Download Quantum supergroups and canonical bases Sean Clark University of Virginia Dissertation Defense

Quantum supergroups and canonical bases
Sean Clark
University of Virginia
Dissertation Defense
April 4, 2014
W HAT IS A QUANTUM GROUP ?
A quantum group is a deformed universal enveloping algebra.
W HAT IS A QUANTUM GROUP ?
A quantum group is a deformed universal enveloping algebra.
Let g be a semisimple Lie algebra (e.g. sl(n), so(2n + 1)).
Π = {αi : i ∈ I} the simple roots.
W HAT IS A QUANTUM GROUP ?
A quantum group is a deformed universal enveloping algebra.
Let g be a semisimple Lie algebra (e.g. sl(n), so(2n + 1)).
Π = {αi : i ∈ I} the simple roots.
Uq (g) is the Q(q) algebra with generators Ei , Fi , Ki±1 for i ∈ I,
Various relations; for example,
I K ≈ qhi , e.g. K E K−1 = qhhi ,αj i E
i
I
i j i
j
quantum Serre, e.g. F2i Fj − [2]Fi Fj Fi + Fj F2i = 0
(here [2] = q + q−1 is a quantum integer)
W HAT IS A QUANTUM GROUP ?
A quantum group is a deformed universal enveloping algebra.
Let g be a semisimple Lie algebra (e.g. sl(n), so(2n + 1)).
Π = {αi : i ∈ I} the simple roots.
Uq (g) is the Q(q) algebra with generators Ei , Fi , Ki±1 for i ∈ I,
Various relations; for example,
I K ≈ qhi , e.g. K E K−1 = qhhi ,αj i E
i
I
i j i
j
quantum Serre, e.g. F2i Fj − [2]Fi Fj Fi + Fj F2i = 0
(here [2] = q + q−1 is a quantum integer)
Some important features are:
I
an involution q = q−1 , Ki = Ki−1 , Ei = Ei , Fi = Fi ;
I
a bar invariant integral Z[q, q−1 ]-form of Uq (g).
C ANONICAL BASIS AND CATEGORIFICATION
Uq (n− ), the subalgebra generated by Fi .
C ANONICAL BASIS AND CATEGORIFICATION
Uq (n− ), the subalgebra generated by Fi .
[Lusztig, Kashiwara]: Uq (n− ) has a canonical basis, which
I
is bar-invariant,
I
descends to a basis for each h. wt. integrable module,
I
has structure constants in N[q, q−1 ] (symmetric type).
C ANONICAL BASIS AND CATEGORIFICATION
Uq (n− ), the subalgebra generated by Fi .
[Lusztig, Kashiwara]: Uq (n− ) has a canonical basis, which
I
is bar-invariant,
I
descends to a basis for each h. wt. integrable module,
I
has structure constants in N[q, q−1 ] (symmetric type).
Relation to categorification:
I
Uq (n− ) categorified by quiver Hecke algebras
[Khovanov-Lauda, Rouquier]
I
canonical basis ↔ indecomp. projectives (symmetric type)
[Varagnolo-Vasserot].
L IE SUPERALGEBRAS
g: a Lie superalgebra (everything is Z/2Z-graded).
e.g. gl(m|n), osp(m|2n)
L IE SUPERALGEBRAS
g: a Lie superalgebra (everything is Z/2Z-graded).
e.g. gl(m|n), osp(m|2n)
Example: osp(1|2) is the set of 3 × 3 matrices of the form

 

0 0 0
0 a b
A =  0 c d  +  −b 0 0 
0 e −c
a 0 0
|
{z
} |
{z
}
A0
A1
with the super bracket; i.e. the usual bracket, except
[A1 , B1 ] = A1 B1 +B1 A1 .
(Note: The subalgebra of the A0 is ∼
= to sl(2).)
O UR QUESTION
Quantized Lie superalgebras have been well studied
(Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane, ...)
O UR QUESTION
Quantized Lie superalgebras have been well studied
(Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane, ...)
