Sean Clark University of Virginia Dissertation Defense ... · WHAT IS A QUANTUM GROUP? A quantum...

Quantum supergroups and canonical bases

Sean ClarkUniversity of Virginia

Dissertation DefenseApril 4, 2014

WHAT 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, K±1i for i ∈ I,

Various relations; for example,

I Ki ≈ qhi , e.g. KiEjK−1i = q〈hi,αj〉Ej

I quantum Serre, e.g. F2i Fj − [2]FiFjFi + FjF2

i = 0(here [2] = q + q−1 is a quantum integer)

Some important features are:I an involution q = q−1, Ki = K−1

i , Ei = Ei, Fi = Fi;I a bar invariant integral Z[q, q−1]-form of Uq(g).

CANONICAL 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].

LIE 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 00 c d0 e −c

︸︷︷︸

0 a b−b 0 0a 0 0

︸︷︷︸

with the super bracket; i.e. the usual bracket, except[A1,B1] = A1B1+B1A1.

(Note: The subalgebra of the A0 is ∼= to sl(2).)

LIE 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 00 c d0 e −c

︸︷︷︸

0 a b−b 0 0a 0 0

︸︷︷︸

with the super bracket; i.e. the usual bracket, except[A1,B1] = A1B1+B1A1.

(Note: The subalgebra of the A0 is ∼= to sl(2).)

OUR 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 a la Lusztig, Kashiwara?

Some potential obstructions are:I Existence of isotropic simple roots: (αi, αi) = 0I 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!

OUR QUESTION

INFLUENCE 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)

INSIGHT FROM [HW]

Key Insight [HW]: use a parameter π2 = 1 for super signse.g. a super commutator AB + BA becomes AB− πBA

I π = 1 non-super case.I π = −1 super case.

There is a bar involution on Q(q)[π] given by q 7→ πq−1.

[n] =(πq)n − q−n

πq− q−1 , e.g. [2] = πq + q−1.

Note πq + q−1 has positive coefficients. (vs. −q + q−1)

(Important for categorification: e.g. F2i = (πq + q−1)F(2)

INSIGHT FROM [HW]

[n] =(πq)n − q−n

πq− q−1 , e.g. [2] = πq + q−1.

INSIGHT FROM [HW]

[n] =(πq)n − q−n

πq− q−1 , e.g. [2] = πq + q−1.

INSIGHT FROM [HW]

[n] =(πq)n − q−n

πq− q−1 , e.g. [2] = πq + q−1.

ANISOTROPIC 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 (diaij = djaji);I (anisotropy) aij ∈ 2Z for i ∈ I1;I (bar-compatibility) di = p(i) mod 2, where i ∈ Ip(i)

EXAMPLES (FINITE AND AFFINE)(•=odd root)

• ◦ ◦ · · ·< ◦ ◦ ◦ (osp(1|2n))

• ◦ ◦ · · ·< ◦ ◦ ◦<

• ◦ ◦ · · ·< ◦ ◦ •>

• ◦ ◦ · · ·< ◦ ◦◦vvvv◦◦H

◦ • ◦> <

• ◦<

• •<>

FINITE TYPE

The only finite type covering algebras have Dynkin diagrams

• ◦ ◦ · · ·< ◦ ◦ ◦

This diagram corresponds toI 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]

FINITE TYPE

• ◦ ◦ · · ·< ◦ ◦ ◦

FINITE TYPE

• ◦ ◦ · · ·< ◦ ◦ ◦

FINITE TYPE

• ◦ ◦ · · ·< ◦ ◦ ◦

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]

FINITE TYPE

• ◦ ◦ · · ·< ◦ ◦ ◦

RANK 1[CW]: Uq(osp(1|2))/Q(q) can be tweaked to get all reps.

EF− πFE =1K − K−1

πq− q−1︸︷︷︸even h.w.

orπK − K−1

πq− q−1︸︷︷︸odd h.w.

New definition: generators E, F, K±1, J, relations

J2 = 1, JK = KJ,

JEJ−1 = E, KEK−1 = q2E, JFJ−1 = F, KFK−1 = q−2F,

EF− πFjEi =JK − K−1

πq− q−1 ; (∗′)

(If h is the Cartan element, K = qh and J = πh.)

