−} are also not equivalent. This problem disappears if we leave the world of abelian categorifications, since the derived categories D^{b}(O^{μ}) and D^{b}(O^{μ′}) are equivalent [48]. The equivalence is based on the geometric description of O^{μ} and O^{μ′} in terms of complexes of sheaves on partial flag varieties.

4. Self-dual projectives in a parabolic block categorify irreducible representations of the symmetric group.

Let I_{μ} be the representation of ℤ[S_{n}] induced from the trivial representation of the subgroup S_{μ}. Up to isomorphism, it only depends on the partition λ associated with μ. Partitions of n naturally index the isomorphism classes of irreducible representations of S_{n} over any field of characteristic zero (we use ℚ here). Denote by S_{ℚ}(λ) the irreducible (Specht) module associated with λ. It is an irreducible representation defined as the unique common irreducible summand of I_{λ} ⊗ ℚ and I_{λ^{∗}}^{−} ⊗ ℚ, where λ^{∗} is the dual partition of λ. Passing to duals, we see that S_{ℚ}(λ^{∗}) is the unique common irreducible summand of I_{λ^{∗}} ⊗ ℚ and I_{λ}^{−} ⊗ ℚ.

We have already categorified the representation I_{λ}^{−} (in several ways) via the parabolic categories O_{0}^{μ}, where μ is any decomposition for λ. It’s natural to try to realize a categorification of some integral lift S(λ^{∗}) of the irreducible representation S_{ℚ}(λ^{∗}) via a suitable subcategory of some O_{0}^{μ} stable under the action of projective endofunctors.

The correct answer, presented in [50], is to pass to a subcategory generated by those projective objects in O_{0}^{μ} which are also injective. Note that these modules are neither projective nor injective in O_{0} (unless if O_{o}^{μ} = O_{0}).

Any projective object in O_{0}^{μ} is isomorphic to a direct sum of indecomposable projective modules P^{μ}(w), for w ∈ (S_{μ}\S_{n})_{short}. Let J ⊂ (S_{μ}\S_{n})_{short} be the subset indexing indecomposable projectives modules that are also injective: w ∈ J if and only if P^{μ}(w) is injective. Projective endofunctors θ_{w}, w ∈ S_{n}, take projectives to projectives and injectives to injectives. Therefore, they take projective-injective modules (modules that are both projective and injective, also called self-dual projective, for instance, in Irving [37]) to projective-injective modules.

The category of projective-injective modules is additive, not abelian. To remedy this, consider the full subcategory C^{μ} of O_{0}^{μ} consisting of modules M admitting a resolution

P_{1} ⟶ P_{0} ⟶ M ⟶ 0  (8)

with projective-injective P_{1} and P_{0}. The category C^{μ} is abelian and stable under all endofunctors θ_{w} for w ∈ S_{n}, see [50].

Irving [37] classified projective-injective modules in O_{0}^{μ}. His results were interpreted in [50] in the language of categorification:

2.4. Theorem. The action of the projective endofunctors θ_{w}, w ∈ S_{n}, on the abelian category C^{μ} categorifies (after tensoring the Grothendieck group with ℚ over ℤ) the irreducible representation S_{ℚ}(λ^{∗}) of the symmetric group S_{n}.

The Grothendieck group K(C^{μ}) is a module over the integral group ring of S_{n}, with s_{i} acting by [θ_{i}] − Id, and the theorem says that K(C^{μ}) ⊗_{ℤ}ℚ is an irreducible representation of the symmetric group corresponding to the partition λ^{∗}. Several explicit examples of categorifications via C^{μ} will be given in Section 3.

Remark: Suppose μ and ν are two decompositions of the same partition λ. It’s shown in [65] (Theorem 5.4.(2)) that the categories C^{μ} and C^{ν} are equivalent, through an equivalence which commutes with the action of the projective functors θ_{u} on these categories (the equivalence is given by a non-trivial composition of derived Zuckerman functors). Therefore, the categorification of S(λ^{∗}) does not depend on the choice of the decomposition μ that represents λ, and we can denote the category C^{μ} by C^{λ}. (This should be compared with the remark after Theorem 2.3.)

Remark: Theorem 2.3 and Theorem 2.4 can be generalized to arbitrary semi-simple complex finite-dimensional Lie algebras, see [66]. However, in the general case Theorem 2.4 does not categorify simple modules for the corresponding Weyl group but rather the Kazhdan-Lusztig cell modules from [41]. This can be used to describe the so-called “rough” structure of generalized Verma modules, which shows that “categorification theoretic” ideas can lead to new results in representation theory.

Remark: The inclusion of categories C^{μ} ⊂ O_{0}^{μ} is not an exact functor; however, it is a part of a very natural filtration of the category O_{0}^{μ} which can be defined using the Gelfand-Kirillov dimension of modules, see [66, 6.9]. To get the inclusion of Grothendieck groups analogous to the inclusion of representations from the irreducible Specht module into the induced sign representation, we pass to the subgroup K′(C^{μ}) of K(C^{μ}) generated by the images of projective modules in C^{μ}. This additional technicality is necessary as the category C^{μ} does not have finite homological dimension in general. The subgroup K′(C^{μ}) is always a finite index subgroup, stable under the action of the [θ_{w}]’s. We denote this subgroup by S′(λ^{∗}) :

S′(λ^{∗}) =^{def} K′(C^{μ}) ⊂ K(C^{μ}) ≅ S(λ^{∗}).

