Kac-Moody quantum superalgebras and global crystal bases Sean Clark - - PowerPoint PPT Presentation

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

Kac-Moody quantum superalgebras and global crystal bases

Sean Clark Joint work with D. Hill and W. Wang University of Virginia Workshop on Super Representation Theory Academia Sinica, Taiwan May 10, 2013

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

QUANTUM GROUPS

Let g be a Kac-Moody algebra with quantum group Uq(g) Uq(n−) ⊂ Uq(g) is related to Hall algebras, quantized shuffles. Uq(g) has a symmetry : q → q−1.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

CANONICAL BASES

The algebra Uq(n−) has an extraordinarily nice basis. It is

◮ suitably independent of choice, ◮ bar-invariant, ◮ well-behaved on the playground of integrable modules, ◮ “almost-orthogonal” (and can be characterized by this), ◮ categorifiable (cf. [Rouquier, Khovanov-Lauda]), ◮ just generally awesome.

For all these reasons (and more!), it deserves the honorific The Canonical Basis

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

FINDING THE CANONICAL BASES

These bases were discovered through the work of Lusztig and Kashiwara. Lusztig: perverse sheaves. Kashiwara: the crystal basis at “q = 0”.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

QUANTUM GROUPS FOR SUPER

Is there a super version of this picture? It isn’t clear what geometry could be used. There are various crystal structures in modules:

◮ osp(1|2n) [Musson-Zou] ◮ gl(m|n) [Benkart-Kang-Kashiwara], [Kwon] ◮ q(n) [Grantcharov-Jung-Kang-Kashiwara-Kim] ◮ for KM superalgebra with “even” weights [Jeong]

Until recently, there was doubt of existence of canonical bases.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

INSPIRATION FROM CATEGORIFICATION

[Wang, Ellis-Lauda-Khovanov], [Kang-Kashiwara-Tsuchioka] provide a fertile setting for categorification. [W, EKL]: spin (nil)Hecke algebras [KKT]: Hecke quiver superalgebras These categorify certain quantum half KM (super)algebras and integrable modules. [Hill-Wang, Kang-Kashiwara-Oh] This is strong evidence for a canonical basis for KM super.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

WHICH KAC-MOODY SUPERALGEBRAS?

We consider a KM superalgebra with GCM A indexed by I = I0 I1 (simple roots) and satisfying:

◮ aij ∈ Z, aii = 2, aij ≤ 0 ◮ there exist positive symmetrizing coefficients di

(diaij = djaji)

◮ (non-isotropy) aij ∈ 2Z for i ∈ I1 ◮ (bar-compatibility) di ≡2 p(i)

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

EXAMPLES

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INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

π-QUANTUM INTEGERS

There is a bar involution on Q(q) given by q → πq−1 (π2 = 1)

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

We have bar-invariant quantum integers: [n] = (πq)n − q−n πq − q−1 , [n]!, n a

• ∈ Z[q, q−1].

These allow us to define quantum divided powers: ∗(n) = ∗n [n]! .

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

THE RANK 1 CASE

Let U be the Q(q)-algebra generated by E, F, K±1 such that KEK−1 = q2E, KFK−1 = q−2F, EF − πFE = K − K−1 πq − q−1 . The bar involution is given by E = E, F = F, K = K−1, q = πq−1.

◮ π = 1 Uq(sl(2)) ◮ π = −1 “quantum osp(1|2)”

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

COMPARING SUPER VS NON-SUPER

There are many nice similarities that can be deduced without choosing π.

◮ U has a triangular decomposition U = E ⊗

• K±1

⊗ F.

◮ U is a Hopf (super)algebra. ◮ U has a quasi-R-matrix and Casimir-type element. ◮ U has semi-simple finite-dimensional modules.

The commutation formulas are even almost the same: E(m)F(n) =

• i

πmn−(i+1

2 )F(m−i)

K; 2i − n − m i

• E(n−i)

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

COMPARING SUPER VS NON-SUPER

The largest difference is in their categories of f.d. modules. Facts:

◮ (π = 1) there is a simple Uq(sl(2))-module of dimension n

for each n ≥ 0.

◮ (π = −1) there is a simple “quantum osp(1|2)”-module of

dimension n for each odd n ≥ 0. This causes headaches for crystals.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

ADJUSTING THE DEFINITION

Fact (Zou): “quantum osp(1|2)” has simple modules of all even dimensions if we extend the field to Q(√π, q). Q: How can we account for all modules over Q(q)? A: Modify the definition of U to get an algebra U′ where EF − πFE = πK − K−1 πq − q−1 When π = −1, U′ has only even-dimensional simples/ Q(q)!