Uq (n− ): algebra generated by Fi satisfying super Serre relations.
Is there a canonical basis à la Lusztig, Kashiwara?
O UR QUESTION
Quantized Lie superalgebras have been well studied
(Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane, ...)
Uq (n− ): algebra generated by Fi satisfying super Serre relations.
Is there a canonical basis à la Lusztig, Kashiwara?
Some potential obstructions are:
I
Existence of isotropic simple roots: (αi , αi ) = 0
I
No integral form, bar involution (e.g. quantum osp(1|2))
I
Lack of positivity due to super signs
Experts did not expect canonical bases to exist!
I NFLUENCE OF CATEGORIFICATION
I
[KL,R] (’08): quiver Hecke categorify quantum groups
I NFLUENCE OF CATEGORIFICATION
I
[KL,R] (’08): quiver Hecke categorify quantum groups
I
[KKT11]: introduce quiver Hecke superalgebras (QHSA)
(Generalizes a construction of Wang (’06))
I NFLUENCE OF CATEGORIFICATION
I
[KL,R] (’08): quiver Hecke categorify quantum groups
I
[KKT11]: introduce quiver Hecke superalgebras (QHSA)
(Generalizes a construction of Wang (’06))
I
[KKO12]: QHSA’s categorify quantum groups
(Generalizes a rank 1 construction of [EKL11])
I NFLUENCE OF CATEGORIFICATION
I
[KL,R] (’08): quiver Hecke categorify quantum groups
I
[KKT11]: introduce quiver Hecke superalgebras (QHSA)
(Generalizes a construction of Wang (’06))
I
[KKO12]: QHSA’s categorify quantum groups
(Generalizes a rank 1 construction of [EKL11])
I
[HW12]: QHSA’s categorify quantum supergroups
(assuming no isotropic roots)
I NSIGHT FROM [HW]
Key Insight [HW]: use a parameter π 2 = 1 for super signs
e.g. a super commutator AB + BA becomes AB − πBA
I NSIGHT FROM [HW]
Key Insight [HW]: use a parameter π 2 = 1 for super signs
e.g. a super commutator AB + BA becomes AB − πBA
I
π=1
I
π = −1
non-super case.
super case.
I NSIGHT FROM [HW]
Key Insight [HW]: use a parameter π 2 = 1 for super signs
e.g. a super commutator AB + BA becomes AB − πBA
I
π=1
I
π = −1
non-super case.
super case.
There is a bar involution on Q(q)[π] given by q 7→ πq−1 .
I NSIGHT FROM [HW]
Key Insight [HW]: use a parameter π 2 = 1 for super signs
e.g. a super commutator AB + BA becomes AB − πBA
I
π=1
I
π = −1
non-super case.
super case.
There is a bar involution on Q(q)[π] given by q 7→ πq−1 .
[n] =
(πq)n − q−n
, e.g. [2] = πq + q−1 .
πq − q−1
Note πq + q−1 has positive coefficients. (vs. −q + q−1 )
(2)
(Important for categorification: e.g. F2i = (πq + q−1 )Fi .)
A NISOTROPIC KM
I = I0
`
I1 (simple roots), parity p(i) with i ∈ Ip(i) .
Symmetrizable generalized Cartan matrix (aij )i,j∈I :
I
aij ∈ Z, aii = 2, aij ≤ 0;
I
positive symmetrizing coefficients di (di aij = dj aji );
I
(anisotropy) aij ∈ 2Z for i ∈ I1 ;
I
(bar-compatibility) di = p(i) mod 2, where i ∈ Ip(i)
E XAMPLES (F INITE AND A FFINE )
(•=odd root)
• < ◦
◦ ··· ◦
◦
• < ◦
◦ ··· ◦
◦ < ◦
• < ◦
◦ ··· ◦
◦ > •
• < ◦
◦ ··· ◦
◦ > • < ◦
• < ◦
• <> •
◦
◦
vvv
H
◦HvHH
◦
(osp(1|2n))
F INITE TYPE
The only finite type covering algebras have Dynkin diagrams
• < ◦
◦ ··· ◦
◦
◦
F INITE TYPE
The only finite type covering algebras have Dynkin diagrams
• < ◦
◦ ··· ◦
This diagram corresponds to
I
the Lie superalgebra osp(1|2n)
◦
◦
F INITE TYPE
The only finite type covering algebras have Dynkin diagrams
• < ◦
◦ ··· ◦
This diagram corresponds to
I
the Lie superalgebra osp(1|2n)
I
the Lie algebra so(1 + 2n)
◦
◦
F INITE TYPE
The only finite type covering algebras have Dynkin diagrams
• < ◦
◦ ··· ◦
◦
◦
This diagram corresponds to
I
the Lie superalgebra osp(1|2n)
I
the Lie algebra so(1 + 2n)
These algebras have similar representation theories.
I
osp(1|2n) irreps ↔ half of so(2n + 1) irreps.
F INITE TYPE
The only finite type covering algebras have Dynkin diagrams
• < ◦
◦ ··· ◦
◦
◦
This diagram corresponds to
I
the Lie superalgebra osp(1|2n)
I
the Lie algebra so(1 + 2n)
These algebras have similar representation theories.
I
osp(1|2n) irreps ↔ half of so(2n + 1) irreps.
I
Uq (osp(1|2n))/C(q) ↔ all of Uq (so(2n + 1)) irreps. [Zou98]
R ANK 1
[CW]: Uq (osp(1|2))/Q(q) can be tweaked to get all reps.
EF − πFE =
1K − K−1
πq − q−1
| {z }
even h.w.
or
πK − K−1
πq − q−1
| {z }
odd h.w.
R ANK 1
[CW]: Uq (osp(1|2))/Q(q) can be tweaked to get all reps.
EF − πFE =
π h K − K−1
πq − q−1
(h the Cartan generator)
(∗)
R ANK 1
[CW]: Uq (osp(1|2))/Q(q) can be tweaked to get all reps.
EF − πFE =
π h K − K−1
πq − q−1
(h the Cartan generator)
(∗)
New definition: generators E, F, K±1 , J, relations
J2 = 1,
JEJ−1 = E,
KEK−1 = q2 E,
JK = KJ,
JFJ−1 = F,
EF − πFj Ei =
KFK−1 = q−2 F,
JK − K−1
;
πq − q−1
(If h is the Cartan element, K = qh and J = π h .)
(∗0 )
D EFINITION OF QUANTUM COVERING GROUPS
Let A be a symmetrizable GCM. U is the Q(q)[π]-algebra with
generators Ei , Fi , Ki±1 , Ji and relations
Ji 2 = 1,
Ji Ki = Ki Ji ,
Ji Jj = Jj Ji
Ji Ej Ji−1 = π aij Ej , Ji Fj Ji−1 = π −aij Fj .
Ei Fj − π
p(i)p(j)
Jidi Kidi − Ki−di
;
Fj Ei = δij
(πq)di − q−di
and others (super quantum Serre, usual K relations).
D EFINITION OF QUANTUM COVERING GROUPS
Let A be a symmetrizable GCM. U is the Q(q)[π]-algebra with
generators Ei , Fi , Ki±1 , Ji and relations
Ji 2 = 1,
Ji Ki = Ki Ji ,
Ji Jj = Jj Ji
Ji Ej Ji−1 = π aij Ej , Ji Fj Ji−1 = π −aij Fj .
Ei Fj − π
p(i)p(j)
Jidi Kidi − Ki−di
;
Fj Ei = δij
(πq)di − q−di
and others (super quantum Serre, usual K relations).
Bar involution: q = πq−1 , Ki = Ji Ki−1 , Ei = Ei , Fi = Fi
Can also define a bar-invariant integral Z[q, q−1 , π]-form!
R ELATION TO QUANTUM ( SUPER ) GROUPS
By specifying a value of π, we have maps
w
π=−1 www
w
ww
w
{w
U|π=−1
UE
EE
EEπ=1
EE
E"
U|π=1
I
U|π=1 is a quantum group (forgets Z/2Z grading).
I
U|π=−1 is a quantum supergroup.
R EPRESENTATIONS
X: integral weights, X+ : dominant integral weights.
A weight module is a U-module M =
n
Mλ = m ∈ M : Ki m = qhhi ,λi m,
L
Mλ , where
o
Ji m = π hhi ,λi m .
λ∈X
R EPRESENTATIONS
X: integral weights, X+ : dominant integral weights.
A weight module is a U-module M =
n
Mλ = m ∈ M : Ki m = qhhi ,λi m,
L
Mλ , where
o
Ji m = π hhi ,λi m .
λ∈X
Example: Uq (osp(1|2)), X = Z, X+ = N and M =
Jm = π n m,
L
Km = qn m (m ∈ Mn )
n∈Z Mn .
R EPRESENTATIONS
Can define highest-weight (h.w.) and integrable (int.) modules.
Theorem (C-Hill-Wang)
For each λ ∈ X+ , there is a unique simple (“π-free”) module V(λ) of
highest weight λ. Any (“π-free”) h.wt. int. M is a direct sum of these
V(λ).
(π-free: π acts freely)
R EPRESENTATIONS
Can define highest-weight (h.w.) and integrable (int.) modules.
Theorem (C-Hill-Wang)
For each λ ∈ X+ , there is a unique simple (“π-free”) module V(λ) of
highest weight λ. Any (“π-free”) h.wt. int. M is a direct sum of these
V(λ).
(π-free: π acts freely)
Example: Uq (osp(1|2)) has simple π-free modules V(n), which
are free Q(q)[π]-modules of rank n + 1. (Like sl(2)!)
V(n) = V(n)|π=1 ⊕ V(n)|π=−1
| {z } | {z }
dimQ(q) =n+1
dimQ(q) =n+1
A PPROACHES TO CANONICAL BASES
Two potential approaches to constructing a canonical basis:
I
[Lusztig] using geometry
I
[Kashiwara] algebraically using crystals (“q = 0”)
A PPROACHES TO CANONICAL BASES
Two potential approaches to constructing a canonical basis:
I
[Lusztig] using geometry
I
[Kashiwara] algebraically using crystals (“q = 0”)
Analogous geometry for super is unknown.
A PPROACHES TO CANONICAL BASES
Two potential approaches to constructing a canonical basis:
I
[Lusztig] using geometry
I
[Kashiwara] algebraically using crystals (“q = 0”)
Analogous geometry for super is unknown.
There are various crystal structures in modules:
I
osp(1|2n) [Musson-Zou] (’98)
I
gl(m|n) [Benkart-Kang-Kashiwara] (’00), [Kwon] (’12)
I
for KM superalgebra with “even” weights [Jeong] (’01)
No examples of canonical bases.
W HY BELIEVE ?
No examples despite extensive study, experts don’t believe.
Why should canonical bases exist?
W HY BELIEVE ?
No examples despite extensive study, experts don’t believe.
Why should canonical bases exist?
Because now we have
I
a better definition of U (all h. wt. modules /Q(q));
W HY BELIEVE ?
No examples despite extensive study, experts don’t believe.
Why should canonical bases exist?
Because now we have
I
a better definition of U (all h. wt. modules /Q(q));
I
a good bar involution;
I
a bar-invariant integral form;
W HY BELIEVE ?
No examples despite extensive study, experts don’t believe.
Why should canonical bases exist?
Because now we have
I
a better definition of U (all h. wt. modules /Q(q));
I
a good bar involution;
I
a bar-invariant integral form;
I
a categorical canonical basis.
This motivates us to try again generalizing Kashiwara.
C RYSTALS
We can define Kashiwara operators ẽi , f̃i .
Let A ⊂ Q(q)[π] be the ring of functions with no pole at q = 0.
V(λ) is said to have a crystal basis (L, B) if
I
L is a A-lattice of V(λ) closed under ẽi , f̃i
and B ⊂ L/qL satisfies
I
B is a π-basis of L/qL; (i.e. signed at π = −1: B = B ∪ πB)
I
ẽi B ⊆ B ∪ {0} and f̃i B ⊆ B ∪ {0};
I
For b ∈ B, if ẽi b 6= 0 then b = f̃i ẽi b.
As in the π = 1 case, the crystal lattice/basis is unique.
C RYSTALS
We can define Kashiwara operators ẽi , f̃i .
Let A ⊂ Q(q)[π] be the ring of functions with no pole at q = 0.
V(λ) is said to have a crystal basis (L, B) if
I
L is a A-lattice of V(λ) closed under ẽi , f̃i
and B ⊂ L/qL satisfies
I
B is a π-basis of L/qL; (i.e. signed at π = −1: B = B ∪ πB)
I
ẽi B ⊆ B ∪ {0} and f̃i B ⊆ B ∪ {0};
I
For b ∈ B, if ẽi b 6= 0 then b = f̃i ẽi b.
As in the π = 1 case, the crystal lattice/basis is unique.
C RYSTALS
We can define Kashiwara operators ẽi , f̃i .
Let A ⊂ Q(q)[π] be the ring of functions with no pole at q = 0.
V(λ) is said to have a crystal basis (L, B) if
I
L is a A-lattice of V(λ) closed under ẽi , f̃i
and B ⊂ L/qL satisfies
I
B is a π-basis of L/qL; (i.e. signed at π = −1: B = B ∪ πB)
I
ẽi B ⊆ B ∪ {0} and f̃i B ⊆ B ∪ {0};
I
For b ∈ B, if ẽi b 6= 0 then b = f̃i ẽi b.
As in the π = 1 case, the crystal lattice/basis is unique.
C ANONICAL B ASIS
We set
V(λ) ⊃ L(λ) =
X
Af̃i1 . . . f̃in vλ ,
n
o
B(λ) = π f̃i1 . . . f̃in vλ + qL(λ)
(λ ∈ X+ ∪ {∞} , V(∞) = U− )
C ANONICAL B ASIS
We set
V(λ) ⊃ L(λ) =
X
Af̃i1 . . . f̃in vλ ,
n
o
B(λ) = π f̃i1 . . . f̃in vλ + qL(λ)
(λ ∈ X+ ∪ {∞} , V(∞) = U− )
Theorem (C-Hill-Wang)
The pairs (L(λ), B(λ)) for λ ∈ X+ ∪ {∞} are crystal bases.
Moreover, there exist maps G : L(λ)/qL(λ) → L(λ) such that
G(B(λ)) is a bar-invariant π-basis of V(λ).
We call G(B(λ)) the canonical basis of V(λ).
(π = −1: first canonical bases for quantum supergroups!)
M AIN OBSTACLE IN PROOF
Most of Kashiwara’s arguments generalize (with extra signs).
M AIN OBSTACLE IN PROOF
Most of Kashiwara’s arguments generalize (with extra signs).
Kashiwara’s construction of G requires ρ(L(∞)) ⊂ L(∞) where
ρ is an anti-automorphism of U− .
Super signs cause non-positivity problems ⇒ usual proof fails.
M AIN OBSTACLE IN PROOF
Most of Kashiwara’s arguments generalize (with extra signs).
Kashiwara’s construction of G requires ρ(L(∞)) ⊂ L(∞) where
ρ is an anti-automorphism of U− .
Super signs cause non-positivity problems ⇒ usual proof fails.
New idea: a twistor (from work with Fan, Li, Wang [CFLW]).
∼
=
U− |π=1 ⊗ C −→ U− |π=−1 ⊗ C
which is almost an algebra isomorphism.
Good enough: the ρ-invariance at π = 1 transports to π = −1.
W HY MUST THE BASIS BE SIGNED ?
Example: I = I1 = {i, j} such that aij = aji = 0.
Fi Fj = πFj Fi
Should Fi Fj or Fj Fi be in B(∞)? No preferred canonical choice.
W HY MUST THE BASIS BE SIGNED ?
Example: I = I1 = {i, j} such that aij = aji = 0.
Fi Fj = πFj Fi
Should Fi Fj or Fj Fi be in B(∞)? No preferred canonical choice.
This is not a bad thing!
I
A π-basis is an honest Q(q)-basis (for π-free modules)!
I
Categorically: represents “spin states” of QHSA modules.
C ANONICAL BASES AND THE WHOLE QUANTUM
GROUP
Can the canonical basis on U− be extended to U?
Not directly: U0 makes such a construction difficult.
C ANONICAL BASES AND THE WHOLE QUANTUM
GROUP
Can the canonical basis on U− be extended to U?
Not directly: U0 makes such a construction difficult.
The ‘right’ construction is to explode U0 into idempotents.
(Beilinson-Lusztig-McPherson (type A), Lusztig)
1
X
λ∈X
1λ with 1λ 1η = δλ,η 1λ ,
Ki
X
λ∈X
qhhi ,λi 1λ
C ANONICAL BASES AND THE WHOLE QUANTUM
GROUP
Can the canonical basis on U− be extended to U?
Not directly: U0 makes such a construction difficult.
The ‘right’ construction is to explode U0 into idempotents.
(Beilinson-Lusztig-McPherson (type A), Lusztig)
1
X
λ∈X
1λ with 1λ 1η = δλ,η 1λ ,
Ki
X
qhhi ,λi 1λ
λ∈X
U̇ is the algebra on symbols x1λ = 1λ+|x| x for x ∈ U, λ ∈ X.
x1λ = projection to λ-wt. space followed by the action of x.
R ANK 1
U̇q (osp(1|2)) is the algebra given by
Generators: E1n = 1n+2 E,
F1n = 1n−2 F,
1n
Relations: 1n 1m = δnm 1n , (E1n−2 )(F1n ) − (F1n+2 )(E1n ) = [n]1n
R ANK 1
U̇q (osp(1|2)) is the algebra given by
Generators: E1n = 1n+2 E,
F1n = 1n−2 F,
1n
Relations: 1n 1m = δnm 1n , (E1n−2 )(F1n ) − (F1n+2 )(E1n ) = [n]1n
Theorem (C-Wang)
U̇q (osp(1|2)) admits a canonical basis
n
o
Ḃ = E(a) 1n F(b) , π ab F(b) 1n E(a) | a + b ≥ n .
R ANK 1
U̇q (osp(1|2)) is the algebra given by
Generators: E1n = 1n+2 E,
F1n = 1n−2 F,
1n
Relations: 1n 1m = δnm 1n , (E1n−2 )(F1n ) − (F1n+2 )(E1n ) = [n]1n
Theorem (C-Wang)
U̇q (osp(1|2)) admits a canonical basis
n
o
Ḃ = E(a) 1n F(b) , π ab F(b) 1n E(a) | a + b ≥ n .
We conjectured U̇q (osp(1|2)) admits a categorification, and
Ellis and Lauda (’13) recently verified our conjecture.
C ANONICAL B ASIS
Theorem (C)
U̇ admits a π-signed canonical basis generalizing the basis for U− .
For π = 1, this specializes to Lusztig’s canonical basis for U̇|π=1 .
C ANONICAL B ASIS
Theorem (C)
U̇ admits a π-signed canonical basis generalizing the basis for U− .
For π = 1, this specializes to Lusztig’s canonical basis for U̇|π=1 .
Idea of proof (generalizing Lusztig):
Consider modules N(λ, λ0 ) → U̇1λ−λ0 as λ, λ0 → ∞.
C ANONICAL B ASIS
Theorem (C)
U̇ admits a π-signed canonical basis generalizing the basis for U− .
For π = 1, this specializes to Lusztig’s canonical basis for U̇|π=1 .
Idea of proof (generalizing Lusztig):
Consider modules N(λ, λ0 ) → U̇1λ−λ0 as λ, λ0 → ∞.
Define epimorphisms t : N(λ + λ00 , λ00 + λ0 ) → N(λ, λ0 ).
({N(λ, λ0 )} with t forms a projective system)
C ANONICAL B ASIS
Theorem (C)
U̇ admits a π-signed canonical basis generalizing the basis for U− .
For π = 1, this specializes to Lusztig’s canonical basis for U̇|π=1 .
Idea of proof (generalizing Lusztig):
Consider modules N(λ, λ0 ) → U̇1λ−λ0 as λ, λ0 → ∞.
Define epimorphisms t : N(λ + λ00 , λ00 + λ0 ) → N(λ, λ0 ).
({N(λ, λ0 )} with t forms a projective system)
Construct suitable bar involution, canonical basis on N(λ, λ0 ).
C ANONICAL B ASIS
Theorem (C)
U̇ admits a π-signed canonical basis generalizing the basis for U− .
For π = 1, this specializes to Lusztig’s canonical basis for U̇|π=1 .
Idea of proof (generalizing Lusztig):
Consider modules N(λ, λ0 ) → U̇1λ−λ0 as λ, λ0 → ∞.
Define epimorphisms t : N(λ + λ00 , λ00 + λ0 ) → N(λ, λ0 ).
({N(λ, λ0 )} with t forms a projective system)
Construct suitable bar involution, canonical basis on N(λ, λ0 ).
The canonical basis is stable under the projective limit
⇒ induces a bar-invariant canonical basis on U̇.
F URTHER DIRECTIONS
I
Construction of braid group action à la Lusztig
I
Forthcoming work with D. Hill
F URTHER DIRECTIONS
I
Construction of braid group action à la Lusztig
I
I
Forthcoming work with D. Hill
Canonical bases for other Lie superalgebras
I
I
gl(m|1), osp(2|2n) using quantum shuffles [CHW3]
Open question in general; e.g. gl(2|2).
F URTHER DIRECTIONS
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Construction of braid group action à la Lusztig
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Canonical bases for other Lie superalgebras
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Forthcoming work with D. Hill
gl(m|1), osp(2|2n) using quantum shuffles [CHW3]
Open question in general; e.g. gl(2|2).
Categorification for covering quantum groups
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Connection to odd link homologies (Khovanov)
Tensor modules?
Higher rank?
S OME R ELATED PAPERS
Lusztig, Canonical bases arising from quantized enveloping algebras,
J. Amer. Math. Soc. 3 (1990), pp. 447–498
Kashiwara, On crystal bases of the Q-analogue of universal enveloping algebras,
Duke Math. J. 63 (1991), pp. 456–516.
Lusztig, Canonical bases in tensor products,
Proc. Nat. Acad. Sci. U.S.A. 89 (1992), pp. 8177–8179
Ellis, Khovanov, Lauda, The odd nilHecke algebra and its diagrammatics,
IMRN 2014 pp. 991–1062
Hill and Wang, Categorication of quantum Kac-Moody superalgebras,
arXiv:1202.2769, to appear in Trans. AMS.
Fan and Li, Two-parameter quantum algebras, canonical bases and
categorifications, arXiv:1303.2429
C and Wang, Canonical basis for quantum osp(1|2),
arXiv:1204.3940, Lett. Math. Phys. 103 (2013), 207–231.
C, Hill, and Wang, Quantum supergroups I. Foundations,
arXiv:1301.1665, Trans. Groups. 18 (2013), 1019–1053.
C, Hill, and Wang, Quantum supergroups II. Canonical Basis, arXiv:1304.7837.
C, Fan, Li, and Wang, Quantum supergroups III. Twistors,
arXiv:1307.7056, to appear in Comm. Math. Phys.
C, Quantum supergroups IV. Modified form, arXiv:1312.4855.
C, Hill, and Wang, Quantum shuffles and quantum supergroups of basic type,
arXiv:1310.7523.
Slides available at
http://people.virginia.edu/˜sic5ag/
Thank you for your attention!