The inclusion of categories C^{μ} ⊂ O_{0}^{μ} induces the inclusion

S′(λ^{∗}) ⊂ K(O_{0}^{μ}) ≅ I_{μ}^{−}

of ℤ[S_{n}]-modules, hence realising the integral lift S′(λ^{∗}) of the Specht module as a subrepresentation of I_{μ}^{−}.

5. Categorification of the induced representations I_{μ} via projectively presentable modules.

Let P^{μ} denote the category of all modules M admitting a resolution (8) in which each indecomposable direct summand of both P_{0} and P_{1} has the form P (w), where w is a longest left coset representative for S_{μ} in S_{n} (we will write w ∈ (S_{μ}\S_{n})_{long}). Such modules are called p_{μ}-presentable modules, see [62]. As in the previous example, the category P^{μ} is stable under all endofunctors θ_{w}, w ∈ S_{n}.

By definition, P^{μ} is a subcategory of O_{0}, but just as in the example above, the natural inclusion functor is not exact. The category P^{μ} does not have finite homological dimension in general, so we again pass to the subgroup K′(P^{μ}) of K(P^{μ}), generated by the images of indecomposable projective modules in P^{μ}. The latter are (up to isomorphism) the P (u), u ∈ (S_{μ}\S_{n})_{long}. This is a finite index subgroup, stable under the action of the [θ_{w}]’s and we have the following statement proved in [62]:

2.5. Theorem. The action of projective endofunctors on the abelian category P^{μ} categorifies (after tensoring with ℚ over ℤ) the induced representation (I_{μ})_{ℚ} of the group algebra of the symmetric group S_{n} (with the basis a = {C_{w}}_{w∈S_{n}}).

Consider the diagram of ℚ[S_{n}]-modules

(I_{λ})_{ℚ} ⟶^{ι_{1}} ℚ[S_{n}] ⟶^{p_{1}} (I_{λ^{∗}}^{−})_{ℚ}.

The map ι_{1} is the symmetrization inclusion map, while p_{1} is the antisymmetrization quotient map. We have

S_{ℚ}(λ) =^{def} p_{1}ι_{1}((I_{λ})_{ℚ}).

The map ι_{1} is categorified as the inclusion of P^{μ} to O_{0}. The map p_{1} is categorified as the projection of O_{0} onto O_{0}^{μ^{∗}}, where μ^{∗} is some decomposition corresponding to λ^{∗}. Unfortunately, the composition of the two functors categorifying these two maps will be trivial in general. To repair the situation we first project P^{μ} onto the full subcategory of P^{μ} given by simple objects of minimal possible Gelfand-Kirillov dimension. It is easy to see that the image category contains enough projective modules, and using the equivalence constructed in [66, Theorem I], these projective modules can be functorially mapped to projective modules in C^{μ^{∗}}, where μ^{∗} is a (good choice of) composition with associated partition λ^{∗}. The latter category embeds into O_{0}^{μ^{∗}} as was explained in the previous example.

6. Categorification of the group algebra The representation theory of groups like GL(n, ℂ), considered as a real Lie group, naturally leads to the notion of Harish-Chandra bimodules. A Harish-Chandra bimodule over 𝔰𝔩_{n} is a finitely-generated module over the universal enveloping algebra U(𝔰𝔩_{n} × 𝔰𝔩_{n}) which decomposes into a direct sum of finitedimensional U(𝔰𝔩_{n})-modules with respect to the diagonal copy {(X, −X)|X ∈ 𝔰𝔩_{n}} of 𝔰𝔩_{n}. Let HC_{0,0} be the category of Harish-Chandra bimodules which are annihilated, on both sides, by some power of the maximal ideal I_{0} of the center Z of U(𝔰𝔩_{n}). Here I_{0} is the annihilator of the trivial U(𝔰𝔩_{n})-module considered as a Z-module. Thus, M ∈ HC_{0,0} if and only if xM = 0 = Mx for all x ∈ I_{0}^{N} for N large enough.

By [11, Section 5] there exists an exact and fully faithful functor

O_{0} ⟶ HC_{0,0}.

Moreover, this functor induces an isomorphism of Grothendieck groups

K(O_{0}) ≅ K(HC_{0,0}).

Since the former group is isomorphic to ℤ[S_{n}], we can identify the Grothendieck group of HC_{0,0} with ℤ[S_{n}] as well.

The advantage of bimodules is that we now have two sides and can tensor with a finitedimensional 𝔰𝔩_{n}-module both on the left and on the right. In either case, we preserve the category of Harish-Chandra bimodules. Taking all possible direct summands of these functors and restricting to endofunctors on the subcategory HC_{0,0} leads to two sets of commuting projective functors, {θ_{r,w}}_{w∈S_{n}} and {θ_{l,w}}_{w∈S_{n}} which induce endomorphisms on the Grothendieck group K(HC_{0,0}) ≅ ℤ[S_{n}] given by left and right multiplication with {C_{w}}_{w∈S_{n}} respectively. Summarizing, we have

2.6. Theorem. The action of the functors {θ_{r,w}}_{w∈S_{n}} and {θ_{l,w}}_{w∈S_{n}} on the category HC_{0,0} of Harish-Chandra bimodules for 𝔰𝔩_{n} with generalized trivial character on both sides categorifies ℤ[S_{n}], viewed as a bimodule over itself. The functors {θ_{r,w}}_{w∈S_{n}} induces the left multiplication with C_{w} on the Grothendieck group, whereas the functors {θ_{l,w}}_{w∈S_{n}} induces the right multiplication with C_{w} on the Grothendieck group,

The first half of the theorem follows at once from [11], the second half from [77]. The category of Harish-Chandra bimodules is more complicated than the category O. For example, HC_{0,0} does not have enough projectives, and is not Koszul with respect to the natural grading, in contrast to O_{0} (for the Koszulity of O_{0} see [9]). The study of translation functors on Harish-Chandra modules goes back to Zuckerman [85].