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

GLUING

Theorem (C-Wang)

For π = −1, the algebra U = U ⊕ U′

◮ is a Hopf (super)algebra; ◮ has finite-dimensional simple modules of each dimension; ◮ has a semisimple finite dimensional representation theory;

This has a trivial canonical basis for U−. But does the modified quantum group have a canonical basis?

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

MODIFIED QUANTUM ALGEBRA

Throw in idempotents 1n to obtain a non-unital algebra. (1

• n

1n) The algebra ˙ U is generated by 1n, E1n, F1n such that 1m1m = δnm1n, 1n+2E = E1n, 1n−2F = F1n−2 (EF − πFE)1n = [n]1n

Theorem (C-Wang)

The algebra ˙ U has a canonical basis

• E(a)1−nF(b),

πabF(b)1nE(a) : n ≥ a + b

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

MISSING LINK

U (?) U′ ˙ U 12n ˙ U ˙ U 12n+1

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

EXPANDING THE CARTAN

The difference between U and U′ is EF − πFE = πpK − K−1 πq − q−1 where p is the parity of the “allowed weights”. If “K = qh”, by analogy we define “J = πh”. Adding these elements, we obtain the definition in [C-Hill-Wang]:

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

DEFINITION [CHW1]

Let g be a KM superalgebra, A its symmetrizable GCM. Let Uq(g) be the Q(q)-algebra with generators Ei, Fi, K±1

i

, Ji such that Ji

2 = 1,

JiKi = KiJi, JiJj = JjJi, KiKj = KjKi, JiEjJ−1

i

= πaijEj, KiEjK−1

i

= qaijEj, JiFjJ−1

i

= π−aijFj, KiFjK−1

i

= q−aijFj, EiFj − πp(i)p(j)FjEi = δij Jdi

i Kdi i − K−di i

(πq)di − q−di ;

1−aij

• k=0

(−1)kπp(k;i,j)E

(1−aij−k) i

EjE(k)

i

=

1−aij

• k=0

(−1)kπp(k;i,j)F

(1−aij−k) i

FjF(k)

i

= 0, where p(k; i, j) = kp(i)p(j) + 1

2k(k − 1)p(i).

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

THE BAR INVOLUTION AND COPRODUCT

We extend q → πq−1 to Uq(g) by setting Ei = Ei, Fi = Fi, Ki = JiK−1

i

, Ji = Ji. We can also define a (super) coproduct ∆ by ∆(Ei) = Ei ⊗ K−di

i

+ Jdi

i ⊗ Ei

∆(Fi) = Fi ⊗ 1 + Kdi

i ⊗ Fi

∆(Ki) = Ki ⊗ Ki ∆(Ji) = Ji ⊗ Ji

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

REPRESENTATIONS

Let P (P+) be the set of (dominant) weights of g. A weight module is a Uq(g)-module M =

λ∈P Mλ, where

Mλ =

• m ∈ M : Kim = qhi,λm,

Jim = πhi,λm

• .

We can define highest-weight and integrable modules as usual to obtain a category Oint. Simple modules: V(λ) for all λ ∈ P+

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

PROPERTIES

Proposition [CHW1]. For π = ±1,

◮ Uq(g) = U+ ⊗ U0 ⊗ U−. ◮ Uq(g) is a Hopf (super)algebra. ◮ There is a quasi-R-matrix and quantum Casimir element. ◮ Each M ∈ Oint is completely reducible.

The question remains: is there a canonical basis for U−?

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

KASHIWARA OPERATORS

There is a left derivation operator e′

i and a bilinear form (−, −)

• n U− such that:

(1, 1) = 1, (Fix, y) = (x, e′

i(y)).

Each u ∈ U− can be written x = F(n)

i

xn such that e′

i(xn) = 0.

We can define Kashiwara operators ˜ fix =

• F(n+1)

i

xn, ˜ eix =

• F(n−1)

i

xn.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

CRYSTAL FOR U−

Let A ⊂ Q(q) be the ring of functions with no pole at 0. U− is said to have a crystal basis (L, B) if L is a A-lattice of U− closed under ˜ ei,˜ fi and B ⊂ L/qL satisfies

◮ B is a signed basis of L/qL; ◮ ˜

eiB ⊆ B ∪ {0} and ˜ fiB ⊆ B;

◮ For b ∈ B, if ˜

eib = 0 then b = ˜ fi˜ eib. As in the π = 1 case, the crystal lattice/basis is unique and V(λ) ⊃ L(λ) =

• A˜

fi1 . . .˜ finvλ, B(λ) =

• ˜

fi1 . . .˜ finvλ + qL

• (λ ∈ P+ ∪ {∞} , V(∞) = U−)

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

WHY MUST THE BASIS BE SIGNED?

