2-Verma modules. Grégoire Naisse Joint work with Pedro Vaz. 22 November Université catholique de Louvain

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1 2-Verma modules Grégore Nasse Jont work wth Pedro Vaz Unversté catholque de Louvan 22 November 2017

2 Hghest weght representatons g s a (symmetrzable) quantum KacMoody algebra. There are 3 knds of hghest weght modules :

3 Hghest weght representatons g s a (symmetrzable) quantum KacMoody algebra. There are 3 knds of hghest weght modules : nte-dmensonal V (β) (where β s ntegral), F acts as nlpotent operator on any w V (β) ;

4 Hghest weght representatons g s a (symmetrzable) quantum KacMoody algebra. There are 3 knds of hghest weght modules : nte-dmensonal V (β) (where β s ntegral), F acts as nlpotent operator on any w V (β) ; Verma modules, Fj nw F j m w for w 0 and n m ;

5 Hghest weght representatons g s a (symmetrzable) quantum KacMoody algebra. There are 3 knds of hghest weght modules : nte-dmensonal V (β) (where β s ntegral), F acts as nlpotent operator on any w V (β) ; Verma modules, Fj nw F j m w for w 0 and n m ; parabolc Verma modules, a mx n-between (F locally nlpotent for some 's and `nnte' for others). (+tensor products...)

6 Hghest weght representatons : the pcture p g s a (standard) parabolc subalgebra ; V (β) U q (p)-module of hghest weght β ; Hghest weght module M p (β) = U q (g) Uq(p) V (β). E.g. g = sl 3 = E 1, F 1, E 2, F 2, K γ and p = E 1, E 2, F 2, K γ, β = (β 1, 1) M p (β) Q -mod U q (g) = F s.

7 Categorcaton of U q (g) KLR-algebras = brad-lke algebras R(k) wth k-strands labeled by smple roots (+dots+relatons) R(k) R(k + 1) :... k... F : R(k) -mod R(k + 1) -mod = nducton... k... Theorem (KhovanovLauda) ( ) K 0 R(k) -mod = Uq (g) k as modules ober U q (g) (and even more...) KLR-algebras = good start to categorfy H.W.M.

8 Some hnts τ : U q (g) U op q (g) ant-automorphsm s.t. τ(f ) E

9 Some hnts τ : U q (g) U op q (g) ant-automorphsm s.t. τ(f ) E t denes a sequlnear form, on M p (β) s.t. F,, E.

10 Some hnts τ : U q (g) U op q (g) ant-automorphsm s.t. τ(f ) E t denes a sequlnear form, on M p (β) s.t. F,, E. t looks lke gdm (=decategorcaton) of some HOM, and HOM(F, ) HOM(, E ).

11 Some hnts τ : U q (g) U op q (g) ant-automorphsm s.t. τ(f ) E t denes a sequlnear form, on M p (β) s.t. F,, E. t looks lke gdm (=decategorcaton) of some HOM, and HOM(F, ) HOM(, E ). E should be rght adjont of F, hence restrcton functor.

12 Restrcton functor We get a decomposton... k = k+1 a=1 p 0... k... p a. bascally E on R(k + 1) -mod acts multplcaton by where q k q qk l 1 q 2 1 = 1 + q 2 + q q 2 we need to modfy the KLR-algebras.

13 Fnte case If F s locally nlpotent (=> H.W. ntegral), E should acts as q k 1 [m 1 ] + + q k l [m l ], where [m s ] = q ms q 1 ms.

14 Fnte case If F s locally nlpotent (=> H.W. ntegral), E should acts as q k 1 [m 1 ] + + q k l [m l ], where [m s ] = q ms q 1 ms. KLR s too bg we take quotent : the cyclotomc quotent β = 0.

15 Innte/Verma case For F `nnte', E should acts as q k qm 1 λ 1 q m 1 λ q k l qm1 λ q m1 λ 1, 1 q 2 1 q 2 where λ = q β (=formal parameter).

16 Innte/Verma case For F `nnte', E should acts as q k qm 1 λ 1 q m 1 λ q k l qm1 λ q m1 λ 1, 1 q 2 1 q 2 where λ = q β (=formal parameter). KLR s too small add gradng λ and superstructure (party), wth new generator, of q-degree = 0, λ -degree = 2 and party = 1 (they antcommute).

17 Some facts For F locally nlpotent, E, F realze (categorcal) sl(2)-commutator as drect sum, otherwse there s a natural SES : 0 F E E F 1 q q 1 (K ΠK 1 ) 0.

18 Some facts For F locally nlpotent, E, F realze (categorcal) sl(2)-commutator as drect sum, otherwse there s a natural SES : 0 F E E F 1 q q 1 (K ΠK 1 ) 0. Also, there s a derental d n = β so that the homology s a cyclotomc quotent. we got a dg-enhanchement of cyclotomc-klr, t allows to compute many thngs easly n these.

19 A bt of topology HOMFLY polynomal arses n parabolc Verma modules of gl(2n) (by Queelec-Sartor work), KR-HOMFLY homology arses naturally n the parabolc 2-Verma modules, we have new tools to compute stu about KR homology.

20 A bt of topology HOMFLY polynomal arses n parabolc Verma modules of gl(2n) (by Queelec-Sartor work), KR-HOMFLY homology arses naturally n the parabolc 2-Verma modules, we have new tools to compute stu about KR homology. Theorem (N., Vaz, 2017) d N nduces a spectral sequence on KR-homology to sl(n)-homology, whch agree wth Rasmussen's one. Moreover, the SS converges at the second page. Idea : lft the complex n the `total' dg-enhancement and observe t s homototopc to a complex concentrated n degree 0.