Lie algebras. 7. In the following we will mention several instances of categorifications of modules over Lie algebras. Our definition of categorification required an associative algebra rather than a Lie algebra, so one should think of this construction as a categorification of representations of the associated universal enveloping algebra.

Let V be the fundamental two-dimensional representation of the complex Lie algebra 𝔰𝔩_{2}. Denote by {e, f, h} the standard basis of 𝔰𝔩_{2}. The n-th tensor power of V decomposes into a direct sum of weight spaces:

V^{⊗n} = ⊕_{k=0}^{n}V^{⊗n}(k),

where hx = (2k − n)x for x ∈ V^{⊗n}(k).

A categorification of V^{⊗n} was constructed in [10]. The authors considered certain singular blocks O_{k,n−k} of the category O for 𝔰𝔩_{n}. More precisely O_{k,n−k} is the choice of an integral block O_{ν}, where the stabilizer of ν is isomorphic to S_{k} ×S_{n−k}. Note that different choices give equivalent blocks. The Grothendieck group of this block has rank \binom{n}{k} equal to the dimension of the weight space V^{⊗n}(k), and there are natural isomorphisms

K(O_{k,n−k}) ⊗_{ℤ} ℂ ≅ V^{⊗n}(k).

The Grothendieck group of the direct sum

O_{n} = ⊕_{k=0}^{n}O_{k,n−k}

is isomorphic to V^{⊗n} (after tensoring with ℂ over ℤ). Suitable translation functors E, F in O_{n} lift the action of the generators e, f of 𝔰𝔩_{2} on V^{⊗n}.

To make this construction compatible with Definition 2.1 one should switch to Lusztig’s version U^{·} of the universal enveloping algebra U(𝔰𝔩_{2}) (see [60], [10]) and set q = 1. Instead of the unit element, the ring U^{·} contains idempotents 1_{n}, n ∈ ℤ, which can be viewed as projectors onto integral weights. The Lusztig basis 𝔹^{·} in U^{·} has the positivity property required by Definition 2.1, and comes along with an integral version V_{ℤ}^{⊗n} of the tensor power representation. The triple (U^{·}, 𝔹^{·}, V_{ℤ}^{⊗n}) is categorified using the above-mentioned category O_{n} and projective endofunctors of it. In fact, each element of 𝔹^{·} either corresponds to an indecomposable projective endofunctor on O_{n} or acts by 0 on V_{ℤ}^{⊗n}. We refer the reader to [10] for details, to [27] for an axiomatic development of 𝔰𝔩_{2} categorifications, and to [10] and [81] for a categorification of the Temperley-Lieb algebra action on V_{ℤ}^{⊗n} via projective endofunctors on the category Koszul dual to O_{n} (see [9] and [61] for details on Koszul duality).

The Lie algebra 𝔰𝔩_{2} has one irreducible (n + 1)-dimensional representation V_{n}, for each n ≥ 0 (V_{1} ≅ V, of course). A categorification of arbitrary tensor products V_{n_{1}} ⊗ ⋯ ⊗ V_{n_{m}} is described in [31]. This tensor product is a submodule of V^{⊗n}, where n = n_{1} + ⋯ + n_{m}. Knowing that O_{n} categorifies V^{⊗n}, we find a ”subcategorification,” a subcategory of O_{n} stable under the action of projective functors, with the Grothendieck group naturally isomorphic to V_{n_{1}} ⊗ ⋯ ⊗ V_{n_{m}}. The subcategory has an intrinsic description via Harish-Chandra modules similar to the one from Example 5.

8. A categorification of arbitrary tensor products of fundamental representations Λ^{i}V, where V is the k-dimensional 𝔰𝔩_{k}-representation and 1 ≤ i ≤ k −1 was found by J. Sussan [84], see also [64] for a more combinatorial approach. A tensor product Λ^{i_{1}}V ⊗ ⋯ ⊗ Λ^{i_{r}}V decomposes into weight spaces Λ^{i_{1}}V ⊗ ⋯ ⊗ Λ^{i_{r}}V (ν), over various integral weights ν of 𝔰𝔩_{k}. Each weight space becomes the Grothendieck group of a parabolic-singular block of the highest weight category for 𝔰𝔩_{N}, where N = i_{1} + ⋯ + i_{r}. For the parabolic subalgebra one takes the Lie algebra of traceless N × N matrices which are (i_{1}, …, i_{r}) block uppertriangular. The choice of the singular block is determined by ν. For the precise dictionary how to determine ν we refer to [64].