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

˜ fi ˜ fjvλ = π˜ fj ˜ fivλ Should ˜ fi ˜ fj vλ or ˜ fj ˜ fi vλ be the basis element?

◮ Tensor Product Rule:

˜ fi(b ⊗ b′) = ˜ fib ⊗ b′ if φi(b) > ǫi(b′), πp(i)p(b)b ⊗ ˜ fib′

• therwise.

˜ ei(b ⊗ b′) =

• ˜

eib ⊗ b′ if φi(b) ≥ ǫi(b′), πp(i)p(b)b ⊗ ˜ eib′

• therwise.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

EXAMPLE

g = osp(1|2) ∆(F(k)) =

r+s=k(πq)−rsF(r)Ks ⊗ F(s)

Looking at V(n) ⊗ V(1), (by convention, p(vλ) = 0) F(k)(vn ⊗ v1) = F(k)vn ⊗ v1 + π1−kqn+1−kF(k−1)vn ⊗ Fv1. Then ˜ f (n+1)(vn ⊗ v1) = πn(˜ f nvn) ⊗ (˜ fv1) mod q.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

REMARK ON THE TENSOR PRODUCT RULE

Signs in the tensor product rule is not a new idea ([BKK]). In earlier work, versions without the sign appear.

◮ In [MZ], the signs are absorbed into factors of

√ −1.

◮ In [Jeong], some signs were forgotten.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

SIGNED ORTHONORMALITY

As usual, there is a polarization bilinear form (−, −) which descends to a bilinear form (−, −)0 on L/qL. When π = −1, B(λ) is a signed orthonomal basis; that is, (b, b)0 = 1 or π. (when π = 1, the basis is honestly orthonormal.) Example: g = osp(1|4) with simple roots α1 (short, odd) and α2 (long, even): x = ˜ f 4

1 ˜

f2 · 1, y = ˜ f 3

1 ˜

f2 ˜ f1 · 1 (x, x) ∈ 1 + q2A, (y, y) ∈ π + q2A, (x, y) ∈ q2A.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

SIGNED ORTHONORMALITY

Proposition.

◮ [Kashiwara] For π = 1,

L(∞) =

• x ∈ U−|(x, x) ∈ A
• .

◮ [CHW2] For π = −1,

L(∞) =

• x ∈ U−|(x, x) ∈ A
• .

Example Continued: Set z = ˜ f 4

1 ˜

f2 · 1 + ˜ f 3

1 ˜

f2 ˜ f1 · 1 ∈ L(∞). Then (z, z) ∈ q2A, so (q−1z, q−1z) ∈ A despite q−1z / ∈ L(∞).

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

GLOBALIZING THE CRYSTAL BASIS

We want to find a map G : L/qL → U−

Z ∩ L ∩ L

such that G(B) is a global (= canonical!) basis for U−. Essential to the argument in [K] is that L is invariant under σ : U− → U− with σ(xy) = σ(y)σ(x), σ(Fi) = Fi. Since (−, −) is σ-invariant, the π = 1 case follows from L =

• x ∈ U−|(x, x) ∈ A
• .

But when π = −1, L =

• x ∈ U−|(x, x) ∈ A
• ,

so how to prove invariance?

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

TWO CURIOUS SOLUTIONS

There are two ways to work around this issue, each interesting in its own way. Categorification: We can realize the crystal in a categorification. Two-Parameter: We can directly connect the crystal bases at π = ±1 by passing through a two-parameter version as developed by [Fan-Li].

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

END OF THE TALE

Theorem (C-Hill-Wang)

The half quantum group U− and its integrable modules admit canonical bases.

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

RELATED PAPERS

[HW] Categorication of quantum Kac-Moody superalgebras, arXiv:1202.2769, to appear in Trans. AMS. [CW] Canonical basis for quantum osp(1|2), arXiv:1204.3940,

• Lett. Math. Phys. 103 (2013), 207–231.

[CHW1] Quantum supergroups I. Foundations, arXiv:1301.1665. [CHW2] Quantum supergroups II. Canonical Basis, arXiv:1304.7837.

Slides available at http://people.virginia.edu/˜sic5ag/

INTRODUCTION BACKGROUND The rank 1 case KM quantum supergroups Crystals

Thank you for your attention!