A braid group action on a p-DG homotopy category
aa r X i v : . [ m a t h . QA ] D ec A braid group action on a p -DG homotopy category You Qi, Joshua Sussan, Yasuyoshi YonezawaJanuary 1, 2021
Abstract
We construct a braid group action on a homotopy category of p -DG modules of a deformed Websteralgebra. Contents p -complexes 2 p -DG algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 A basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References 33
An approach to categorification of quantum groups, their representations, and quantum invariants at aprime p root of unity was outlined in [Kho16] and further developed in [Qi14]. These works suggest thatone should look for p -differentials on structures categorifying objects at generic values of the quantumparameter. There has been some progress in this program for quantum sl [EQ16a, EQ16b, KQ15, KQS17,QS16, QS17, QS18].In particular, a categorification of the braid group action on the Burau representation at a prime rootof unity was constructed in [QS16]. The authors considered a p -DG structure on the algebra A ! n whichdescribes a singular block of category O ( gl n ) corresponding to the Young subgroup S × S n − . Ignoringthe p -DG structure, this categorical action is a consequence of Koszul duality and Khovanov and Seidel’saction of the braid group on the homotopy category of modules over the zigzag algebra A n . Whereasthe projective modules play the role of the Temperley-Lieb algebra in [KS02], the simple objects form anexceptional sequence of objects on the other side of Koszul duality. As a result, one must find projectiveresolutions of the simple objects of A ! n in order to construct the braid group action directly. In the context ofthe p -differential, one must find cofibrant replacements of the simple objects. This was the main technicalstep in [QS16]. cknowledgements. 2 Partially motivated by the construction in [QS16], a deformation W = W ( n, of A ! n was consideredin [KS18] and the authors showed that there is a categorical braid group action on the homotopy categoryof W -modules. This result was extended in [KLSY18] to a categorical braid group action on the homotopycategory of a deformation of more general Webster algebras for sl , for which W is a special case (hence thenotation).In this note, we return to the simplified setting of W . There is a p -DG structure on this algebra (see[Yon]). We show that there are braiding complexes in a homotopy category of p -DG W -modules using keyideas from [KR16]. Using some results from p -DG theory we extend the main result of [KS18] to the p -DGsetting. Theorem.
There is a categorical action of the braid group on n strands on the relative p -DG homotopycategory of W .Deforming the algebra A ! n allows us to replace the p -DG derived category in [QS16] with the relative p -DG homotopy category here. This theorem should be compared to [QS16, Theorem 5.14], albeit in theweaker context of the relative homotopy category. However, the authors believe that the results will becomeuseful towards builting a p -DG link homology theory, as proposed in [QS20]. These further questions willbe addressed in subsequent works of the authors.In [KLSY18] a braid group action on the homotopy category of deformed Webster algebras was con-structed by exhibiting an action of the Khovanov-Lauda-Rouquier -category [KL10, Rou08], and then us-ing symmetric Howe duality. We expect the main result here holds on the level of generality of [KLSY18].One would need to consider a p -DG version of the Khovanov-Lauda-Rouquier category and show thebraiding of Rickard complexes [CK12] holds in the presence of the p -differential.In recent work [Web], the main result of [KLSY18] was proved by relating the deformed algebra tovarious objects in Lie theory and geometry. It would be interesting to import the p -DG structure to thecategories of Gelfand-Tsetlin modules and perverse sheaves studied there. Y. Q. is partially supported by the NSF grant DMS-1947532. J. S. is partially supported by the NSF grantDMS-1807161 and PSC CUNY Award 63047-00 51. Y. Q. and Y. Y. were partially supported by the Re-search Institute for Mathematical Sciences, an International Joint Usage/Research Center located in KyotoUniversity. p -complexes Let A be an algebra over the ground field k of finite characteristic p > . We equip A with the trivial p -differential graded structure by declaring ∂ A := 0 . In this subsection, we study a functor relating the usualhomotopy category C ( A, d ) of A with its p -DG homotopy category C ( A, ∂ ) .To do this, recall that a chain complex of A -modules consists of a collection of A -modules and homor-phisms d i : M i −→ M i +1 called coboundary maps · · · d i − / / M i − d i − / / M i d i / / M i +1 d i +1 / / M i +2 d i +2 / / · · · , satisfying d i ◦ d i − = 0 for all i ∈ Z . A null-homotopic map is a squence of A -module maps f i : M i −→ N i , i ∈ Z , of A -modules, as depicted in the diagram below, · · · d i − / / M i − h i − ②②②② | | ②②②② d i − / / f i − (cid:15) (cid:15) M ih i ①①① { { ①①① d i / / f i (cid:15) (cid:15) M i +1 h i +1 ①①①① { { ①①①① d i +1 / / f i +1 (cid:15) (cid:15) M i +2 d i +2 / / h i +2 ✈✈✈✈ z z ✈✈✈✈ f i +2 (cid:15) (cid:15) · · · h i +3 ②②② | | ②②② · · · d i − / / N i − d i +1 / / N i d i / / N i +1 d i +1 / / N i +2 d i +2 / / · · · elativehomotopycategories 3 which satisfy f i = d i +1 ◦ h i + h i +1 ◦ d i for all i ∈ Z . The homotopy category C ( A, d ) , by construction, is thequotient of chain complexes over A by the ideal of null-homotopic morphisms.We define the p -extension functor P : C ( A, d ) −→C ( A, ∂ ) (2.1)as follows. Given a complex of A -modules, we repeat every term sitting in odd homological degrees ( p − times while keeping terms in even homological degrees unaltered. More explicitly, for a given complex · · · d k − / / M k − d k − / / M k d k / / M k +1 d k +1 / / M k +2 d k +2 / / · · · , the extended complex looks like · · · d k − / / M k − · · · M k − d k − / / / / M k EDBCGF φ k @A / / ❵❵❵❵❵❵❵ M k +1 · · · M k +1 d k +1 / / M k +2 d k +2 / / · · · . Likewise, for a chain-map · · · d k − / / M k − d k − / / f k − (cid:15) (cid:15) M k − d k − / / f k − (cid:15) (cid:15) M k d k / / f k (cid:15) (cid:15) · · ·· · · d k − / / N k − d k − / / N k − d k − / / N k d k / / · · · the obtained morphism of p -DG A -modules is given by · · · d k − / / M k − d k − / / f k − (cid:15) (cid:15) M k − f k − (cid:15) (cid:15) · · · M k − d k − / / f k − (cid:15) (cid:15) M k d k / / f k (cid:15) (cid:15) · · ·· · · d k − / / N k − d k − / / N k − · · · N k − d k − / / N k d k / / · · · This is clearly a functor from the abelian category of cochain complexes over A into the category of p -complexes of A -modules, which we call b P . Lemma 2.1. [QS20, Lemma 2.2] The functor b P preserves the ideal of null-homotopic morphism.This lemma implies that b P descends to a functor P : C ( A, d ) −→C ( A, ∂ ) . (2.2)which we call the p -extension functor . Proposition 2.2. [QS20, Proposition 2.3] The p -extension functor P is exact. For any graded or ungraded algebra B over k , denote by d the zero super differential and by ∂ the zero p -differential on B , while letting B sit in homological degree zero. When B is graded, the homologicalgrading is independent of the internal grading of B . For a graded module M over a graded algebra B , welet M { n } denote the module M , where the internal grading has been shifted up by n .Suppose ( A, ∂ A ) is a p -DG algebra, i.e., a graded algebra equipped with a differential ∂ A of degree two,satisfying ∂ pA ( a ) ≡ , ∂ A ( ab ) = ∂ A ( a ) b + a∂ A ( b ) , (2.3) for all a, b ∈ A . In other words, A is an algebra object in the module category of the graded Hopf algebra H = k [ ∂ ] / ( ∂ p ) , where the primitive degree-two generator ∂ ∈ H acts on A by the differential ∂ A .Then, we may form the smash product algebra A H in this case. As a k -vector space, A H is isomorphicto A ⊗ H , subject to the multiplication rule determined by ( a ⊗ ∂ )( b ⊗ ∂ ) = ab ⊗ ∂ + a∂ A ( b ) ⊗ ∂. (2.4)Notice that, by construction, A ⊗ and ⊗ H sit in A H as subalgebras.There is an exact forgetful functor between the usual homotopy categories of chain complexes of graded A H -modules F d : C ( A H, d ) −→C ( A, d ) . An object K • in C ( A H, d ) lies inside the kernel of the functor if and only if, when forgetting the H -module structure on each term of K • , the complex of graded A -modules F d ( K • ) is null-homotopic. Thenull-homotopy map on F d ( K • ) , though, is not required to intertwine H -actions.Likewise, there is an exact forgetful functor F ∂ : C ( A H, ∂ ) −→C ( A, ∂ ) . Similarly, an object K • in C ( A H, ∂ ) lies inside the kernel of the functor if and only if, when forgettingthe H -module structure on each term of K • , the p -complex of A -modules F ( K • ) is null-homotopic. Thenull-homotopy map on F ( K • ) , though, is not required to intertwine H -actions. Definition 2.3.