Translation functors between singular blocks, restricted to the parabolic category, provide an action of the generators E_{j} and F_{j} of the Lie algebra 𝔰𝔩_{k}. Relations in the universal enveloping algebra lift to functor isomorphisms. Conjecturally, Sussan’s categorification satisfies the framework of Definition 2.1 above, with respect to Lusztig’s completion U^{·} of the universal enveloping algebra of 𝔰𝔩_{k} and Lusztig’s canonical basis there.

9. Ariki, in a remarkable paper [2], categorified all finite-dimensional irreducible representations of 𝔰𝔩_{m}, for all m, as well as integrable irreducible representations of affine Lie algebras 𝔰̂𝔩_{r}. Ariki considered certain finite-dimensional quotient algebras of the affine Hecke algebra Ĥ_{n,q}, known as Ariki-Koike cyclotomic Hecke algebras, which depend on a number of discrete parameters. He identified the Grothendieck groups of blocks of these algebras, for generic values of q ∈ ℂ, with the weight spaces V_{λ}(μ) of finite-dimensional irreducible representations

V_{λ} = ⊕_{μ}V_{λ}(μ)

of 𝔰𝔩_{m}. Direct summands of the induction and restriction functors between cyclotomic Hecke algebras for n and n + 1 act on the Grothendieck group as generators e_{i} and f_{i} of 𝔰𝔩_{m}.

Specializing q to a primitive r-th root of unity, Ariki obtained a categorification of integrable irreducible representations of the affine Lie algebra 𝔰̂𝔩_{r}.

We conjecture that direct summands of arbitrary compositions of Ariki’s induction and restriction functors correspond to elements of the Lusztig canonical basis 𝔹^{·} of Lusztig’s completions U^{·} of these universal enveloping algebras. This conjecture would imply that Ariki’s categorifications satisfy the conditions of Definition 1.

Lascoux, Leclerc and Thibon, in an earlier paper [57], categorified level-one irreducible 𝔰̂𝔩_{r}-representations, by identifying them with the direct sum of Grothendieck groups of finite-dimensional Hecke algebras H_{n,q}, over all n ≥ 0, with q a primitive r-th root of unity. Their construction is a special case of Ariki’s. We also refer the reader to related works [3], [33]. Categorifications of the adjoint representation and of irreducible 𝔰𝔩_{m}-representations with highest weight ω_{j} + ω_{k} are described explicitly in [34], [35] and [24].

Another way to categorify all irreducible finite-dimensional representations of 𝔰𝔩_{m}, for all m, was found by Brundan and Kleshchev [15], via the representation theory of Walgebras. There is a good chance that their categorification is equivalent to that of Ariki, and that an equivalence of two categorifications can be constructed along the lines of Arakawa-Suzuki [1] and Brundan-Kleshchev [16].

Biadjointness. Definition 1 of (weak) categorifications was minimalistic. Categorifications in the above examples share extra properties, the most prominent of which is biadjointness: there exists an involution a_{i} → a_{i′} on the basis a of A such that the functor F_{i′} is both left and right adjoint to F_{i}. This is the case in the examples 2 through 9, while in example 1 the functors are almost biadjoint. Namely, the induction functor F_{x} lifting the action of x is left adjoint to the restriction functor F_{∂} (which lifts the action of ∂) and right adjoint to F_{∂} conjugated by an involution.

A conceptual explanation for the pervasiveness of biadjointness in categorifications is given by the presence of the Hom bifunctor in any abelian category. The Hom bifunctor in B descends to a bilinear form on the Grothendieck group B of B, via

([M], [N]) =^{def} dim Hom_{B}(M, N),

where M is projective or N is injective, and some standard technical conditions are satisfied. When a representation naturally comes with a bilinear form, the form is usually compatible with the action of A: there exists an involution a → a′ on A such that (ax, y) = (x, a′y) for x, y ∈ B. A categorification of this equality should be an isomorphism

Hom(F_{a}M, N) ≅ Hom(M, F_{a′}N)

saying that the functor lifting the action of a′ is right adjoint to the functor lifting the action of a. If the bilinear form is symmetric, we should have the adjointness property in the other direction as well, leading to biadjointness of F_{a} and F_{a′}.

A beautiful approach to 𝔰𝔩_{2} categorifications via biadjointness was developed by Chuang and Rouquier [27]. The role of biadjointness in TQFTs and their categorifications is clarified in [47, Section 6.3]. An example how the existence of a categorification with a bilinear form can be used to determine dimensions of hom-spaces can be found in [82].

Grading and q-deformation. In all of the above examples, the data (A, a, B) that is being categorified admits a natural q-deformation (A_{q}, a_{q}, B_{q}). Here A_{q} is a ℤ[q, q^{−1}]- algebra, B_{q} an A_{q}-module, and a_{q} a basis of A_{q}. We assume that both A_{q} and B_{q} are free ℤ[q, q^{−1}]-modules, that the multiplication in A_{q} in the basis a_{q} has all coefficients in ℕ[q, q^{−1}], and that taking the quotient by the ideal (q − 1) brings us back to the original data:

A = A_{q}/(q − 1)A_{q}, B = B_{q}/(q − 1)B_{q}, a_{q} ⟶^{q=1} a.

