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Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

CANONICAL BASES AND QUANTUM SHUFFLE

SUPERALGEBRAS OF BASIC TYPE

Sean Clark University of Virginia (joint with David Hill and Weiqiang Wang) arXiv:1310.7523 12/20/13

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

QUANTUM GROUPS AND CANONICAL BASES

Let g = n− ⊕ h ⊕ n be a simple Lie algebra. [Lusztig, Kashiwara]: Uq(n) admits a canonical basis B; that is,

◮ B is invariant under q → ¯

q = q−1;

◮ B equals (any choice of) a PBW basis mod q; ◮ B is orthonormal (mod q) with respect to a bilinear form.

This basis holds a remarkable amount of information about

◮ geometry; ◮ representation theory; ◮ categorification.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

LIE SUPERALGEBRAS

Question: do quantized Lie superalgebras admit canonical bases? (e.g. gl(m|n), osp(m|2n), Kac-Moody superalgebras) No adequate setting for geometric construction à la Lusztig

Theorem (C-Hill-Wang)

The half-quantum supergroup Uq(n) associated to a Kac-Moody superalgebra with no isotropic roots admits a canonical basis. Example: osp(1|2n) (finite type). Sketch:

◮ Set a parameter π2 = 1 and a bar involution ¯

q = πq−1

◮ Use π to interpolate between super and non-super ◮ Use Kashiwara’s algebraic (crystal) approach to obtain CB

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

BASIC TYPE

An important class are Lie superalgebras of basic type (gl, osp, simple Lie algebras)

◮ Isotropic simple roots ⇒ no root strings

◮ In particular, Kashiwara’s algebraic strategy fails

◮ [Benkart-Kang-Kashiwara, Kwon] Many interesting

gl(m|n)-modules admit crystal bases.

◮ [Khovanov] gl(1|2) admits a categorification

Conclusion: We will have to try some new methods

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

A QUANTUM SHUFFLE APPROACH

Non-super Uq(n) has been studied using quantum shuffles [Ram-Lalonde, Rosso, Green, Leclerc, . . . ] Algebra structure combinatorics of words.

◮ Order on simple roots induces lexicographic order. ◮ Certain words (dominant Lyndon) ↔ positive roots. ◮ Get distinguished bases from word combinatorics.

Fact: [Leclerc] Canonical bases can be constructed using quantum shuffles

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

SUPER CANONICAL BASES

Goal: Use quantum shuffles to construct PBW/canonical bases. For this, we need:

◮ a quantum shuffle presentation of the quantum group; ◮ to study super word combinatorics; ◮ a PBW basis for any order on roots; ◮ a suitable integral form;

This is a nontrivial generalization:

◮ Super shuffles lack positivity. ◮ Super word combinatorics are less well behaved. ◮ Leclerc’s construction is not self-contained (need Lusztig’s

PBW).

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

LIE SUPERALGEBRAS

Let g = g¯

0 ⊕ g¯ 1 be a simple Lie superalgebra of basic type (A − G). ◮ [x, y] = xy − (−1)p(x)p(y)yx.

Fix a triangular decomposition g = n− ⊕ h ⊕ n+:

◮ Root system:

Φ = Φ¯

0 ⊔

Φ¯

1 = {α ∈ h∗ | gα = 0};

◮

Φ¯

1 =

Φiso ⊔ Φn−iso;

◮ Simple roots: Π = Π¯ 0 ⊔ Π¯ 1 = {αi | i ∈ I}, ◮ Dynkin diagram: (Γ, I),

◮ I = I¯

0 ⊔ I¯ 1, I¯ 1 = Iiso ⊔ In−iso.