Given a p -DG algebra ( A, ∂ A ) , the relative homotopy category is the Verdier quotient C ∂ ( A, d ) := C ( A H, d )Ker( F d ) . Likewise, the relative p -homotopy category is the Verdier quotient C ∂ ( A, ∂ ) := C ( A H, ∂ )Ker( F ∂ ) . The subscripts in the definitions are to remind the reader of the H -module structures on the objects.The categories C ∂ ( A, d ) and C ∂ ( A, ∂ ) are triangulated. By construction, there is a factoraization of theforgetful functor C ( A H, d ) F d / / ' ' ◆◆◆◆◆◆◆◆◆◆◆ C ( A, d ) C ∂ ( A, d ) rrrrrrrrrr , C ( A H, ∂ ) F ∂ / / ' ' ◆◆◆◆◆◆◆◆◆◆◆ C ( A, ∂ ) C ∂ ( A, ∂ ) rrrrrrrrrr . Proposition 2.4. [QS20, Proposition 2.13] The p -extension functor P : C ( A H, d ) −→C ( A H, ∂ ) descendsto a functor, still denoted P , between the relative homotopy categories: P : C ∂ ( A, d ) −→C ∂ ( A, ∂ ) . p -DG algebra We begin by recalling the definition of a particular deformed Webster algebra W ( n, . More general ver-sions of these algebras W ( s , n ) can be found in [KS18, KLSY18]. The p -DG structures on these algebras wereintroduced in [Yon]. The algebras are deformations of the algebras introduced by Webster in [Web17]. he p -DGalgebra 5Definition 3.1. Let n ≥ be an integer. Let Seq(1 n , be the set of all sequences i = ( i , ..., i n +1 ) where n ofthe entries are and the other entry is the symbol b . Therefore, we have | Seq(1 n , | = n + 1 . Denote by i j the j -th entry of i . Each transposition σ j ∈ S n +1 naturally acts on the set of sequences. W = W ( n, is the graded algebra over the ground field k generated by e ( i ) , where i ∈ Seq(1 n , , x j ,where ≤ j ≤ n , y and ψ j , where ≤ j ≤ n , satisfying the relations below. X i ∈ Seq(1 n , e ( i ) = 1 (3.1) e ( i ) e ( j ) = δ i , j e ( i ) (3.2) ψ j e ( i ) = e ( σ j ( i )) ψ j (3.3) ψ j e ( i ) = 0 if i j = i j +1 = 1 (3.4) ψ j ψ ℓ = ψ ℓ ψ j if | j − ℓ | > (3.5) x j and y are central (3.6) ψ j e ( i ) = ( x j − y ) e ( i ) if ( i j , i j +1 ) = (1 , b ) , ( b , (3.7)The degrees of the generators are deg( e ( i )) = 0 , deg( x j ) = 2 , deg( y ) = 2 , deg( ψ j ) = 1 . In some contexts, it is natural to impose the so-called cyclotomic relation e ( i ) = 0 if i = b . (3.8)Quotienting W by the cyclotomic relation yields an algebra denoted by W .We now recall the diagrammatic description of the deformation of the Webster algebra W = W ( n, .Consider collections of smooth arcs in the plane connecting n red points and black point on one horizontalline with n red points and black point on another horizontal line. The red points correspond to the ’sin the sequence i of e ( i ) and the black point corresponds to b in the sequence i . The n red points and black point on the line appear in the order in which they appear in i of e ( i ) . The arcs are colored ina manner consistent with their boundary points. Arcs are assumed to have no critical points (in otherwords no cups or caps). Arcs are allowed to intersect (as long as they are both not solid red), but no tripleintersections are allowed. Arcs can carry dots. Two diagrams that are related by an isotopy that does notchange the combinatorial types of the diagrams or the relative position of crossings are taken to be equal.We give W ( n, the structure of an algebra by concatenating diagrams vertically as long as the colors ofthe endpoints match. If they do not, the product of two diagrams is taken to be zero. For two diagrams D and D , their product D D is realized by stacking D on top of D .The generator e ( i ) is represented by vertical strands comprised of n red strands and one black strand.For instance, in the case i = (1 , , b , , the generator e ( i ) is represented byThe generator y is represented by a black dot and x j is represented by a red dot on the j th red strand: , . The generator ψ j e ( i ) is represented by a black-red crossing on the left of the diagram below if i j = 1 and i j +1 = b or a black-red crossing on the right if i j = b and i j +1 = 1 : , . Note that a red-red crossing does not appear due to relation (3.4). basis 6
Far away generators commute. The relations (3.6) and (3.7) involving red-black strands are = , = , (3.9) = , = , (3.10) = − , = − . (3.11)The cyclotomic relation (3.8) translates to: a black strand, appearing on the far left of any diagram, annihi-lates the entire picture: · · · = 0 . (3.12)Note that setting red dots to be zero, we recover Webster’s algebra, and so we may think of the polynomialalgebra generated by red dots as a polynomial deformation space of the Webster algebra.For i = 0 , . . . , n , there is a sequence (1 i , b , n − i ) . Denote by e i the idempotent e (1 i , b , n − i ) ; e i = e (1 i , b , n − i ) = · · · i i +1 · · · n . (3.13)Over a base field k of finite characteristic p > , a p -differential graded ( p -DG) algebra structure wasintroduced on a generalization of W in [Yon]. Definition 3.2.