An automorphism τ of an abelian category B (more accurately, an invertible endofunctor on B) induces a ℤ[q, q^{−1}]-module structure on the Grothendieck group K(B). Multiplication by q corresponds to the action of τ:

[τ(M)] = q[M], [τ^{−1}(M)] = q^{−1}[M].

In many of the examples, B will be the category of graded modules over a graded algebra, and τ is just the functor which shifts the grading. To emphasize this, we denote τ by {1} and its n-th power by {n}.

2.7. Definition. A (weak) abelian categorification of (A_{q}, a_{q}, B_{q}) consists of an abelian category B equipped with an invertible endofunctor {1}, an isomorphism of ℤ[q, q^{−1}]- modules φ : K(B) ⟶^{∼} B_{q} and exact endofunctors F_{i} : B ⟶ B that commute with {1} and such that the following hold

(qC-I) F_{i} lifts the action of a_{i} on the module B_{q}, i.e. the action of [F_{i}] on the Grothendieck group corresponds to the action of a_{i} on B_{q}, under the isomorphism ϕ, in the sense that the diagram below is commutative.

K(B) ⟶^{[F_{i}]} K(B)
φ↓  ↓^{φ}
B_{q} ⟶^{a_{i}} B_{q}

(qC-II) There are isomorphisms of functors

F_{i}F_{j} ≅ ⨁_{k} F_{k}^{c_{ij}^{k}},

i.e., the composition F_{i}F_{j} decomposes as the direct sum of functors F_{k} with multiplicities c_{ij}^{k} ∈ ℕ[q, q^{−1}]

The graded versions are well-known in all of the examples above up to Example 8. In Example 1 the nilCoxeter algebra R_{n} is naturally graded with deg(Y_{i}) = 1. The inclusion R_{n} ⊂ R_{n+1} induces induction and restriction functors between categories of graded R_{n} and R_{n+1}-modules. In the graded case, induction and restriction functors satisfy the isomorphism

DX ≅ XD{1} ⊕ Id

which lifts the defining relation ∂x = qx∂ + 1 of the q-Weyl algebra (see [46] for more detail).

An accurate framework for graded versions of examples 2–8 is a rather complicated affair. To construct a canonical grading on a regular block of the highest weight category [9] requires étale cohomology, perverse sheaves [7], and the Beilinson-Bernstein-Brylinski-Kashiwara localization theorem [6], [19]. Soergel’s approach to this grading is more elementary [76], [78], [77], but still relies on these hard results. Extra work is needed to show that translation or projective functors can be lifted to endofunctors in the graded category [80].

Ariki’s categorification of irreducible integrable representations (Example 9 above) should admit a graded version as well.

3. Four examples of categorifications of irreducible representations

In the example 4 above we categorified an integral lift of the irreducible representation S_{ℚ}(λ^{∗}) of the symmetric group via the abelian category C^{λ} built out of projective-injective modules in a parabolic block of O. The category C^{λ} is equivalent to the category of finite-dimensional representations over a finite-dimensional algebra A^{λ}, the algebra of endomorphisms of the direct sum of indecomposable projective-injective modules P^{μ}(w). Under this equivalence, projective functors θ_{i} turn into the functors of tensoring with certain A^{λ}-bimodules. It’s not known how to describe A^{λ} and these bimodules explicitly, except in a few cases, four of which are discussed below.

a. The sign representation. The sign representation of the symmetric group (over ℤ) is a free abelian group ℤv on one generator v, with s_{i}v = −v for all i. It corresponds to the partition (1^{n}) of n, which in our notation is λ^{∗} for λ = (n). The parabolic category O_{0}^{(n)} has as objects exactly the finite-dimensional modules from O_{0} since the parabolic subalgebra in this case is all of 𝔰𝔩_{n}.

Actually, O_{0} has only one simple module with this property, the one-dimensional trivial representation ℂ. In our notation, this is the module L(e), the simple quotient of the Verma module M(e) assigned to the unit element of the symmetric group.

Consequently, any object of O_{0}^{(n)} is isomorphic to a direct sum of copies of L(e), and the category is semisimple. Furthermore, the category C^{(n)} is all of O_{0}^{(n)}. Thus, C^{(n)} is equivalent to the category of finite-dimensional ℂ-vector spaces. Projective functors θ_{w} act by zero on C^{(n)} for all w ∈ S_{n}, w ≠ e, while θ_{e} is the identity functor.

The graded version C_{gr}^{(n)} is equivalent to the category of graded finite-dimensional ℂvector spaces. Again, projective functors θ_{w}, w ≠ e, act by zero, and θ_{e} is the identity functor.

Thus, our categorification of the sign representation is rather trivial.

b. The trivial representation. The trivial representation ℤz of ℤ[S_{n}] is a free abelian group on one generator z, with the action wz = z, w ∈ S_{n}. The corresponding partition is (n), with the dual partition λ = (1^{n}). Only one decomposition μ = (1^{n}) corresponds to the dual partition; the parabolic subalgebra associated with (1^{n}) is the positive Borel subalgebra, and the parabolic category O_{0}^{(1^{n})} is all of O_{0}.

The unique self-dual indecomposable projective P in O_{0} is usually called the big projective module. Its endomorphism algebra End_{O}(P) is isomorphic to the cohomology ring H_{n} of the full flag variety Fl of ℂ^{n}, see [76].