RANK 1

[CW]: Uq(osp(1|2))/Q(q) can be tweaked to get all reps.

EF− πFE =πhK − K−1

πq− q−1 (h the Cartan generator) (∗)

J2 = 1, JK = KJ,

πq− q−1 ; (∗′)

RANK 1

[CW]: Uq(osp(1|2))/Q(q) can be tweaked to get all reps.

EF− πFE =πhK − K−1

πq− q−1 (h the Cartan generator) (∗)

J2 = 1, JK = KJ,

πq− q−1 ; (∗′)

DEFINITION OF QUANTUM COVERING GROUPS

Let A be a symmetrizable GCM. U is the Q(q)[π]-algebra withgenerators Ei, Fi, K±1

i , Ji and relations

Ji2 = 1, JiKi = KiJi, JiJj = JjJi

JiEjJ−1i = πaijEj, JiFjJ−1

i = π−aijFj.

EiFj − πp(i)p(j)FjEi = δijJdii Kdi

i − K−dii

(πq)di − q−di;

and others (super quantum Serre, usual K relations).

Bar involution: q = πq−1, Ki = JiK−1i , Ei = Ei, Fi = Fi

Can also define a bar-invariant integral Z[q, q−1, π]-form!

DEFINITION OF QUANTUM COVERING GROUPS

Let A be a symmetrizable GCM. U is the Q(q)[π]-algebra withgenerators Ei, Fi, K±1

i , Ji and relations

Ji2 = 1, JiKi = KiJi, JiJj = JjJi

JiEjJ−1i = πaijEj, JiFjJ−1

i = π−aijFj.

EiFj − πp(i)p(j)FjEi = δijJdii Kdi

i − K−dii

(πq)di − q−di;

and others (super quantum Serre, usual K relations).

Bar involution: q = πq−1, Ki = JiK−1i , Ei = Ei, Fi = Fi

Can also define a bar-invariant integral Z[q, q−1, π]-form!

RELATION TO QUANTUM (SUPER)GROUPS

By specifying a value of π, we have maps

Eπ=−1

{{wwwwwwwww

U|π=−1 U|π=1

I U|π=1 is a quantum group (forgets Z/2Z grading).I U|π=−1 is a quantum supergroup.

REPRESENTATIONS

X: integral weights, X+: dominant integral weights.

A weight module is a U-module M =⊕

λ∈X Mλ, where

Mλ ={

m ∈M : Kim = q〈hi,λ〉m, Jim = π〈hi,λ〉m}.

Example: Uq(osp(1|2)), X = Z, X+ = N and M =⊕

n∈Z Mn.

Jm = πnm, Km = qnm (m ∈Mn)

REPRESENTATIONS

X: integral weights, X+: dominant integral weights.

A weight module is a U-module M =⊕

λ∈X Mλ, where

Mλ ={

m ∈M : Kim = q〈hi,λ〉m, Jim = π〈hi,λ〉m}.

Example: Uq(osp(1|2)), X = Z, X+ = N and M =⊕

n∈Z Mn.

Jm = πnm, Km = qnm (m ∈Mn)

REPRESENTATIONS

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) ofhighest weight λ. Any (“π-free”) h.wt. int. M is a direct sum of theseV(λ).(π-free: π acts freely)

Example: Uq(osp(1|2)) has simple π-free modules V(n), whichare free Q(q)[π]-modules of rank n + 1. (Like sl(2)!)

V(n) = V(n)|π=1︸︷︷︸dimQ(q)=n+1

⊕V(n)|π=−1︸︷︷︸dimQ(q)=n+1

REPRESENTATIONS

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) ofhighest weight λ. Any (“π-free”) h.wt. int. M is a direct sum of theseV(λ).(π-free: π acts freely)

Example: Uq(osp(1|2)) has simple π-free modules V(n), whichare free Q(q)[π]-modules of rank n + 1. (Like sl(2)!)

V(n) = V(n)|π=1︸︷︷︸dimQ(q)=n+1

⊕V(n)|π=−1︸︷︷︸dimQ(q)=n+1

APPROACHES TO CANONICAL BASES

Two potential approaches to constructing a canonical basis:I [Lusztig] using geometryI [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.