Unlike Lie algebras, (Γ, I) depends on h, and Φ may be unreduced (e.g. type BC). Reduced root system: Φ = {α ∈ Φ | 1

2α /

∈ Φ}

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

(Types A-D) gl(m|n)

• · · ·

⊗ ⊗ · · ·

• ⊙

⊙ · · · ⊙ ⊙ ⊙ ⊙ · · · ⊙ ⊙

• sp(2m + 1|2n)

⊙ ⊙ · · · ⊙ ⊙ ⊙ ⊙ · · · ⊙

• >

⊙ ⊙ · · · ⊙ ⊙ ⊙

• <
• sp(2n|2m)

⊙ ⊙ · · · ⊙ ⊙ ⊙ ⊙ · · · ⊙

• ✈

✈ ✈

⊙

• ❍

❍ ❍

⊙ ⊙ · · · ⊙ ⊙ ⊙ ⊙ · · · ⊙ ⊗

✈ ✈ ✈

⊙ ⊗

❍ ❍ ❍

⊗ ⊗

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

THE HALF-QUANTUM SUPERGROUP

Let Uq = Uq(n+) = Q(q)ei | i ∈ I;

◮ This is a bialgebra under the multiplication:

(u ⊗ v)(x ⊗ y) = (−1)p(v)p(x)q−(|v|,|x|)(ux ⊗ vy), where |v|, |x| ∈ Q+ = ⊕i∈IZ+αi;

◮ [Yamane] Subject to (exotic) Serre-type relations determined by

subdiagrams of Γ. Example: ⊗

• 1

2 3 e1e2e3e2 + e3e2e1e2 + e2e1e2e3 + e2e3e2e1 = (q + q−1)e2e1e3e2

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

BILINEAR FORM

F = Q(q)I, the free algebra on I, i = (i1i2 . . . id) = i1 · i2 · · · id, |i| = αi1 + . . . + αid There is a canonical surjection F − → Uq, i → ei. Facts: [Lusztig, Yamane]

◮ Uq ∼

= F/Rad(·, ·);

◮ Uq is equipped with a nondegenerate bilinear form (·, ·)

satisfying

• 1. (ei, ej) = δij;
• 2. (xy, z) = (x ⊗ y, ∆(z));

◮ the bilinear form on Uq ⊗ Uq can be given by

(u ⊗ v, x ⊗ y) = (u, x)(v, y).

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

QUANTUM SHUFFLE EMBEDDING

Coproduct on F induces a product on F∗. Dualizing F ։ Uq induces an injective homomorphism Ψ : Uq ∼ = U∗

q ֒

→ F∗ ∼ = (F, ⋄ ) where ⋄ is the quantum shuffle product: (i·i) ⋄ (j·j) = (i ⋄ (j·j))·i+(−1)(p(i)+p(i))p(j)q−(|i|+αi,αj)((i·i) ⋄ j)·j. (quantum shuffles approach is dual to Lusztig’s bialgebra approach) We shall study Uq through its image U = Ψ(Uq) ⊂ F.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

EXAMPLE:

gl(3|2)

• ⊗
• 1

2 3 4 I¯

0 = {1, 2, 4},

I¯

1 = Iiso = {3},

A =     2 − 1 −1 2 −1 −1 1 1 −2     Ψ(e1e2) = 1 ⋄ 2 = (21) + q(12) Ψ(e3e3) = 3 ⋄ 3 = (33) + (−q0)(33) = 0 4 ⋄ (13) = (134) + q−1(143) + q−1(413) (In fact, (13) ∈ U: Ψ(e1e3) = (13) + (31))

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

TOWARDS PBW BASES

Theorem: [Yamane] Any basic type Lie superalgebra admits a PBW basis for a particular (standard) ordering of the roots. But we need PBW bases associated to an arbitary ordering of the simple roots. Idea: Order on simple roots induces a lexicographic order on words. We can then use the combinatorics of the word basis of F.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

COMBINATORICS OF WORDS

F has a word basis W = ⊔n≥0In ⊂ F, so we can learn things about elements of U by writing them in this basis, e.g. Ψ(eiej) = i ⋄ j = (ji) + (−1)p(i)p(j)q−(αi,αj)(ij).

◮ Fix the lexicographic order on W relative to some ordering

(I, ≤).

◮ Let W+ be the set of dominant words, i.e words of the form

i = max(u) for some u ∈ U. e.g. if i < j, then max(i ⋄ j) = (ji).

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

MONOMIAL BASIS

Proposition: Let i = (i1, . . . , id) ∈ W and εi = Ψ(ei1 . . . eid). Then {εi = i1 ⋄ · · · ⋄ id | i = (i1, . . . , id) ∈ W+} is a basis of U. We want to refine this basis. Let L denote the set of Lyndon words: i = (i1, . . . , id) ∈ L ⇔ i < (ik, . . . , id) for k > 1. Let L+ = L ∩ W+.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

DOMINANT LYNDON WORDS AND POSITIVE ROOTS

Theorem: [C-Hill-Wang]

• 1. The map

i = (i1, . . . , id) → αi1 + · · · + αid = |i| is a bijection L+ − → Φ+;

• 2. Every i ∈ W+ has a canonical factorization i = i1 · · · in, where

i1, . . . , in ∈ L+, i1 ≥ · · · ≥ in and ik > ik+1 if |ik| ∈ Φ+

iso.

Sketch:

◮ Induct on height of roots ◮ Yamane’s PBW theorem gives dimensions of root spaces ◮ If no bijection, then ii ∈ W+ with |i| ∈ Φiso ◮ This yields a contradition.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

EXAMPLE:

gl(2|1)

• ⊗

1 2 L+ = {(1), (12), (2)} W+ = {(2)a(12)b(1)c : c ∈ N, a, b ∈ {0, 1}} Words which are dominant:

◮ (2121) ◮ (21111) ◮ (121) ◮ (1111)

Words which are not dominant:

◮ (112) ◮ (22111) ◮ (1212) ◮ (1121)

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

ROOT VECTORS

To construct PBW, we need root vectors q-commutators. Need some prescribed order to inductively build them. Define the co-standard factorization of i ∈ L+ to be i = i1i2, where i1 ∈ L+ is of maximal length (i2 ∈ L+, too!).

Example

◮ (1234) = (123)(4); ◮ (12332) = (1233)(2); ◮ (1231234) = (123)(1234).

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

Definition: For i ∈ I, set Ei = i. Otherwise, define Ei, i ∈ L+, inductively by Ei = κ−1

i

[Ei2, Ei1]q−1 = κ−1

i

• Ei2⋄Ei1 − (−1)p(i1)p(i2)q−(|i1|,|i2|)Ei1⋄Ei2
• where

◮ i = i1i2 is the co-standard factorization; ◮ κi is an explicit quantum integer defined in terms of Φ+.

For i ∈ W+ with canonical factorization i = ik1

1 · · · ikn n ,

it ∈ L+ s.t. i1 > · · · > in, we set Ei = E(kn)

in

⋄ · · · ⋄ E(k2)

i2

⋄ E(k1)

i1

.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

EXAMPLE

gl(3|2)

• ⊗
• 1

2 3 4 L+ = {(i, . . . , j) : 1 ≤ i ≤ j ≤ 4} E(34) = (4) ⋄ (3) − q−1(3) ⋄ (4) = (34) + q−1(43) − q−1(43) − q−2(34) = (1 − q−2)(34) E(23) = (1 − q2)(23) E(234) = (1 − q−2) ((34) ⋄ (2) − q(2) ⋄ (34)) = (1 − q−2)

• (234) + q(324) + q(342) − q(342) − q(324) − q2(234)
• = −(q − q−1)2(234)

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

Theorem: [C-Hill-Wang]

• 1. The set PBW = {Ei | i ∈ W+} is a basis of U:

◮ Ei = ετ(i) +

j>i aijετ(j);

• 2. for i ∈ W+, max(Ei) = i;
• 3. for i ∈ L+,

∆(Ei) = Ei ⊗ 1 +    

• |j|+|k|=|i|

j<i<k

ϑi

j,k Ek ⊗ Ej

    + 1 ⊗ Ei;

• 4. PBW is orthogonal: (Ei, Ej) = 0 if i = j.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

INTEGRAL FORM

If we want to construct a canonical basis, we need a bar-invariant integral form. This integral form should be spanned by the PBW bases. The classical choice is to take the lattice of divided powers. This is too naive and won’t work in general. (Silly example: gl(n|n) with all simples isotropic) However, it does work for certain standard root systems.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

RESTRICTED SIMPLE SYSTEMS

From now on, we shall only consider the following systems: gl(m, n)

• · · ·

⊗ ⊗ · · ·

• sp(1, 2n)
• · · ·
• >
• sp(2|2n)

⊗ · · ·

• <

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

INTEGRAL FORM

Theorem: [C-Hill-Wang] For these types, PBW = {Ei | i ∈ W+} is a basis for UA = Ae(n)

i

| i ∈ I, n ≥ 0, (A = Z[q, q−1]). Remark: Even for standard systems, there are examples where this fails.