The category C^{(1^{n})} is equivalent to the category of finite-dimensional H_{n}-modules. The unique (up to isomorphism) simple H_{n}-module generates the Grothendieck group K_{0}(H − mod) ≅ ℤ.

To describe how the functors θ_{i} act on C^{λ} consider generalized flag varieties

Fl_{i} = {0 ⊂ L_{1} ⊂ L_{2} ⊂ ⋯ ⊂ L_{n−1} ⊂ ℂ^{n}, L′_{i}|
dim(L_{j}) = j, dim(L′_{i}) = i, L_{i−1} ⊂ L′_{i} ⊂ L_{i+1}}.

This variety is a ℙ^{1}-bundle over the full flag variety Fl in two possible ways, corresponding to forgetting L_{i}, respectively L′_{i}. These two maps from Fl_{i} onto F induce two ring homomorphisms

H_{n} = H(Fl, ℂ) ⟶ H(Fl_{i}, ℂ)

which turn H(F_{i}, ℂ) into an H_{n}-bimodule. The functor θ_{i} : C^{(1^{n})} ⟶ C^{(1^{n})} is given by tensoring with this H_{n}-bimodule (under the equivalence C^{(1^{n})} ≅ H_{n}−mod).

To describe functors θ_{w} for an arbitrary w ∈ S_{n}, we recall that Fl = G/B where G = SL(n, ℂ) and B the Borel subgroup of G. The orbits of the natural left action of G on Fl × Fl are in natural bijection with elements of the symmetric group. Denote by O_{w} the orbit associated with w and by IC(\overline{O}_{w}) the simple perverse sheaf on the closure of this orbit. The cohomology of IC(\overline{O}_{w}) is an H_{n}-bimodule, and the functor

θ_{w} : H_{n}−mod ⟶ H_{n}−mod

takes a module M to the tensor product

H(IC(\overline{O}_{w}), ℂ) ⊗_{H_{n}} M.

Notice that all cohomology rings above have a canonical grading (by cohomological degree). The graded version of C^{(1^{n})} is the category of finite-dimensional graded H_{n}-modules and the graded version of θ_{w} tensors a graded module with the graded H_{n}-bimodule H(IC(\overline{O}_{w}), ℂ).

It is surprising how sophisticated the categorification of the trivial representation is, especially when compared with the categorification of the sign representation. Both the trivial and the sign representation are one-dimensional, but their categorifications have amazingly different complexities. All of the complexity is lost when we pass to the Grothendieck group, which has rank one.

c. Categorification of the Burau representation. Consider the partition λ^{∗} = (2, 1^{n−2}) and the dual partition λ = (n − 1, 1). The category C^{(n−1,1)} admits an explicit description, as follows. For n > 3 let A_{n−1} be the quotient of the path algebra of the graph from Figure 1 by the relations

(i|i + 1|i + 2) = 0,
(i|i − 1|i − 2) = 0,
(i|i − 1|i) = (i|i + 1|i)

Figure 1: Quiver diagram of A_{n−1}

Also, let A_{1} be the exterior algebra on one generator, and A_{2} be the quotient of the path algebra of the graph from Figure 1 (for n = 2) by the relations (1|2|1|2) = 0 = (2|1|2|1). The ℂ-algebra A_{n−1} is finite-dimensional.

3.1. Proposition. The category C^{(n−1,1)} is equivalent to the category of finite-dimensional left A_{n−1}-modules.

This is a well-known result, see e.g. [79] for n = 2 and [83] for the general case.

Denote by P_{i} the indecomposable left projective A_{n−1}-module A_{n−1}(i). This module is spanned by all paths that end in vertex i. Likewise, let _{i}P stand for the indecomposable right projective A_{n−1}-module (i)A_{n−1}. Under the equivalence between C^{(n−1,1)} and the category A_{n−1}−mod of finite-dimensional A_{n−1}-modules, the functor θ_{i} becomes the functor of tensoring with the bimodule

P_{i} ⊗ _{i}P.

The functors θ_{w} are zero for most w ∈ S_{n}. They are nonzero only when the corresponding composition of θ_{i}’s is nonzero (which rarely happens, note that already θ_{i}θ_{j} = 0 for |i − j| > 1).

The algebras A_{n−1}, as well as the modules P_{i}, _{i}P are naturally graded by the length of paths. The categories of finite dimensional graded modules over these algebras provide a categorification of the reduced Burau representation of each of the corresponding braid groups. For more information about the algebras A_{n−1} and their uses we refer the reader to the papers [52], [72], [75], [82], [34].

d. Categorification of the 2-column irreducible representation (partition (2^{n})).

Let λ^{∗} = (2^{n}) and λ = (n, n). The irreducible representation S_{ℚ}(λ^{∗}) has the following explicit description. The basis of the representation consists of crossingless matchings of 2n points positioned on the x-axis by n arcs lying in the lower half-plane, as depicted below.

The element 1 + s_{i} acts on a basis element by concatenating it with the diagram

If the concatenation contains a circle, we remove it and multiply the result by 2, see Figure 2.

