2-Verma modules

2-Verma modules

Grégoire Naisse Joint work with Pedro Vaz

Université catholique de Louvain

22 November 2017

Highest weight representations

gis a (symmetrizable) quantum KacMoody algebra.

There are 3 kinds of highest weight modules :

nite-dimensional V(β) (whereβ is integral),

→ F_i acts as nilpotent operator on anyw ∈V(β); Verma modules,

→ F_jⁿw 6=F_j^mw forw 6=0 and n6=m; parabolic Verma modules,

→ a mix in-between (F_i locally nilpotent for somei's and

`innite' for others). (+tensor products...)

Highest weight representations

gis a (symmetrizable) quantum KacMoody algebra.

There are 3 kinds of highest weight modules : nite-dimensional V(β) (whereβ is integral),

→ F_i acts as nilpotent operator on anyw ∈V(β);

Verma modules,

→ F_jⁿw 6=F_j^mw forw 6=0 and n6=m; parabolic Verma modules,

→ a mix in-between (F_i locally nilpotent for somei's and

`innite' for others). (+tensor products...)

Highest weight representations

gis a (symmetrizable) quantum KacMoody algebra.

There are 3 kinds of highest weight modules : nite-dimensional V(β) (whereβ is integral),

→ F_i acts as nilpotent operator on anyw ∈V(β); Verma modules,

→ F_jⁿw 6=F_j^mw forw 6=0 and n6=m;

parabolic Verma modules,

→ a mix in-between (F_i locally nilpotent for somei's and

`innite' for others). (+tensor products...)

Highest weight representations

gis a (symmetrizable) quantum KacMoody algebra.

There are 3 kinds of highest weight modules : nite-dimensional V(β) (whereβ is integral),

→ F_i acts as nilpotent operator on anyw ∈V(β); Verma modules,

→ F_jⁿw 6=F_j^mw forw 6=0 and n6=m; parabolic Verma modules,

→ a mix in-between (F_i locally nilpotent for somei's and

`innite' for others).

(+tensor products...)

Highest weight representations : the picture

p⊂g is a (standard) parabolic subalgebra ; V(β) U_q(p)-module of highest weightβ; Highest weight module

M^p(β) =Uq(g)⊗_Uq(p)V(β).

E.g.g=sl₃=hE₁,F₁,E₂,F₂,K_γiandp=hE₁,E₂,F₂,K_γi, β= (β₁,1).

•

• •

• • 0

• • •

. . . 0

⇒M^p(β)^Q,^-→^modU_q⁻(g) =hF_i⁰si.

Categorication of U

(g)

KLR-algebras = braid-like algebrasR(k)with k-strands labeled by simple roots (+dots+relations)

R(k),→R(k+1) : k . . .

. . .

7→ k . . .

. . . i Fi :R(k) -mod→R(k+1) -mod=induction

Theorem (KhovanovLauda)

K₀ M

k

R(k) -mod

!

∼=U_q⁻(g)

as modules oberU_q⁻(g) (and even more...)

⇒KLR-algebras = good start to categorify H.W.M.

Some hints

τ :Uq(g)→Uq^op(g) anti-automorphism s.t.τ(F_i)∼E_i

⇒it denes a sequilinear form h−,−ion M^p(β) s.t. hF_i−,−i ∼ h−,E_i−i.

⇒it looks like gdim (=decategorication) of some HOM, and HOM(Fi−,−)∼HOM(−,Ei−).

⇒Ei should be right adjoint of Fi, hence restriction functor.

Some hints

τ :Uq(g)→Uq^op(g) anti-automorphism s.t.τ(F_i)∼E_i

⇒it denes a sequilinear form h−,−ion M^p(β) s.t.

hF_i−,−i ∼ h−,E_i−i.

⇒it looks like gdim (=decategorication) of some HOM, and HOM(Fi−,−)∼HOM(−,Ei−).

⇒Ei should be right adjoint of Fi, hence restriction functor.

Some hints

τ :Uq(g)→Uq^op(g) anti-automorphism s.t.τ(F_i)∼E_i

⇒it denes a sequilinear form h−,−ion M^p(β) s.t.

hF_i−,−i ∼ h−,E_i−i.

⇒it looks like gdim (=decategorication) of some HOM, and HOM(Fi−,−)∼HOM(−,Ei−).

⇒Ei should be right adjoint of Fi, hence restriction functor.

Some hints

τ :Uq(g)→Uq^op(g) anti-automorphism s.t.τ(F_i)∼E_i

⇒it denes a sequilinear form h−,−ion M^p(β) s.t.

hF_i−,−i ∼ h−,E_i−i.

⇒it looks like gdim (=decategorication) of some HOM, and HOM(Fi−,−)∼HOM(−,Ei−).

⇒Ei should be right adjoint of Fi, hence restriction functor.

Restriction functor

We get a decomposition

k+1 . . . . . .

∼=

k+1

M

a=1

M

p≥0

k . . .

. . . a

p .

⇒basically Ei on R(k+1) -mod acts multiplication by q^k1 1

1−q² +· · ·+q^k` 1 1−q² where _1−q¹2 =1+q²+q⁴+. . .

⇒we need to modify the KLR-algebras.

Finite case

If Fi is locally nilpotent (=> H.W. integral), Ei should acts as q^k1[m₁] +· · ·+q<sup>k`</sup>[m<sub>`</sub>],

where[ms] =q^ms−1+· · ·+q^1−ms .

⇒KLR is too big →we take quotient : the cyclotomic quotient

i

β_i · · · = 0.

Finite case

If Fi is locally nilpotent (=> H.W. integral), Ei should acts as q^k1[m₁] +· · ·+q<sup>k`</sup>[m<sub>`</sub>],

where[ms] =q^ms−1+· · ·+q^1−ms .

⇒KLR is too big→ we take quotient : the cyclotomic quotient

i

β_i · · · = 0.

Innite/Verma case

For Fi `innite', Ei should acts as q^k1q^m1λi −q^−m1λ⁻_i ¹

1−q² +· · ·+q^k`q^m1λi−q^−m1λ⁻_i ¹ 1−q² , whereλi =q^βi (=formal parameter).

⇒KLR is too small → add gradingλ_i and superstructure (parity), with new generator

i

i · · · ,

ofq-degree=0,λi-degree=2 and parity =1 (they anticommute).

Innite/Verma case

For Fi `innite', Ei should acts as q^k1q^m1λi −q^−m1λ⁻_i ¹

1−q² +· · ·+q^k`q^m1λi−q^−m1λ⁻_i ¹ 1−q² , whereλi =q^βi (=formal parameter).

⇒KLR is too small → add gradingλ_i and superstructure (parity), with new generator

i

i · · · ,

ofq-degree=0,λi-degree=2 and parity=1 (they anticommute).

Some facts

For Fi locally nilpotent, Ei,Fi realize (categorical)

sl(2)-commutator as direct sum, otherwise there is a natural SES : 0→FiEi →EiFi → 1

q−q⁻¹(K ⊕ΠK⁻¹)→0.

Also, there is a dierential

d_ni







 i

i · · ·









= i

β_i · · · ·

so that the homology is a cyclotomic quotient.

⇒we got a dg-enhanchement of cyclotomic-KLR,

⇒it allows to compute many things easily in these.

Some facts

For Fi locally nilpotent, Ei,Fi realize (categorical)

sl(2)-commutator as direct sum, otherwise there is a natural SES : 0→FiEi →EiFi → 1

q−q⁻¹(K ⊕ΠK⁻¹)→0. Also, there is a dierential

dni







 i

i · · ·









= i

β_i · · · ·

so that the homology is a cyclotomic quotient.

⇒we got a dg-enhanchement of cyclotomic-KLR,

⇒it allows to compute many things easily in these.

A bit of topology

HOMFLY polynomial arises in parabolic Verma modules ofgl(2n) (by Queelec-Sartori work),

⇒KR-HOMFLY homology arises naturally in the parabolic 2-Verma modules,

⇒we have new tools to compute stu about KR homology.

Theorem (N., Vaz, 2017)

d_N induces a spectral sequence on KR-homology to

sl(N)-homology, which agree with Rasmussen's one. Moreover, the SS converges at the second page.

→Idea : lift the complex in the `total' dg-enhancement and observe it is homototopic to a complex concentrated in degree 0.

A bit of topology

HOMFLY polynomial arises in parabolic Verma modules ofgl(2n) (by Queelec-Sartori work),

⇒KR-HOMFLY homology arises naturally in the parabolic 2-Verma modules,

⇒we have new tools to compute stu about KR homology.

Theorem (N., Vaz, 2017)

d_N induces a spectral sequence on KR-homology to

sl(N)-homology, which agree with Rasmussen's one. Moreover, the SS converges at the second page.

→Idea : lift the complex in the `total' dg-enhancement and observe it is homototopic to a complex concentrated in degree 0.