WHY BELIEVE?

No examples despite extensive study, experts don’t believe.Why should canonical bases exist?

Because now we haveI 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.

WHY BELIEVE?

Because now we haveI 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.

WHY BELIEVE?

Because now we haveI 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.

WHY BELIEVE?

Because now we haveI 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.

CRYSTALS

We can define Kashiwara operators ei, fi.

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) ifI L is a A-lattice of V(λ) closed under ei, fi

and B ⊂ L/qL satisfiesI B is a π-basis of L/qL; (i.e. signed at π = −1: B = B ∪ πB)I eiB ⊆ B ∪ {0} and fiB ⊆ B ∪ {0};I For b ∈ B, if eib 6= 0 then b = fieib.

As in the π = 1 case, the crystal lattice/basis is unique.

CRYSTALS

CANONICAL BASIS

We set

V(λ) ⊃ L(λ) =∑Afi1 . . . finvλ, B(λ) =

{πε fi1 . . . finvλ + 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 thatG(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!)

CANONICAL BASIS

We set

V(λ) ⊃ L(λ) =∑Afi1 . . . finvλ, B(λ) =

{πε fi1 . . . finvλ + 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 thatG(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!)

MAIN OBSTACLE IN PROOF

Most of Kashiwara’s arguments generalize (with extra signs).

Kashiwara’s construction of G requires ρ(L(∞)) ⊂ L(∞) whereρ 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.

U−|π=1 ⊗ C∼=−→ U−|π=−1 ⊗ C

WHY MUST THE BASIS BE SIGNED?

Example: I = I1 = {i, j} such that aij = aji = 0.

FiFj = πFjFi

Should FiFj or FjFi 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.

WHY MUST THE BASIS BE SIGNED?

Example: I = I1 = {i, j} such that aij = aji = 0.

FiFj = πFjFi

Should FiFj or FjFi 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.

CANONICAL BASES AND THE WHOLE QUANTUM

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 ∑λ∈X

1λ with 1λ1η = δλ,η1λ, Ki ∑λ∈X

q〈hi,λ〉1λ

U 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.

1 ∑λ∈X

q〈hi,λ〉1λ

1 ∑λ∈X

q〈hi,λ〉1λ

RANK 1

Uq(osp(1|2)) is the algebra given byGenerators: E1n = 1n+2E, F1n = 1n−2F, 1nRelations: 1n1m = δnm1n, (E1n−2)(F1n)− (F1n+2)(E1n) = [n]1n

Theorem (C-Wang)Uq(osp(1|2)) admits a canonical basis

E(a)1nF(b), πabF(b)1nE(a) | a + b ≥ n}.

We conjectured Uq(osp(1|2)) admits a categorification, andEllis and Lauda (’13) recently verified our conjecture.

RANK 1

CANONICAL BASIS

Theorem (C)U admits a π-signed canonical basis generalizing the basis for U−.For π = 1, this specializes to Lusztig’s canonical basis for U|π=1.

Idea of proof (generalizing Lusztig):Consider modules N(λ, λ′)→ U1λ−λ′ as λ, λ′ →∞.

Define epimorphisms t : N(λ+ λ′′, λ′′ + λ′)→ N(λ, λ′).({N(λ, λ′)}with t forms a projective system)

Construct suitable bar involution, canonical basis on N(λ, λ′).

The canonical basis is stable under the projective limit⇒ induces a bar-invariant canonical basis on U.

CANONICAL BASIS

FURTHER DIRECTIONS

I Construction of braid group action a la LusztigI Forthcoming work with D. Hill

I Canonical bases for other Lie superalgebrasI gl(m|1), osp(2|2n) using quantum shuffles [CHW3]I Open question in general; e.g. gl(2|2).

I Categorification for covering quantum groupsI Connection to odd link homologies (Khovanov)I Tensor modules?I Higher rank?

FURTHER DIRECTIONS

SOME RELATED 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 andcategorifications, 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 athttp://people.virginia.edu/˜sic5ag/

Thank you for your attention!

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