A categorification of this representation and of its quantum deformation was described in [47], in the context of extending a categorification of the Jones polynomial to tangles. The basis elements b corresponding to crossingless matchings become indecomposable projective modules P_{b} over a certain finite-dimensional algebra H^{n}. The space of homs Hom_{H^{n}}(P_{a}, P_{b}) between projective modules is given by gluing crossingless matchings a and b along their endpoints and applying a 2-dimensional TQFT to the resulting 1manifold. The TQFT is determined by a commutative Frobenius algebra, which is just the cohomology of the 2-sphere. Spaces of these homs together with compositions

Hom_{H^{n}}(P_{a}, P_{b}) × Hom_{H^{n}}(P_{b}, P_{c}) ⟶ Hom_{H^{n}}(P_{a}, P_{c})

Figure 2: The product (1 + s_{i})b, for a basis element b, is either another basis element (top diagram) or the same basis element times 2.

determine H^{n} uniquely. The above geometric action of 1 + s_{i} lifts to the action on the category of H^{n}−mod of finite-dimensional H^{n}-modules given by tensoring with a certain H^{n}-module. This results in a very explicit categorification of the 2-column irreducible representation of S_{n} (and of the corresponding representation of the Hecke algebra) via the category of H^{n}-modules.

It was shown in [83] that H^{n}−mod is equivalent to the category C^{(n,n)} generated by projective-injective modules in the parabolic block O_{0}^{(n,n)}. Subquotient algebras of H^{n} considered in [83], [25], [24] can be used to categorify other 2-column representations of the Hecke algebra and the symmetric group. These subquotient algebras provide also a graphical description of the whole category O_{0}^{μ} for any composition μ_{1} + μ_{2} = n of n ([83]).

4. Miscellaneous

Braid group actions. Graded versions of projective functors θ_{w} categorify the action of the Hecke algebra H_{n,q} on its various representations. There is a homomorphism from the braid group on n strands to the group of invertible elements in H_{n,q}. This homomorphism, too, admits a categorification. The categorification should be at least an action of the braid group on a category, and this action is indeed well-known. To define it we need to pass to one of the triangulated extensions of the highest weight category: there does not seem to exist any interesting braid group actions on abelian categories, due to the positivity imposed by the abelian structure (see discussion in Section 6 of [47]).

The translation through the wall functors θ_{i}, i ≤ 1 ≤ n − 1, for the regular block O_{0} are compositions of two projective endofunctors (on and off the wall) of O, which are biadjoint to each other. This results in natural transformations θ_{i} ⟶ Id and Id ⟶ θ_{i}. Let D(O_{0}) be the bounded derived category of O_{0}. The complexes R_{i} and R′_{i} of functors 0 ⟶ θ_{i} ⟶ Id ⟶ 0 and 0 ⟶ Id ⟶ θ_{i} ⟶ 0 can be viewed as endofunctors of D(O_{0}) (we normalize the above functors so that Id sits in cohomological degree 0).

4.1. Proposition. The functors R_{i} define a braid group action on D(O_{0}). The functor R′_{i} is the inverse of R_{i}.

We distinguish between weak and genuine group actions; the terminology can be found in [52], [71], [53]. That R′_{i} and R_{i} are inverses of each other follows from a more general result of J. Rickard. That R_{i} define a weak braid group action follows from [62], [63, Proposition 10.1].

The Koszul duals of the functor R_{i} and its inverse are described in [61] in terms of the so-called twisting and completion functors on O_{0}. A geometric description of these functors can be found in [8] and [71].

The functors θ_{i} restrict to exact endofunctors of the parabolic categories O_{0}^{μ} and of the categories C^{μ}. Hence, the functors R_{i} and R′_{i} first induce endofunctors on D^{b}(O_{0}^{μ}), and define braid group actions there and then restrict to endofunctors of the subcategory given by complexes of projective-injective modules in O_{0}^{μ}.

The braid group acts by functors respecting the triangulated structure of the involved categories, resulting in a categorification of parabolic braid group modules as well as those irreducible representations of the braid group that factor through the Hecke algebra. The two commuting actions of projective functors on the category of Harish-Chandra bimodules as described in example 6 of Section 2 give rise to two commuting actions on the braid group on the derived category of the category of Harish-Chandra bimodules. For more examples of braid group actions on triangulated categories and a possible framework for these actions see [53].

Invariants of tangle cobordisms. In several cases, braid group actions on triangulated categories can be extended to representations of the 2-category of tangle cobordisms. The objects of this 2-category (when 2-tangles are not decorated) are non-negative integers, morphisms from n to m are tangles with n bottom and m top boundary components, and 2-morphisms are isotopy classes of tangle cobordisms. A representation of the 2-category of tangle cobordisms associated a triangulated category K_{n} to the object n, an exact functor K_{n} ⟶ K_{m} to a tangle, and a natural transformation of functors to a tangle cobordism. Such representations can be derived from examples 7 and 8 of Section 2 (see [82], [84], [64]) and from example d of Section 3 (see [47]). Example 6 is related to at least braid cobordisms (if not tangle cobordisms) via the construction of [49]. We expect that a categorification of tensor products of representations of quantum 𝔰𝔩_{2}, mentioned at the end of example 7 extends (after passing to derived categories, suitable functors, and natural transformations) to a representation of the 2-category of tangle cobordisms colored by irreducible representations of quantum 𝔰𝔩_{2}. Such an extension would give a categorification of the colored Jones polynomial.