• sp(3|2)

⊗

• 1

2 >

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

PSEUDO-CANONICAL BASES

Lemma: [C-Hill-Wang] For i ∈ W+, write Ei =

• j∈W+

aijEj, for aij ∈ Q(q). Then, aii = 1 for all i ∈ W+ and aij = 0 if i > j. (This holds for arbitrary type.) For our allowed diagrams, UA = UA so all aij ∈ A. Therefore, by standard argument there exists a unique bar-invariant basis of the form bi = Ei +

• j>i

θijEj, θij ∈ qZ[q] Call such a basis a pseudo-canonical basis.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

CANONICAL BASES

A priori, this pseudo-canonical basis depends on the PBW basis, hence on an ordering. A pseudo-canonical basis will be called a canonical basis if it is almost orthogonal in the sense that, for all i, j ∈ W+,

• 1. (bi, bj) ∈ Z[q], and
• 2. (bi, bj) = ±δij (mod q) for some θ ∈ {0, 1}.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

Theorem:[C-Hill-Wang] For U corresponding to the standard system of type gl(m|n),

• sp(1|2n), or osp(1|2n)

◮ U has a pseudo-canonical basis. ◮ The pseudo-canonical basis is canonical except for gl(m|n) with

both m, n > 1 Remark: Our approach is entirely self contained!

◮ Though we have focused on super, this works for non-super. ◮ This is a new self-contained construction of non-super CBs.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

WHAT ABOUT OTHER TYPES?

Arbitrary root system: is there a good integral form?

◮ Do PBW lattices coincide? ◮ Are these lattices bar-invariant?

gl(m|n) for m, n > 1: We have a pseudo-canonical basis.

◮ Basis isn’t compatible with Uq(gl(n) ⊕ gl(m)) ◮ Unlikely to behave well with representations

Crucial case to understand is gl(2|2).

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

WHAT ABOUT REPRESENTATIONS?

Question: Does the canonical basis descend to a basis on simple modules? No choice but to compute the image of basis elements. We will do this for the following system: gl(2|1)

• ⊗

1 2

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

CANONICAL BASIS OF gl(2|1)

Let us work with Uq(n−) instead of Uq(n). Using our construction, we compute the canonical basis B = {F(r)

1 , F(r) 1 F2, F2F(r+1) 1

, F2F(r+1)

1

F2} (This is equal to Khovanov’s categorical CB for gl(2|1)) Set

◮ λ = aǫ1 + bǫ2 + cǫ3: weight for gl(2|1) with b − a ∈ Z≥0. ◮ K(λ): the Kac module (maximal finite-dim. parabolic Verma). ◮ L(λ): the unique simple quotient of K(λ). ◮ vK λ, vL λ: some highest weight vectors of K(λ) and L(λ).

How does B act on K(λ) and L(λ)?

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

DESCENT OF CANONICAL BASIS

For b ∈ M, let B(m) = {bm = 0 | b ∈ B}. Proposition:

• 1. B(vK

λ) is a basis of K(λ).

• 2. B(vL

λ) is a basis of L(λ) if and only if a = −1 − c.

• 3. When a = −1 − c, the spans of F(r)

1 F2vL λ and F2F(r) 1 vL λ coincide

and are nonzero for 1 ≤ r ≤ a − b + 1. In particular, B(vL

λ) is a basis for any “polynomial” weight.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

A CONJECTURE

Now consider gl(n|1) and extend our earlier notations. Conjecture:

• 1. B(vK

λ) is a basis of K(λ) for all weights λ.

• 2. B(vL

λ) is a basis of L(λ) for all polynomial weights λ.

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

SOME QUESTIONS

K(λ) and L(λ) have crystal bases [Benkart-Kang-Kashiwara, Kwon]. Question: How are the canonical and crystal bases related? Existence of canonical bases suggests a connection to categorification. Question: Can gl(n|1) or osp(2|2n) be categorified?

Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory

Thank you for your attention!

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