The p -derivation ∂ : W → W of degree satisfying the Leibniz rule: ∂ ( ab ) = ∂ ( a ) b + a∂ ( b ) for any a, b ∈ W is defined on the generators of the algebra W by ∂ ( e i ) = 0 , ∂ ( x i ) = x i , ∂ ( y ) = y , ∂ ( ψ j ) = x j ψ j e j − + yψ j e j and extended by the Leibniz rule to the entire algebra.An easy exercise shows that ∂ p ≡ .In the diagrammatic description, we have ∂ ! = , ∂ ! = , (3.14a) ∂ ! = , ∂ ! = . (3.14b) A basis for the cyclotomic deformed Webster W ( n, was given in [KS18] (see also [Web] and [SW11]). Weslightly modify this basis and a representation of the algebra for the case where the cyclotomic condition isomitted. basis 7Proposition 3.3. Let R n = k [ x , . . . , x n ] , V n,i = R n [ y i ] , V n = n M i =0 V n,i . There is an action of W ( n, on V n determined by e i : f ( x , y j ) ∈ V n,j (cid:26) f ( x , y i ) ∈ V n,i if j = i if j = ix a k k e i : f ( x , y i ) ∈ V n,i x a k k f ( x , y i ) ∈ V n,i ,y a k e i : f ( x , y i ) ∈ V n,i y a k i f ( x , y i ) ∈ V n,i ,ψ i e i : f ( x , y i ) ∈ V n,i f ( x , y i − ) ∈ V n,i − ψ i +1 e i : f ( x , y i ) ∈ V n,i ( y i +1 − x i +1 ) f ( x , y i +1 ) ∈ V n,i +1 . In the diagrammatic description, we have a · · · i a i b i +1 a i +1 · · · n a n : f ( x , y i ) ∈ V n,i x a · · · x a n n y bi f ( x , y i ) ∈ V n,i · · · i − i i +1 · · · n : f ( x , y i ) ∈ V n,i f ( x , y i − ) ∈ V n,i − · · · i i +1 i +2 · · · n : f ( x , y i ) ∈ V n,i ( y i +1 − x i +1 ) f ( x , y i +1 ) ∈ V n,i +1 Proof.
This is a straightforward check.Next we will define elements which form a basis of W ( n, .Let a = ( a , . . . , a n ) ∈ Z n ≥ and b ∈ Z ≥ . For ≤ i ≤ j ≤ n , define N E i,j ( a , b ) := n Y k =1 x a k k y b ψ j ψ j − · · · ψ i +1 e i = a · · · i a i b i +1 a i +1 · · · na n if i = j a · · · i a i bi +1 a i +1 · · · j a j j +1 a j +1 · · · n a n if i < j For ≤ i < j ≤ n , define SE i,j ( a , b ) := n Y k =1 x a k k y b ψ i +1 ψ i · · · ψ j e i = a · · · i a i bi +1 a i +1 · · · j a j j +1 a j +1 · · · n a n The proof of the next proposition is similar to the proof of [KS18, Proposition 2, Corollary 1]. See also[SW11, Proposition 4.9] and [Web, Definition 2.7].
Proposition 3.4.
We have the following facts about W ( n, and its representation V n .1. The action of W ( n, on V n is faithful.2. W ( n, has a basis { N E i,j ( a , b ) , SE i ′ ,j ′ ( a ′ , b ′ ) | ≤ i ≤ j ≤ n, ≤ i ′ < j ′ ≤ n, a , a ′ ∈ Z n ≥ , b, b ′ ∈ Z ≥ } . We recall the ( W, W ) -bimodules W i for i = 1 , . . . , n − , introduced in [KS18], and W i,i +1 for i = 1 , . . . , n − .These bimodules were generalized in [KLSY18].In order to define the bimodules W i and W i,i +1 , we introduce the algebra W ((1 n ) i,k , which is also aparticular deformed Webster algebra and has been shown in earlier works to be isomorphic to a subalgebraof W .Let (1 n ) i,k be the sequence whose n − k entries are and i -th entry from left is k , (1 n ) i,k = (1 i − , k, n − i − k +1 ) , and let Seq((1 n ) i,k , be the set of all sequences i = ( i , ..., i n − k +2 ) composed of the sequence (1 n ) i,k intowhich the symbol b is inserted.For instance, the set Seq((1 ) , , is { ( b , , , , (1 , b , , , (1 , , b , , (1 , , , b ) } . Definition 4.1. W ((1 n ) i,k , is the graded algebra over the ground field k generated by• e ( i ) , where i ∈ Seq((1 n ) i,k , ,• x j , where ≤ j < i and i < j ≤ n − k + 1 ,• y ,• ψ j , where ≤ j ≤ n − k + 1 ,• E ( d ) , where ≤ d ≤ k satisfying the relations below. X i ∈ Seq((1 n ) i,k , e ( i ) = 1 (4.1) e ( i ) e ( j ) = δ i , j e ( i ) (4.2) ψ j e ( i ) = e ( σ j ( i )) ψ j (4.3) ψ j e ( i ) = 0 if ( i j , i j +1 ) = (1 , , (1 , k ) , ( k, (4.4) ψ j ψ ℓ = ψ ℓ ψ j if | j − ℓ | > (4.5) x j , y and E ( d ) are central (4.6) ψ j e ( i ) = ( y − x j ) e ( i ) if ( i j , i j +1 ) = (1 , b ) , ( b , , k X a =0 ( − a E ( a ) y k − a e ( i ) if ( i j , i j +1 ) = ( k, b ) , ( b , k ) . (4.7)The degrees of the generators are deg( e ( i )) = 0 , deg( x j ) = 2 , deg( y ) = 2 , deg( E ( d )) = 2 d, deg( ψ j e ( i )) = a if ( i j , i j +1 ) = ( b , a ) , ( a, b ) , where a is or k . imodules 9 We have a translation from (1 n ) i,k to (1 n ) which is obtained by replacing k of the sequence (1 n ) i,k by (1 , . . . , | {z } k ) . This translation naturally induces the map φ : Seq((1 n ) i,k , → Seq((1 n ) , .We define the inclusion map Φ from W ((1 n ) i,k , to W ((1 n ) , by mapping idempotents e ( i ) for i ∈ Seq((1 n ) i,k , n ) by Φ( e ( i )) = e ( φ ( i )) , mapping generators x j by Φ( x j ) = ( x j if ≤ j < i,x j + k − if i < j ≤ n − k + 1 , mapping generators ψ j by Φ( ψ j ) = ψ j ( e j − + e j ) j < i,ψ j ψ j +1 · · · ψ j + k − e j + k − + ψ j + k − ψ j + k − · · · ψ j e j − j = i,ψ j + k − ( e j + k − + e j + k − ) j > i and mapping generators E ( d ) by Φ( E ( d )) = X d + d + ··· + d k = dd ,...,d k ≥ x d i x d i +1 · · · x d k i + k − . This inclusion map gives rise to a left action of W ((1 n ) i, , on ˆ e i W ((1 n ) , and a right action on W ((1 n ) , e i ,where ˆ e i is P j = i e j . Definition 4.2.
We define the bimodule W i as the tensor product of two deformed Webster algebras W ((1 n ) , over W ((1 n ) i, , . W i := W ((1 n ) , e i ⊗ W ((1 n ) i, , ˆ e i W ((1 n ) , . The following proposition follows directly from definitions.
Proposition 4.3.
In the bimodule W i , we have the following equalities. e k ⊗ e ℓ = 0 if k = ℓ, e j ⊗ e j = e j ⊗ ⊗ e j , y ⊗ ⊗ y, (4.8) x ℓ ⊗ e j = 1 ⊗ x ℓ e j if ℓ = i, i + 1 , ( x i + x i +1 ) ⊗ e j = 1 ⊗ ( x i + x i +1 ) e j , x i x i +1 ⊗ e j = 1 ⊗ x i x i +1 e j , (4.9) ψ ℓ ⊗ e j = 1 ⊗ ψ ℓ e j if ℓ = i, i + 1 , ψ i +1 ψ i ⊗ e j = 1 ⊗ ψ i +1 ψ i e j , ψ i ψ i +1 ⊗ e j = 1 ⊗ ψ i ψ i +1 e j , (4.10)where j = i .A graphical description of the bimodules W i for i = 1 , . . . , n − was introduced in [KS18]. We considercollections of smooth arcs in the plane connecting n red points and black point on one horizontal line with n red points and black point on another horizontal line. The i th and ( i + 1) st red dots on one horizontalline must be connected to the i th and ( i + 1) st red dots on the other horizontal line by a diagram whichhas a thick red strand in the middle, which is given in (4.11). The arcs are colored in a manner consistentwith their boundary points. Arcs are assumed to have no critical points (in other words no cups or caps).Arcs are allowed to intersect, but no triple intersections are allowed. Arcs are allowed to carry dots. Twodiagrams that are related by an isotopy that does not change the combinatorial types of the diagrams or therelative position of crossings are taken to be equal. The elements of the vector space W i are formal linearcombinations of these diagrams modulo the local relations for W along with the relations given in (4.12)and (4.13) which correspond to relations in Proposition 4.3.The elements e j ⊗ e j , where ≤ j < i or i < j ≤ n , are represented by diagrams e j ⊗ e j = · · · j j +1 · · · i − i i +1 i +2 · · · n ∈ W i . (4.11) imodules 10 The second and third equations of (4.9) in terms of diagrams are i i +1 + i i +1 = i i +1 + i i +1 , i i +1 = i i +1 . (4.12)The second and third equations of (4.10) in terms of diagrams are i i +1 = i i +1 , i i +1 = i i +1 . (4.13)The bimodule W i naturally inherits a p -DG structure from W as follows: ∂ W i := ∂ W ⊗ Id + Id ⊗ ∂ W . One may twist the p -DG structure on W i to obtain a new p -DG bimodule W − e i . As bimodules, W i = W − e i but the p -DG structure on the generator is twisted as follows: ∂ (1 ⊗ e j ) = − ( x i + x i +1 ) ⊗ e j (4.14)In terms of a diagrammatic description, this twisted differential is ∂ = − − . (4.15)We use the notation W − e i because the differential is twisted by the first elementary symmetric function inthe dots corresponding to the bimodule generator connected to the thick red strand.We will also need the bimodules W i,i +1 for i = 1 , . . . , n − . Definition 4.4.
We define the bimodule W i,i +1 as the tensor product of two deformed Webster algebras W ((1 n ) , over W ((1 n ) i, , . W i,i +1 := W ((1 n ) , e i,i +1 ⊗ W ((1 n ) i, , ˆ e i,i +1 W ((1 n ) , , where ˆ e i,i +1 = X j = i,i +1 e j .The following proposition follows directly from definitions. Proposition 4.5.
In the bimodule W i,i +1 , we have the following equalities. e k ⊗ e ℓ = 0 if k = ℓ, e j ⊗ e j = e j ⊗ ⊗ e j , x ℓ ⊗ e j = 1 ⊗ x ℓ e j if ℓ = i, i + 1 , i + 2 , (4.16) ψ ℓ ⊗ e j = 1 ⊗ ψ ℓ e j if ℓ = i, i + 1 , i + 2 , y ⊗ ⊗ y, (4.17) X a + b + c = da,b,c ≥ x ai x bi +1 x ci +2 ⊗ e j = 1 ⊗ X a + b + c = da,b,c ≥ x ai x bi +1 x ci +2 e j where d = 1 , , , (4.18) ψ i +2 ψ i +1 ψ i ⊗ e j = 1 ⊗ ψ i +2 ψ i +1 ψ i e j , ψ i ψ i +1 ψ i +2 ⊗ e j = 1 ⊗ ψ i ψ i +1 ψ i +2 e j , (4.19)where j = i, i + 1 . asesforbimodules 11 For a diagrammatic description of W i,i +1 , we consider collections of smooth arcs in the plane connect-ing n red points and black point on one horizontal line with n red points and black point on anotherhorizontal line. The i th, ( i + 1) st and ( i + 2) nd red dots on one horizontal line must be connected to the i th, ( i + 1) st and ( i + 2) nd red dots on the other horizontal line by a diagram which has a thick red strandin the middle, which is given in (4.20). The arcs are colored in a manner consistent with their boundarypoints. Arcs are assumed to have no critical points (in other words no cups or caps). Arcs are allowed tointersect, but no triple intersections are allowed. Arcs are allowed to carry dots. Two diagrams that arerelated by an isotopy that does not change the combinatorial types of the diagrams or the relative positionof crossings are taken to be equal. The elements of the vector space W i,i +1 are formal linear combinationsof these diagrams modulo the local relations for W along with the relations given in (4.21), (4.22), (4.23)and (4.24).The elements e j ⊗ e j , where ≤ j < i or i + 1 < j ≤ n , are represented by e j ⊗ e j = · · · j j +1 · · · i − i i +3 · · · n ∈ W i,i +1 . (4.20)The generating equations in (4.18) in terms of diagrams are + + = + + , (4.21) + + = + + , (4.22) = . (4.23)The equations in (4.19) in terms of diagrams are i i +1 i +2 = i i +1 i +2 , i i +1 i +2 = i i +1 i +2 . (4.24)The bimodule W i,i +1 naturally inherits a p -DG structure from W where one sets the differential on thegenerator (4.20) to be zero. We construct bases of the bimodules just introduced. asesforbimodules 12
Let ℵ ( a , b, c ) = n Y k =1 x a k k y b ψ i ⊗ ψ i x ci e i = a · · · b i c i +1 a i a i +1 · · · n a n ℵ ( a ) = n Y k =1 x a k k ψ i +1 ⊗ ψ i +1 e i = a · · · i i +1 a i a i +1 · · · n a n For ≤ j ≤ n and ≤ ℓ ≤ n unless j = ℓ = i , let ℵ ( a , b, c, j, ℓ ) = Q nk =1 x a k k y b ψ ℓ ψ ℓ − · · · ψ j +1 ⊗ x ci e j if j < ℓ and j = i Q nk =1 x a k k y b ψ ℓ +1 ψ ℓ +2 · · · ψ j ⊗ x ci e j if j > ℓ and j = i Q nk =1 x a k k y b ⊗ x ci e j if j = ℓ, j = i Q nk =1 x a k k y b ⊗ x ci ψ ℓ ψ ℓ − · · · ψ i +1 e i if j < ℓ and j = i Q nk =1 x a k k y b ⊗ x ci ψ ℓ +1 ψ ℓ +2 · · · ψ i e i if j > ℓ and j = i For j < ℓ and j = i , diagramatically ℵ ( a , b, c, j, ℓ ) is given by a · · · b i c i +1 a i a i +1 · · · n a n · · · · · · where on the bottom, the endpoint of the black strand is directly to the right of the j th red strand, and onthe top, the endpoint of the black strand is directly to the right of the ℓ th red strand. Proposition 4.6.
The set {ℵ ( a , b, c ) , ℵ ( a ) , ℵ ( a , b, c, j, ℓ ) | a , . . . , a n , b ∈ Z , c ∈ { , } , ≤ j ≤ n, ≤ ℓ ≤ n } is a spanning set of W i . Proof.
We will only highlight interesting features of the proof, which include cases where the black strandbegins and ends in between the two red strands connected to the thick red strand. First we will show that ψ i +1 ⊗ ψ i +1 x i e i is in the span of the proposed spanning set. The diagram for this element is . (4.25) asesforbimodules 13 Using relations, we have the following equations − = − , (4.26) − = − . (4.27)Subtracting equation (4.27) from equation (4.26) yields − = − . (4.28)Using symmetric function relations on (4.28) we get = + − . Thus a diagram with a portion as in (4.25) is in the span.Consider a portion of a diagram of the form . (4.29)Then using relations we have = + − which shows that a picture (4.29) is in the span of the proposed spanning set since we already showed thefirst term above is already in the proposed spanning set.In order to prove that the elements in Proposition 4.6 are actually a basis of W i , we will construct a ( W ( n, , W ( n, -bimodule homomorphism φ i : W i → Hom k ( V n , V n ) .Consider the divided difference operator D i : k [ y j ][ x , . . . , x n ] → k [ y j ][ x , . . . , x n ] f f − f s i x i − x i +1 where f s i is the polynomial obtained from f by exchanging the variables x i and x i +1 and keeping all othervariables the same. Proposition 4.7.
There is a bimodule homomorphism φ i : W i → Hom k ( V n , V n ) inherited from the represen-tation of W ( n, on V n determined by · · · i i +1 · · · n : f ∈ V n D i ( f ) ∈ V n . asesforbimodules 14 Proof.
This is a routine check.
Proposition 4.8.
The spanning set in Proposition 4.6 is a basis of W i . Proof.
Clearly a collection of vectors in this bimodule could be linearly dependent only if the configurationof boundary points are the same. We will focus on the most interesting case. For simplicity, assume there isa dependence relation of the form k a ,a ,b ba a + k a ,a ,b ba a + k a ,a a a = 0 . (4.30)Applying the bimodule homomorphism φ from above on the element yields k a ,a ,b ( y − x ) x a x a y b + k a ,a x a x a = 0 . This implies that k a ,a ,b = k a ,a = 0 and thus k a ,a ,b = 0 . Therefore the elements in (4.30) are in factlinearly independent.Checking linear independence for the other elements in the spanning set proceeds in a similar manner. Proposition 4.9.
The following elements span W i,i +1 i ( a , b, c i , c i +1 ) = a · · · bc i c i +1 a i +1 a i a i +2 · · · a n b ≥ , c i = 0 , , , c i +1 = 0 , i ( a , c i , c i +1 ) = a · · · c i c i +1 a i +1 a i a i +2 · · · a n c i = 0 , c i +1 = 0 , i ( a , b, c i , c i +1 ) = a · · · bc i c i +1 a i +1 a i a i +2 · · · a n b ≥ , c i = 0 , , , c i +1 = 0 , i ( a , c i , c i +1 ) = a · · · c i c i +1 a i +1 a i a i +2 · · · a n c i = 0 , c i +1 = 0 , i ( a , c i , c i +1 ) = a · · · c i c i +1 a i +1 a i a i +2 · · · a n ( c i , c i +1 ) = (0 , , (1 , , (0 , , (1 , asesforbimodules 15 i ( a , b, c i , c i +1 ) = a · · · bc i c i +1 a i +1 a i a i +2 · · · a n b ≥ , c i = 0 , , , c i +1 = 0 , i ( a , b, c i , c i +1 ) = a · · · c i c i +1 a i +1 a i a i +2 · · · a n b b ≥ , c i = 0 , , , c i +1 = 0 , i ( a , c i , c i +1 ) = a · · · c i c i +1 a i +1 a i a i +2 · · · a n ( c i , c i +1 ) = (0 , , (1 , , (0 , , (2 , i ( a , b, c i , c i +1 , j, ℓ ) = a · · · bc i c i +1 a i +1 a i a i +2 · · · a n b ≥ , c i = 0 , , , c i +1 = 0 , , ≤ j, ℓ ≤ n unless ( j, ℓ ) = ( i, i ) , ( i, i + 1) , ( i + 1 , i ) , ( i + 1 , i + 1) where the diagram in i ( a , b, c i , c i +1 , j, ℓ ) represents any picture where the black strand begins after the j thred strand and ends after the ℓ th red strand and has a minimal number of intersections with the red strands. Proof.
Once again, we highlight features of the proof that only involve cases where the black strand beginsand ends in the segments connecting red boundary points connected to the thick red strands. First notethat by classical arguments (see for example [EK10, Section 2.2]), we could assume that the red dots appearin the following configuration a · · · c i c i +1 a i +1 a i a i +2 · · · a n where all a i ∈ Z ≥ , c i ∈ { , , } , and c i +1 ∈ { , } .Next we will explain why we could assume that in (4.31) that there are no black dots. a · · · c i c i +1 a i +1 a i a i +2 · · · a n (4.31)Using relations we have = + 2 − − , (4.32) asesforbimodules 16 so a black dot produces a linear combination of other elements in the proposed spanning set.Next we will explain why we may assume that c i = 0 in (4.31). As in (4.32) we have a relation = + 2 − − (4.33)Subtracting (4.33) from (4.32) yields = + 1 + 1 − − − + By classical arguments, the third to last and last diagrams above are in the proposed span, and thusis in the span of other elements of the proposed spanning set.Next we will show that we could assume that there are no black dots on diagrams of the form . (4.34)Using relations we have = + − . (4.35)Thus we may assume that there are no black dots on diagrams of the form (4.34).Next we will show that elements of the form (4.36)are already in the span of the proposed spanning set. Using bimodule relations we have − − + = 2 − − + . (4.37) asesforbimodules 17 Using (4.37) and (4.35) repeatedly, we get = + − − − + + . Thus (4.36) is in the span of the proposed spanning set.Similar techniques show that the other elements stated in the proposition complete a spanning set.In order to prove that the elements in Proposition 4.9 are actually a basis of W i,i +1 , we use a ( W ( n, , W ( n, -bimodule homomorphism γ i,i +1 : W i,i +1 → Hom k ( V n , V n ) . Proposition 4.10.
There is a bimodule homomorphism γ i,i +1 : W i,i +1 → Hom k ( V n , V n ) inherited from therepresentation of W ( n, on V n determined by · · · i i +2 · · · ni +1 : f ∈ V n D i D i +1 D i ( f ) ∈ V n . Proof.
This is a routine calculation.
Proposition 4.11.
The spanning set in Proposition 4.9 is a basis of W i,i +1 . Proof. If ǫ ′ v ǫ , ǫ ′ v ǫ ∈ W i,i +1 , where ǫ , ǫ ′ , ǫ , ǫ ′ are idempotents, then there could only be a dependencerelation if ǫ ′ = ǫ ′ and ǫ = ǫ . Using this fact, there are several cases to check. We supply the details fortwo non-trivial cases.First consider a dependence relation of the form k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a (4.38) + k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a + k a ,a ,a a a a + k a ,a ,a a a a = 0 . Apply the homomorphism γ i,i +1 to (4.38) and evaluate on to get k a ,a ,a ,b ( y − x ) x a x a x a y b + k a ,a ,a x a x a x a = 0 . This forces k a ,a ,a ,b = k a ,a ,a = 0 . asesforbimodules 18 Then the dependence relation becomes k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a (4.39) + k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a + k a ,a ,a a a a = 0 . Now apply γ i,i +1 to (4.39) and evaluate on x to get k a ,a ,a ,b ( y − x ) x a x a x a y b = 0 . This implies k a ,a ,a ,b = 0 and the dependence relations becomes k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a (4.40) + k a ,a ,a ,b b a a a + k a ,a ,a a a a = 0 . Now apply γ i,i +1 to (4.40) and evaluate on x to get k a ,a ,a ,b ( y − x ) x a x a x a y b + k a ,a ,a x a x a x a = 0 . This implies that k a ,a ,a ,b = k a ,a ,a = 0 and the dependence relation becomes k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a = 0 . (4.41)Now apply γ i,i +1 to (4.41) and evaluate on x x to get k a ,a ,a ,b ( y − x ) x a x a x a y b = 0 . Thus k = 0 and the dependence relation becomes k a ,a ,a ,b b a a a + k a ,a ,a ,b b a a a = 0 . (4.42) asesforbimodules 19 Apply γ i,i +1 to (4.42) and evaluate on x x to get k a ,a ,a ,b ( y − x ) x a x a x a y b = 0 . Thus k a ,a ,a ,b = 0 which implies k a ,a ,a ,b = 0 . It is straightforward to show that each element in thespanning set gets sent to something non-zero under γ i,i +1 by evaluating on an appropriate element. Thusthese elements of the spanning set are non-zero and linearly independent.Next we will consider a dependence relation of the form k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b (4.43) + k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a a a a + k a ,a ,a a a a + k a ,a ,a a a a + k a ,a ,a a a a = 0 . Applying the homomorphism γ i,i +1 to (4.43) and evaluating on yields the equation k a ,a ,a ,b ( y − x ) x a x a x a y b + k a ,a ,a x a x a x a = 0 . Thus k = k = 0 and the dependence relation becomes k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b (4.44) + k a ,a ,a ,b a a a b + k a ,a ,a a a a + k a ,a ,a a a a + k a ,a ,a a a a = 0 . Now apply γ i,i +1 to (4.44) and evaluate on x to get k a ,a ,a ,b ( y − x ) x a x a x a y b + k a ,a ,a x a x a x a = 0 . asesforbimodules 20 Thus k a ,a ,a ,b = k a ,a ,a = 0 and the dependence relation becomes k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b (4.45) + k a ,a ,a ,b a a a b + k a ,a ,a a a a + k a ,a ,a a a a = 0 . Now apply γ i,i +1 to (4.45) and evaluate on x to get k a ,a ,a ,b ( y − x ) x a x a x a y b − k a ,a ,a x a x a x a = 0 . Thus k a ,a ,a ,b = k a ,a ,a = 0 and the dependence relation becomes k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b + k a ,a ,a a a a = 0 . (4.46)Now apply γ i,i +1 to (4.46) and evaluate on x to get k a ,a ,a ,b ( y − x ) x a x a x a y b + k a ,a ,a x a x a x a = 0 . Thus k a ,a ,a ,b = k a ,a ,a = 0 and the dependence relation becomes k a ,a ,a ,b a a a b + k a ,a ,a ,b a a a b = 0 . (4.47)Finally apply γ i,i +1 to (4.47) and evaluate on x x to get k a ,a ,a ,b ( y − x ) x a x a x a y b = 0 . Thus k a ,a ,a ,b = 0 and consequently k a ,a ,a ,b = 0 . Therefore these spanning elements are also linearlyindependent.The other cases are checked in a similar fashion. Proposition 4.12.
The following elements span the bimodule W i ⊗ W i +1 ⊗ W i . ג ( a , b, r, s, t ) = a · · · l a n · · · i t i +1 i +2 sra i a i +1 a i +2 b a i ∈ Z ≥ , b ∈ Z ≥ , r, s, t ∈ { , } asesforbimodules 21 ג ( a , s ) = a · · · n a n · · · i i +1 i +2 sa i a i +1 a i +2 a i ∈ Z ≥ , s ∈ { , } ג ( a ) = a · · · n a n · · · i i +1 i +2 a i a i +1 a i +2 a i ∈ Z ≥ , ג ( a , b, r, s, t ) = a · · · n a n · · · i t i +1 i +2 sra i a i +1 a i +2 b a i ∈ Z ≥ , b ∈ Z ≥ , r, s, t ∈ { , } ג ( a , r, t ) = a · · · n a n · · · i t i +1 i +2 ra i a i +1 a i +2 a i ∈ Z ≥ , r, t ∈ { , } ג ( a , b, r, s, t ) = a · · · n a n · · · i t i +1 i +2 sra i a i +1 a i +2 b a i ∈ Z ≥ , b ∈ Z ≥ , r, s, t ∈ { , } asesforbimodules 22 ג ( a , r, t ) = a · · · n a n · · · i t i +1 i +2 ra i a i +1 a i +2 a i ∈ Z ≥ , r, t ∈ { , } ג ( a , t ) = a · · · n a n · · · i t i +1 i +2 a i a i +1 a i +2 a i ∈ Z ≥ , t ∈ { , } ג ( a , b, r, s, t ) = a · · · n a n · · · i t i +1 i +2 sra i a i +1 a i +2 b a i ∈ Z ≥ , b ∈ Z ≥ , r, s, t ∈ { , } ג ( a , r, t ) = a · · · n a n · · · i t i +1 i +2 ra i a i +1 a i +2 a i ∈ Z ≥ , r, t ∈ { , } ג ( a , r ) = a · · · n a n · · · i i +1 i +2 ra i a i +1 a i +2 a i ∈ Z ≥ , r ∈ { , } asesforbimodules 23 ג ( a , b, r, s, t, j, ℓ ) = a · · · n a n · · · i t i +1 i +2 sra i a i +1 a i +2 b a i ∈ Z ≥ , b ∈ Z ≥ , r, s, t ∈ { , } ≤ j, ℓ ≤ n unless ( j, ℓ ) = ( i, i ) , ( i, i + 1) , ( i + 1 , i ) , ( i + 1 , i + 1) where the diagram in ג ( a , b, r, s, t, j, ℓ ) represents any picture where the black strand begins after the j thred strand and ends after the ℓ th red strand and has a minimal number of intersections with the red strands.Note that it could equally well be positioned in a northwest-southeast configuration. Proof.
By classical arguments we may assume that any red dot configuration in the part of the pictureconnecting the i , i + 1 , and i + 2 red boundary points is a linear combination of elements t sra i a i +1 a i +2 (4.48)where r, s, t ∈ { , } . Next we will show that we do not need any black dots on a diagram of the form a i a i +1 a i +2 b . (4.49)This follows from the calculation − = 2 − − . asesforbimodules 24 Next note that − = 2 − − . Thusis already in the span of the proposed spanning set. Similarly,is also already in the span of the proposed spanning set.Next we will show that we do not need black dots on a diagram of the form i i +1 i +2 a i a i +1 a i +2 . This follows from the calculation − = 2 − − . asesforbimodules 25 Next note that − = − which shows thatis in the span of the proposed spanning set. The equality − = 2 − − implies thatis in the span of the proposed spanning set. A longer computation using the equality − = − asesforbimodules 26 implies thatis in the proposed span.Now consider elements of the form t sra i a i +1 a i +2 b . By classical arguments one may assume that r, s, t ∈ { , } and a i , a i +1 , a i +2 , b ∈ Z ≥ . Next note that theequation − = − implies that a black dot on a diagramis already in the span of the proposed spanning set. The equation − = − asesforbimodules 27 implies thatis already in the span of the proposed spanning set.By classical arguments, we may assume that dots on diagrams of the following form have the configu-ration i t i +1 i +2 sra i a i +1 a i +2 b where r, s, t ∈ { , } and b, a i , a i +1 , a i +2 ∈ Z ≥ . Next note that − = − implies thatis in the span of the proposed spanning set. The equation − = − asesforbimodules 28 implies thatis in the span.Next note that − = − . Thus a diagram with a black dot onis already in the span of the proposed spanning set. The equation − = − implies that imodulehomomorphisms 29 is in the span. The equation − = − implies thatis in the span.Similar manipulations show that the set of elements in the statement of the proposition, do in fact pro-vide a spanning set for the bimodule.We will prove that the spanning set above is indeed a basis. This will utilize a bimodule homomorphismintroduced in the next section. There is a bimodule homomorphism ǫ i : W i → W determined by (4.50)It is clear that ǫ i commutes with ∂ .There is a bimodule homomorphism ι i : W → W − e i {− } determined by
7→ − ,
7→ − . The fact that ι i is a well-defined bimodule homomorphism is shown in [KS18, Proposition 17].It is clear that ι i : W → W i {− } does not commute with ∂ . However, we do have the following variation. Proposition 4.13.
The bimodule homomorphism ι i : W → W − e i {− } commutes with the action of ∂ . imodulehomomorphisms 30 Proof.
We first check the proposition in the case that there is no black strand in between the i th and ( i + 1) stred strands. By definition ι i ◦ ∂ = 0 . On the other hand, ∂ ◦ ι i = ∂ − = 2 − − − + + 2= 0 where the four last terms come from the twisted p -DG structure on W − e i .The second case we must consider is when the black strand lies between the i th and ( i + 1) st red strands.Once again ι i ◦ ∂ = 0 . Let X = − ∈ W − e i . We must show that ∂ ( X ) = 0 . By definition ∂ ( X ) = + − − − − + + (4.51)where the last four terms comes from the twisting on W − e i . It follows that ∂ ( X ) = − − + (4.52)where the second, fifth, and sixth terms in (4.51) combine to become the second term in (4.52). Using thefirst set of relations of (3.9) and (4.13), we get that the above simplifies to ∂ ( X ) = − = 0 . Proposition 4.14.
There are bimodule homomorphisms α i,i +1 : W i,i +1 → W i ⊗ W i +1 ⊗ W i and α i +1 ,i : W i,i +1 → W i +1 ⊗ W i ⊗ W i +1 that commute with the action of ∂ . imodulehomomorphisms 31 Proof.
The homomorphism α i,i +1 is defined by mapping the generator (4.20) as follows (4.53)The differential ∂ annihilates both the generator (4.20) and its image under the homomorphism.The homomorphism α i +1 ,i is defined by mapping the generator (4.20) as follows The differential ∂ annihilates both the generator (4.20) and its image under the homomorphism.There is a bimodule homomorphism W i ⊗ W i +1 ⊗ W i → W i { } defined as a composition of two mapsconstructed in [KS18]. It is defined on generators in (4.54) and (4.55). , (4.54) , (4.55)The proof of the next result is similar to the proofs of Propositions 4.8 and 4.11, so we just supply thegeneral strategy. Proposition 4.15.
The spanning set in Proposition 4.12 is a basis of W i ⊗ W i +1 ⊗ W i . Proof.
Let B ij be elements in the spanning set which are in the image of multiplication on the left by e i andon the right by e j .In order to prove that the elements of B ii +1 are linearly independent, one writes down a dependencerelation and uses the bimodule homomorphism γ i,i +1 and the bimodule homomorphism defined in (4.54)and (4.55) to conclude that all the coefficients in the dependence relation are zero. One proves in a similarway the linear independence of elements in B ii and B i +1 i . raidgroupaction 32 By applying the element · · · i +1 i +2 · · · n , on top, the linear independence of elements in B i +1 i +1 follows from the linear independence of B ii +1 .Showing that all other sets B ij are linearly independent is routine. As a consequence of the results of Section 4.3, there are complexes of ( W, W ) H -modules Σ i = W i ǫ i / / W , Σ ′ i = W ι i / / W − e i {− } . Lemma 4.16.
There exists an isomorphism of ( W, W ) H -modules W i ⊗ W W i ∼ = W i ⊕ W x i + x i +1 i { } Proof.
First note that black strands play no role in these bimodules as in [KS18, Lemma 5]. Thus the proofreduces to the proof of [KR16, Lemma 4.3] or [QS20, Lemma 3.2].
Lemma 4.17.
There is a short exact sequence of ( W, W ) H -modules which splits as ( W, W ) -bimodules → W i,i +1 → W i ⊗ W i +1 ⊗ W i → W e i → . Proof.
The map W i,i +1 → W i ⊗ W i +1 ⊗ W i is just given by α i,i +1 defined in (4.53).Recall there is a map W i ⊗ W i +1 ⊗ W i → W i defined on generators in (4.54) and (4.55). This is clearly asurjection and a straightforward calculation shows that this surjection is a p -DG map.Note thatmaps to the kernel of W i ⊗ W i +1 ⊗ W i → W e i . By a graded dimension count utilizing the bases of thebimodules from Propositions 4.8, 4.11, 4.15, we get the exactness of the sequence of the lemma.Now we define a splitting map W i ⊗ W i +1 ⊗ W i → W i,i +1 by , . (4.56) , . (4.57) raidgroupaction 33 Note that this splitting does not respect the p -DG structure.Next we define a splitting map W e i → W i ⊗ W i +1 ⊗ W i by , . (4.58)Note that this splitting map does not respect the p -DG structure either.Thus there is a short exact sequence of ( W, W ) H -bimodules which splits as ( W, W ) -bimodules.The next crucial proposition is proved in manner similar to [KR16, Theorems 4.2, 4.4]. Proposition 4.18.
The complexes of ( W, W ) H -modules satisfy the following isomorphisms of functors inthe classical relative homotopy category C ∂ ( W, d ) .1. Σ i ◦ Σ ′ i ∼ = Id ∼ = Σ ′ i ◦ Σ i ,2. Σ i ◦ Σ j ∼ = Σ j ◦ Σ i for | i − j | > ,3. Σ i ◦ Σ j ◦ Σ i ∼ = Σ j ◦ Σ i ◦ Σ j for | i − j | = 1 . Proof.
The first isomorphism follows as in [KR16, Theorem 4.2] or [QS20, Proposition 3.3] which use ver-sions of Lemma 4.16.The second item is clear.The third isomorphism follows as in [KR16, Theorem 4.4] or [QS20, Proposition 3.5] which use versionsof Lemma 4.17.Applying the p -extension functor P , we obtain complexes of p -DG ( W, W ) bimodules T i := P (Σ i ) and T ′ i := P (Σ ′ i ) in the relative p -homotopy category C ∂ ( W, ∂ ) . Explicitly, these complexes look like: T i := (cid:16) W i = −→ · · · = −→ W i ǫ i −→ W (cid:17) (4.59) T ′ i := (cid:16) W ι i −→ W − e i {− } = −→ · · · = −→ W − e i {− } (cid:17) , (4.60)(4.61)where the repeated terms appear p − times. Theorem 4.19.
The complexes of p -DG ( W, W ) -bimodules satisfy the following isomorphisms of functorson the relative p -homotopy category C ∂ ( W, ∂ ) .1. T i ◦ T ′ i ∼ = Id ∼ = T ′ i ◦ T i ,2. T i ◦ T j ∼ = T j ◦ T i for | i − j | > ,3. T i ◦ T j ◦ T i ∼ = T j ◦ T i ◦ T j for | i − j | = 1 . Proof.
This follows from Proposition 4.18 by applying the functor P from Section 2. EFERENCES 34
References [CK12] S. Cautis and Kamnitzer. Braiding via geometric Lie algebra actions.
Compos. Math. , 148(2):464–506, 2012. arXiv:1001.0619.[EK10] B. Elias and M. Khovanov. Diagrammatics for Soergel categories.
Int. J. Math. Math. Sci. , pagesArt. ID 978635, 58, 2010. arXiv:0902.4700.[EQ16a] B. Elias and Y. Qi. A categorification of quantum sl(2) at prime roots of unity.
Adv. Math. , 299:863–930, 2016. arXiv:1503.05114.[EQ16b] B. Elias and Y. Qi. A categorification of some small quantum groups II.
Adv. Math. , 288:81–151,2016. arXiv:1302.5478.[Kho16] M. Khovanov. Hopfological algebra and categorification at a root of unity: The first steps.
J. KnotTheory Ramifications , 25(3):359–426, 2016. arXiv:math/0509083.[KL10] M. Khovanov and A. D. Lauda. A categorification of quantum sl(n).
Quantum Topol. , 2(1):1–92,2010. arXiv:0807.3250.[KLSY18] M. Khovanov, A. Lauda, J. Sussan, and Y. Yonezawa. Braid group actions from categorical sym-metric Howe duality on deformed Webster algebras.
Trans. Groups , 2018. arXiv:1802.05358.[KQ15] M. Khovanov and Y. Qi. An approach to categorification of some small quantum groups.
Quan-tum Topol. , 6(2):185–311, 2015. arXiv:1208.0616.[KQS17] M. Khovanov, Y. Qi, and J. Sussan. p -DG cyclotomic nilhecke algebras. 2017. arXiv:1711.07159.[KR16] M. Khovanov and L. Rozansky. Positive half of the Witt algebra acts on triply graded link ho-mology. Quantum Topol. , 7(4):737–795, 2016. arXiv:1305.1642.[KS02] M. Khovanov and P. Seidel. Quivers, Floer cohomology, and braid group actions.
J. Amer. Math.Soc. , 15:203–271, 2002. arXiv:math/0006056.[KS18] M. Khovanov and J. Sussan. The Soergel category and the redotted Webster algebra.
J. of Pureand Applied Algebra , 222:1957–2000, 2018. arXiv:1605.02678.[Qi14] Y. Qi. Hopfological algebra.
Compos. Math. , 150(01):1–45, 2014. arXiv:1205.1814.[QS16] Y. Qi and J. Sussan. A categorification of the Burau representation at prime roots of unity.
SelectaMath. (N.S.) , 22(3):1157–1193, 2016. arXiv:1312.7692.[QS17] Y. Qi and J. Sussan. Categorification at prime roots of unity and hopfological finiteness. In
Cat-egorification and higher representation theory , volume 683 of
Contemp. Math. , pages 261–286. Amer.Math. Soc., Providence, RI, 2017. arXiv:1509.00438.[QS18] Y. Qi and J. Sussan. p -DG cyclotomic nilHecke algebras II. 2018. arXiv:1811.04372.[QS20] Y. Qi and J. Sussan. On some p -differential graded link homologies. 2020. arXiv:2009.06498.[Rou08] R. Rouquier. 2-Kac-Moody algebras. 2008. arXiv:0812.5023.[SW11] C. Stroppel and B. Webster. Quiver Schur algebras and q-Fock space. 2011. arXiv:1110.1115.[Web] B. Webster. Three perspectives on categorical symmetric Howe duality. arXiv:2001.07584.[Web17] B. Webster. Knot invariants and higher representation theory. Mem. Amer. Math. Soc. ,250(1191):pp. 133, 2017. arXiv:1309.3796.[Yon] Y. Yonezawa. A p -DG deformed Webster algebra of type A . arXiv:1910.04897. EFERENCES 35
Y. Q.:
DepartmentofMathematics,UniversityofVirginia,Charlottesville,VA22904,USA email: [email protected]
J. S.:
DepartmentofMathematics,CUNYMedgarEvers,Brooklyn,NY,11225,USA email: [email protected]
MathematicsProgram,TheGraduateCenter,CUNY,NewYork,NY,10016,USA email: [email protected]
Y. Y.