The Cautis-Kamnitzer invariant of tangle cobordisms [23] is based on a similar framework, but their version of the category K_{n} is the derived category of coherent sheaves on a certain iterated ℙ^{1}-bundle. The Grothendieck group of their category is isomorphic to V^{⊗n}, where V is the fundamental representation of quantum 𝔰𝔩_{2}, just like in the example 7, but these two categorifications of V^{⊗n} are noticeably different. For instance, in the example 7 the category decomposes into the direct sum matching the weight decomposition of the tensor product, while the category in [23] is indecomposable. When the parameter is even, the two categorifications of V^{⊗2n} appear to have a common “core” subcategory, a categorification of the invariants in V^{⊗2n} (the latter isomorphic to S((2^{n}))) briefly reviewed in the example d above.

In the matrix factorization invariant of tangle cobordisms [51], the abelian category remains hidden inside the triangulated category of matrix factorizations.

Determinant of the Cartan matrix. With λ and μ as in example 4, let {P_{a}}_{a∈I} be a collection of indecomposable projectives in C^{μ}, one for each isomorphism class. The Cartan matrix of C^{μ} is an I × I matrix C with the (a, b)-entry being the dimension of Hom(P_{a}, P_{b}), the space of homomorphisms between projective modules P_{a} and P_{b}. Since End(P, P) is a symmetric algebra by [65], where P = ⊕_{a∈I}P_{a}, the Cartan matrix is

symmetric, c_{a,b} = c_{b,a}. These algebras are not commutative, but the center has a nice geometrical description as the cohomology of some Springer fibre ([48], and more general [14], [83]).

What is the determinant of this Cartan matrix? Since C^{μ} depends (up to equivalence) on the partition λ only ([65]), so does the determinant. The answer to the question is obvious in each of the first three cases considered in the previous section: the determinant is equal to 1 for λ = (n), to n! for λ = (1^{n}) and to n for λ = (n − 1, 1). The fourth case, when λ = (2^{n}), requires more work, and follows from the results of [29] and [44]. The determinant equals

∏_{i=1}^{n}(i + 1)^{r_{n,i}}, r_{n,i} = \binom{2n}{n − i} − 2\binom{2n}{n − i − 1} + \binom{2n}{n − i − 2},  (9)

with the convention \binom{j}{s} = 0 if s < 0. The answer for an arbitrary λ is more complicated. However, we want to point out that this determinant of the Cartan matrix is the determinant of the Shapovalov form ([74]) on a certain weight space of some irreducible 𝔰𝔩_{n}-module, as can be obtained from instance from [16]. It can be computed using the so-called Jantzen-Schaper formula [39, Satz 2].

The absolute value of the determinant has an interesting categorical interpretation. C^{μ} is equivalent to the category of finite-dimensional modules over some symmetric ℂ-algebra A^{μ}. Given any symmetric ℂ-algebra A (an algebra with a nondegenerate symmetric trace A ⟶ ℂ), the stable category A−\underline{mod} is triangulated. Objects of A−\underline{mod} are finitedimensional A-modules and the set of morphisms from M to N is the quotient vector space of all module maps modulo those that factor through a projective module. If det(C) ≠ 0 then the Grothendieck group of the stable category is finite abelian of cardinality equal to the absolute value of the determinant.

The graded version of this problem makes sense as well. Modules P_{a} are naturally graded, and to a pair (a, b) we can assign the Laurent polynomial in q which is the graded dimension of the graded vector space Hom(P_{a}, P_{b}). Arrange these polynomials into an I × I matrix (the graded Cartan matrix of C_{gr}^{μ}).

Problem: Find the determinant of the graded Cartan matrix of C^{μ}.

The determinant depends only on λ. Again, the answer is known in the above four cases. In the last case, the determinant of the graded Cartan matrix is given by formula (9), with the quantum integer [i + 1] = 1 + q^{2} + ⋯ + q^{2i} in place of (i + 1) in the product (the proof follows by combining results of [29] and [44]).

The determinant is algorithmically computable, since the entries of the graded Cartan matrix can be computed from the Kazhdan-Lusztig polynomials of the symmetric group. We are almost tempted to conjecture that, for any λ, the determinant (up to a power of q) is a product of quantum integers [j] = q^{j−1} + q^{j−3} + ⋯ + q^{1−j}, for small j, with some multiplicities.

In [38], a q-analogue of the Jantzen-Schaper formula is obtained. Generalizing [14] by working out a graded or q-version, should imply that the determinant is equal to the determinant of the q-analogue of the Shapovalov form on a suitable weight space of an irreducible 𝔰𝔩_{m}-module.

Acknowledgement M.K. was partially supported by the NSF grant DMS-0407784. V. M. was supported by STINT, the Royal Swedish Academy of Sciences, and the Swedish Research Council, C. S. was partially supported by EPSRC grant 32199. We thank all these institutions. We also thank Kenneth Brown for useful discussions and remarks and Jonathan Brundan for remarks on an earlier version of this paper.

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Department of Mathematics, Columbia University, New York, NY 10027, US

Department of Mathematics, Uppsala University, Box 480, 751 06, Uppsala, SWEDEN

Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK

Email: khovanov@math.columbia.edu
      mazor@math.uu.se
      c.stroppel@maths.gla.ac.uk

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THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

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TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, McGill University: gavin seal@fastmail.fm

Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Cuill Inc.: valeria@cuill.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Tom Leinster, University of Glasgow, T.Leinster@maths.gla.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca