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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 892387, 34 pages
The Khovanov-Lauda 2-Category and Categorifications of a Level Two Quantum SL(N) Representation
Department of Mathematics, University of California Berkeley, Berkeley, CA 94720-3840, USA
Received 18 March 2010; Accepted 29 April 2010
Academic Editor: Aaron Lauda
Copyright © 2010 David Hill and Joshua Sussan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We construct 2-functors from a 2-category categorifying quantum sl(n) to 2-categories categorifying the irreducible representation of highest weight .
Khovanov and Lauda introduced a 2-category whose Grothendieck group is . This work generalizes earlier work by Lauda for the case . Rouquier has independently produced a 2-category with similar generators and relations . There have been several examples of categorifications of representations of arising in various contexts. Khovanov and Lauda conjectured that their 2-category acts on various known categorifications via a 2-functor. For example, in their work they construct such a 2-functor to a category of graded modules over the cohomology of partial flag varieties. This 2-category categorifies the irreducible representation of of highest weight where is the first fundamental weight.
In this paper we construct this action for the categorification constructed by Huerfano and Khovanov in . They categorify the irreducible representation of highest weight , by a modification of a diagram algebra introduced in . The objects of 2-category are categories which are module categories over the modified Khovanov algebra. We explicitly construct natural transformations between the functors in  and show that they satisfy the relations in the Khovanov-Lauda 2-category giving the following theorem.
Theorem 1.1. Over a field of characteristic two, there exists a 2-functor
The Huerfano-Khovanov categorification is based on categories used for the categorification of -tangle invariants. This hints that a categorification of may also be obtained on maximal parabolic subcategories of certain blocks of category More specifically, we construct a 2-category whose objects are full subcategories of graded category whose set of objects are those modules which have projective presentations by projective-injective objects. The 1-morphisms of are certain projective functors. We explicitly construct the 2-morphisms as natural transformations between the projective functors by the Soergel functor We then prove the following.
Theorem 1.2. There is a 2-functor
It should be possible to categorify for using categories which appear in various knot homologies. For the module categories in the Huerfano-Khovanov construction should be replaced by suitable categories of matrix factorization based on Khovanov-Rozansky link homology. The categories of matrix factorizations must be generalized from those used in . Khovanov and Rozansky suggest that the categories of matrix factorizations should be taken over tensor products of polynomial rings invariant under the symmetric group. These categories were studied in depth by Yonezawa and Wu [7, 8]. In fact, the isomorphisms of functors categorifying the relations were defined implicitly in . To check that there is a 2-representation of the Khovanov-Lauda 2-category, these isomorphisms would need to be made more explicit. The category approach should be modified as well. Now the objects of the 2-category should be subcategories of parabolic subcategories corresponding to the composition of blocks of and the stabilizer of the dominant integral weight is taken to be where each compare, for example, Section 5 below. Note that a categorification of for arbitrary dominant integral , hence in particular of , is constructed in  using cyclotomic quotients of Khovanov-Lauda-Rouquier algebras.
While this paper was in preparation, two very relevant papers appeared. In , Brundan and Stroppel also defined the appropriate natural transformations and checked relations between them to establish a version of the first theorem above, but for Rouquier's 2-category from  rather than the Khovanov-Lauda 2-category. One of the advantages of their result is that they are able to work over an arbitrary field, while we work over a field of characteristic 2 in constructing the 2-functor to . It is not immediately clear to us how to use their sign conventions to get an action of the full Khovanov-Lauda 2-category in characteristic zero, because they seem to lead to inconsistencies between Propositions 4.7, 4.8, 4.10, and 4.16. Additionally, Brundan and Stroppel categorify using graded category . More precisely, they first categorify the classical limit of at using a certain parabolic category , without mentioning gradings. Then they establish an equivalence between this category and the (ungraded) diagrammatic category. Finally, they observe that both categories are Koszul (by  and , respectively) so, exploiting unicity of Koszul gradings, their categorification at can be lifted to a categorification of the module itself in terms of graded category . Our construction on the graded category side is more explicit, relying heavily on the Soergel functor, the Koszul grading that inherits from geometry, and explicit calculations on the cohomology of flag varieties made in . In the other relevant paper, M. Mackaay  constructs an action of the Khovanov-Lauda 2-category on a category of foams which is the basis of an -knot homology.
2. The Quantum Group
2.1. Root Data
Let denote the Lie algebra of traceless -matrices with standard triangular decomposition . Let be the root system of type with simple system . Let denote the symmetric bilinear form on satisfying where is the Cartan matrix of type : Let be the set of simple roots relative to . Let be the elements satisfying , and let denote the root lattice, positive root lattice, weight lattice, and dominant weight lattice, respectively.
Set , , and . Define , and extend the definition of to all accordingly. Finally, for , let be the sign of .
The quantum group is the associative algebra over with generators for , satisfying the following conditions: (1), and for ,(2), , (3), ,(4), , ,(5), , .
We fix a comultiplication given as follows for all : Via a tensor product of -modules becomes a -module.
In this paper we are interested in the irreducible -modules, with highest weight Therefore, we will identify the weight lattice as follows. Assume that . For each , set
Let denote the set of weights of . It is well known that under this identification each satisfies for all and .
3. The Khovanov-Lauda 2-Category
Let be a field. The -linear 2-category defined here was originally constructed in .
Let , where and denote -fold Cartesian products. Given that , let Given that , let and, for , define . Finally, define
3.1. The Objects
The set of objects for this 2-category is the weight lattice, .
3.2. The 1-Morphisms
For each , let be the identity morphism and, for , set . For each , we define morphisms . Evidently, we have . For , we have where if , and refers to a grading shift. Observe that unless , and .
3.3. The 2-Morphisms
The 2-morphisms are generated by for . We define to be the identity transformation.
For , the degrees of the basic 2-morphisms are given by
Let and Let and Then denote the horizontal composition of these 2-morphisms by which is an element of If denote the vertical composition of and by
For convenience of notation, we define the following 2-morphisms. If , let For each , define the bubble Also, define half-bubbles
We now define the relations satisfied by these basic 2 morphisms. In what follows, we omit the argument when the relation is independent of it.Relations(a)For all , (b)For all , (c)Suppose that and , then (d)Let . If , (e)Let . If , then (f)Let . If , then If , then
Remark 3.1. Note that in 1(e) above the exponent of the bubble may be negative, which is not defined. To make sense of this, for , define these symbols (referred to as fake bubbles in ) inductively by the formula and whenever .The nil-Hecke Relations(a)For each , . (b)For , (c)For , (d)For ,
Remark 3.2. For all , setThe Relations(a)For , (b)For , , (c)For , , (d)For ,
4. The Huerfano-Khovanov 2-Category
4.1. The Khovanov Diagram Algebra
Let This is a -graded algebra with multiplication map such that and . There is a comultiplication map such that and There is a trace map such that and There is also a unit map given by Also, let be given by This algebra gives rise to a two-dimensional TQFT , which is a functor from the category of oriented cobordisms to the category of abelian groups. The functor sends a disjoint union of copies of the circle to For a cobordism from two circles to one circle, For a cobordism , from one circle to two circles, For a cobordism from the empty manifold to For a cobordism , from the empty manifold to
For any nonnegative integer consider marked points on a line. Let be the set of nonintersecting curves up to isotopy whose boundary is the set of the marked points such that all of the curves lie on one side of the line. Then there are elements in this set. The set of crossingless matches for is given in Figure 1.
Let Then is a collection of circles obtained by concatenating with the reflection of in the line. Then applying the two-dimensional TQFT one associates the graded vector space to this collection of circles. Taking direct sums over all crossingless matches gives a graded vector space where the degree component of is the degree component of This graded vector space obtains the structure of an associative algebra via ; compare, for example, .
Let be a tangle from points to points. Let be a crossingless match for points and a crossingless match for points. Then let be the concatenation and See Figure 3 for an example when is the identity tangle.
To any tangle diagram from points to points, there is an -bimodule To any cobordism between tangles and , there is a bimodule map of degree , where is the Euler characteristic of ; compare, for example, Proposition of .
Consider the tangles and in Figure 4. Then there are saddle cobordisms and
Lemma 4.1. Let and be the tangles in Figure 5. (1)There exists an -bimodule homomorphism of degree one. (2)There exists an -bimodule homomorphism of degree one.
Proof. There is a degree zero isomorphism of bimodules Then by  there is a bimodule map of degree one
where denotes the identity map. Finally note that Then is the composition of these maps.
The construction of is similar.
Remark 4.2. One may construct, in a similar way, maps of degree one: and .
Lemma 4.3. Let and be two crossingless matches. Let be the tangle on the right side of Figure 5. Let be the tangle in Figure 4. Consider the homomorphism induced by the cobordism . Let , where corresponds to the circle passing through the point on the top line and corresponds to the remaining circles. Then
Proof. The map is induced by the cobordism On the set of circles, this cobordism is a union of identity cobordisms and a cobordism The result now follows upon applying
Lemma 4.4. Let be the identity tangle from points to points, a tangle from points to points, and a tangle from points to points. Let and be cup diagrams for points (). Consider the map where the first and last maps are isomorphisms and the middle map is Let correspond to the circle passing through point of correspond to the remaining circles, and Then the map above sends
Proof. The map is induced by a cobordism On the set of circles, this cobordism is union of identity cobordisms and a cobordism The result now follows upon applying
4.2. The Huerfano-Khovanov Categorification
Let . Recall that . Hence, for , we have Label collinear points by the integers Those points labeled by or will never be the boundaries of arcs but will rather just serve as place holders. Then define the algebra (as in Section 4.1), where Let be the identity element.
The cobordisms and are saddle cobordisms for Similarly, the cobordisms are saddle cobordisms in the opposite direction. For example, the cobordism is given in Figure 9.
Let be the category of finitely generated, graded -modules, and let be the identity functor. For , set .
Let . Let denote the identity functor which is tensoring with the -bimodule . Let be the functor of tensoring with a bimodule defined as follows: Evidently, for all , and .
For , let be the grading shift functor Finally, set , , , and .
Propositions and of  are that these functors satisfy quantum relations.
Proposition 4.5 (see [4, Propositions ]). One has(1), and for , (2), for , (3) if , (4) if , (5) if , (6)For ,
Now we define the Huerfano-Khovanov 2-category over the field , char.
4.3. The Objects
The objects of are the categories , .
4.4. The 1-Morphisms
For each , is the identity morphism and, for , set as above. For each , we have defined morphisms . Evidently, we have . For , we have where if , and refers to a grading shift. Observe that unless , and .
4.5. The 2-Morphisms
In this section we define natural transformations of functors. These maps were not explicitly defined in . Note that the notation for these 2-morphisms is similar to the 2-morphisms in Section 3 since we will construct a 2-functor mapping one set of 2-morphisms to the other. Recall the convention for .
(1) The Maps ,
Let , and let and be the identity maps.
(2) The Maps
For we define maps of degree 2. Let be the tangle diagram for the functor It depends on the pair Let and be crossingless matches such that is a disjoint union of circles. Thus for some natural number Define where (a)if then the factor in corresponds to the circle passing through the point on the bottom set of dots for tangle in Figure 6,(b)if then the factor in corresponds to the circle passing through the point on the top set of dots for tangle in Figure 6,(c)if then the factor in corresponds to the circle passing through the point on the top set of dots for tangle in Figure 7,(d)if then the factor in corresponds to the circle passing through the point on the bottom set of dots for tangle in Figure 7.
(3) The Map
We define a map There are four nontrivial cases for to consider. (a) The identity functor is induced from the identity tangle . The functor is isomorphic to tensoring with the bimodule which is equal to . Thus in this case is given by the identity map.(b) Then the functor is isomorphic to tensoring with the bimodule . Then is (c) Then the functor is isomorphic to tensoring with the bimodule Then the bimodule map is given by (d) The functor is isomorphic to tensoring with the bimodule As in case 1, this tangle is isotopic to the identity so the map between the functors is the identity map.
(4) The Map .
We define a map There are four non-trivial cases for to consider. (a) The functor is isomorphic to tensoring with the bimodule which is equal to . Thus in this case is given by the identity map. (b) Then the functor is isomorphic to tensoring with the bimodule . Then the homomorphism is (c) Then the functor is isomorphic to tensoring with the bimodule Then the bimodule map is given by (d) The functor is given by tensoring with the bimodule As in case 1, this tangle is isotopic to the identity so the map between the functors is the identity map.
(5) The Maps
We define a map for .
There are four cases for and to consider and then subcases for
(a) In this case, the functors are non-trivial only if and The bimodule for is isomorphic to tensoring with the bimodule Then (b) In this case, the functors and are isomorphic via an isomorphism induced from a cobordism isotopic to the identity so set to the identity map.(c) There are four non-trivial subcases to consider. (i) The bimodule for is The bimodule for is In this case we define the bimodule map to be (ii) The functor is given by tensoring with a bimodule isomorphic to The bimodule for is isomorphic to Then define to be since (iii) The bimodule for is isomorphic to The bimodule for is isomorphic to Then define to be since (iv) The bimodule for is The bimodule for is Then set (d) We essentially just have to read the maps in cases (c)(i)–(iv) above backwards. (i) The functors are just as in case (c)(i). Now the map is (ii) The bimodule for is isomorphic to Then define (iii) The bimodule for is isomorphic to Then define (iv) The functors are just as in case (c)(iv). Now the map is
Proposition 4.6. For all , and , the maps are bimodule homomorphisms.
For convenience of notation, we define the following 2-morphisms. If , let For each , define the bubble and define fake bubbles inductively by the formula and whenever . Also, define half-bubbles Finally, for , define
4.6. The 2-Morphism Relations
In this section we prove certain relations between the 2-morphisms defined in Section 4.5. This will allow us to define a 2-functor from the Khovanov-Lauda 2-category to the Huerfano-Khovanov 2-category. Again, we will often omit the argument when it is clear from context.
Proposition 4.7. For all ,
Proof. The second equality is similar to the first equality. The case is similar to the case so we just compute the map on the bimodule for the functor for . There are four cases to consider.
Suppose that Then the tangle diagrams for the functors and are and and can be found in Figure 12.
The cobordism between the tangles is isotopic to the identity map so in this case the composition is equal to the identity map.
The case is similar to the case.
Now let Then the tangle diagrams for the functors and can be found in Figure 13.
Let be the bimodule for the functor Then the bimodule for is isomorphic to The map is given by the unit map which sends an element to The map is obtained from the cobordism joining the circle to the upper cup which induces the multiplication map. This maps to Thus the composition is equal to the identity.
Finally consider the case The tangle diagrams for the functors and can be found in Figure 14.
Let be the bimodule giving rise to the functor and let be the bimodule giving rise to the functor Let , where is in the tensor factor corresponding to the circle passing through point on the bottom row of the left side of Figure 14 and belongs to the remaining tensor factors.
The cobordism between the two tangle diagrams is a saddle which, on the level of bimodule maps, sends Then the map from to is given by so by considering the two cases or Thus the composition is equal to the identity map.
Proposition 4.8. One has
Proof. We prove only the first equality as the second is similar. There are four cases to consider for which the functor is nonzero.
Suppose that Then the tangle diagrams for the functors and can be found in Figure 12.
Note that the bimodules for and are the same. Denote this bimodule by . Let , where is an element in the tensor factor corresponding to a circle passing through point in the bottom row of Figure 12. Then the first map is given by the identity cobordism and is thus the identity. The second map is multiplication by on all tensor components corresponding to circles passing through the point in the second row of the right side of Figure 12. The final map is also given by the identity cobordism. Thus the composition maps On the other hand,
The case is similar to the previous case.
Suppose that Then the bimodule for the functor is and the tangle diagram for is Let , where is an element of the tensor factor corresponding to the circle passing through the point in the top row of the tangle and is an element in the remaining tensor factors. Then the composition of maps sends This is equal to
Suppose that Then the tangle diagrams for the functors and can be found in Figure 14.
Let be the bimodule for the functor and let be the bimodule for Let , where is an element in the tensor factor corresponding to the circle passing through point on the bottom row of Figure 14 and is an element in the remaining tensor factors. First let Then where the last map is If then
Proposition 4.9. Suppose and , then
Proof. In order that it must be the case that Thus the only possibility is and Then the bimodule for is Thus the map is given by the unit map. The map is given by the trace map. Thus the composition of the maps in the proposition sends an element
Proposition 4.10. If , then
Proof. The only cases to consider are
Consider the case Let Then the bimodule corresponding to is Let Then , , and Thus in this case, the composition is the identity map.
For the case The cobordism between the tangle diagrams for the identity functor and is isotopic to the identity cobordism. Similarly, the cobordism between the tangle diagrams for the functors and the identity functor is isotopic to the identity cobordism. Thus the bimodule map is equal to the identity.
The case is the same as the case
Proposition 4.11. Let . If , then
Proof. There are three cases to consider:
For the case the first term on the right-hand side is zero since that map passes through the functor which is zero for this The summation on the right-hand side reduces to by definition (4.17) of the fake bubbles. This map is a composition This composition of maps is the identity.
The case is similar to the case.
For the case the first term on the right-hand side is zero as in the previous two cases. The summation on the right-hand side consists of three terms, which simplifies by (4.17) to Let Then the bimodule for is Then Under this composition of maps, maps to zero since the first map is given by a trace map on the first component. The element gets mapped to as follows: where the first map is the trace map, the second map is the unit map, and the third map is multiplication by Similarly, Under this composition, and Finally, the map is zero because the middle term is zero. Thus the right-hand side is the identity as well.
Proposition 4.12. Let . (1)If , then (2)If , then
Proof. We prove , the proof of () being similar. Since the maps on both sides pass through the functor