A canonical basis for covering quantum groups Sean Clark Joint - - PowerPoint PPT Presentation

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

A canonical basis for covering quantum groups

Sean Clark Joint work with D. Hill and W. Wang University of Virginia AMS Fall Western Sectional Meeting University of California, Riverside November 2, 2013

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

QUANTUM GROUPS

q: a generic parameter; g: a Kac-Moody algebra with simple roots Π = {αi : i ∈ I}. Uq(g) is the Q(q) algebra with generators Ei, Fi, K±1

i

for i ∈ I. Uq(n−), the subalgebra generated by Fi. Uq(n−) has many interesting properties, e.g.

◮ Lusztig-Kashiwara canonical basis; ◮ categorifications of Khovanov-Lauda and Rouquier;

Uq(g) admits a categorification for its modified form [L, KL, R].

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

HALF QUANTUM SUPERGROUPS

g: an anisotropic Kac-Moody superalgebra with Z/2Z-graded simple roots Π = Π0 ⊔ Π1 = {αi : i ∈ I} Uq(n−): algebra generated by Fi satisfying super Serre relations. Was not expected to admit a canonical basis. Super KLR= quiver Hecke superalgebras (Ellis-Khovanov-Lauda in rank 1, Kang-Kashiwara-Tsuchioka independently defined the general construction) [Hill-Wang] Uq(n−) is categorified by QHSA’s. ⇒ It has a categorical canonical basis. Is there an intrinsic canonical basis ` a la Lusztig, Kashiwara?

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

INSIGHT FROM [HW]

Anisotropic super and non-super are formally similar Key Insight [HW]: use a parameter π2 = 1 for super signs

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

There is a bar involution on Q(q)π = Q(q, π)/(π2 − 1) given by q → πq−1 (π2 = 1) and quantum integers [n] = (πq)n − q−n πq − q−1 , [n]!, n a

• ∈ Z[q, q−1].

giving Uq(n−) a suitable bar-invariant integral form.

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

ANISOTROPIC KM

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)

◮ (anisotropy) aij ∈ 2Z for i ∈ I1

We call these “of anisotropic type”. We will also impose:

◮ (bar-compatibility) di ≡2 p(i)

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

EXAMPLES

(•=odd root)

• · · ·

<

• (osp(1|2n))
• · · ·

<

• <
• · · ·

<

• >
• · · ·

<

• ✈

✈ ✈ ✈

• ❍

❍ ❍ ❍

• >

<

• <
• <

>

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

KNOWN FACTS ABOUT KM SUPER

Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Musson, Zou, ...) Some key coincidences exist for anisotropic KM:

◮ osp(1|2n) reps “=” half of so(2n + 1) reps

(R.B. Zhang, Lanzmann)

◮ Over C(q), Uq(osp(1|2n)) miraculously has the missing

• reps. (Musson-Zou)

[CW]: Uq(osp(1|2))/Q(q) can be tweaked to get all reps. EF − πFE = K − K−1 πq − q−1

• even h.w.
• r

πK − K−1 πq − q−1

• dd h.w.

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

DEFINITION [CHW1]

Let g be a KM superalgebra of anisotropic type, A its symmetrizable GCM. Let U = 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 MOTIVATION COVERING QUANTUM GROUPS Modified Form

RANK 1

For Uq(osp(1|2))) 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.) We call this covering quantum osp(1|2) or sl(2) (π = 1 case)

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

FINITE TYPE

The only finite type covering algebras have Dynkin diagrams

• · · ·

<

• This diagram corresponds to

◮ the Lie superalgebra osp(1|2n) ◮ the Lie algebra so(1 + 2n)

(NB. There is no ”covering sl(n)” in this construction)

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

STRUCTURES IN A COVERING QUANTUM GROUP

U has all the nice features you could hope for:

◮ U = U− ⊗ U0 ⊗ U+; ◮ U− admits a nondegenerate bilinear form; ◮ there is a Hopf superalgebra structure (super sign → π); ◮ there is a bar involution (K → JK−1); ◮ there is a quasi-R-matrix and Casimir-type operator;

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

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 semi-simple category Oint. Simple modules: V(λ) for all λ ∈ P+ (Same character as in classical case)

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

CRYSTALS

To construct a CB, we use the algebraic approach with crystals. Specifically, we construct a covering analogue for

◮ Kashiwara operators ˜

ei,˜ fi;

◮ the crystal lattice; ◮ the action of the q-Boson algebra; ◮ the polarizations (= deformed Shapovalov forms); ◮ the tensor product rule;

Kashiwara’s grand loop argument can be extended to the covering case. Moreover, this crystal basis admits globalization to a canonical basis.

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

CANONICAL BASIS

Theorem (C-Hill-Wang)

U− and the integrable modules admit compatible canonical bases. Let B be the canonical basis of U−.

◮ If vλ is the highest weight vector of V(λ),

bvλ = 0 or is a CB element.

◮ B|π=1 = the Lusztig-Kashiwara CB ◮ B is typically π-signed: b ∈ B implies πb ∈ B.

Example: aij = 0, p(i) = p(j) = 1 FiFj = πFjFi (Categorically: M is not isomorphic to its parity shift ΠM.)

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

MODIFIED FORM

Basic idea: 1

λ∈P 1λ with 1λ1η = δλ,η1λ

For x ∈ U, let |x| be the weight. ˙ U is the algebra on symbols x1λ = 1λ+|x|x for x ∈ U, λ ∈ P satisfying (xy)1λ = x1λ+|y|y1λ, JµKν1λ = πµ,λqν,λ1λ Any weight U-module M is a ˙ U module: x1λ acts as projection to Mλ followed by the U-action of x.

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

SOME PROPERTIES

˙ U has some additional useful properties:

◮ Automorphisms of U extend to ˙

U;

◮ ˙

U1λ

v.s.

= U− ⊗ U+;

Theorem (C.)

There is a non-degenerate symmetric bilinear form on ˙ U which:

◮ extends the form on U−; ◮ is invariant under our favorite maps; ◮ is a limit of polarizations;

For π = 1, this is Lusztig’s form on ˙ U.

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

RANK 1

˙ Uq(osp(1|2)) is the algebra given by Generators: E1n = 1n+2E, F1n = 1n−2F, 1n Relations: 1n1m = δnm1n and EF1n − FE1n = [n]1n

Theorem (C-Wang)

˙ Uq(osp(1|2)) admits a canonical basis ˙ B =

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

(In rank 1, the basis need not be π-signed) Ellis and Lauda have categorified ˙ Uq(osp(1|2)).

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

CONSTRUCTING THE CB

◮ ˙

U1λ−λ′ projects “nicely” onto N(λ, λ′) (highest weight ⊗ lowest weight);

◮ N(λ, λ′) has a bar involution (Lusztig quasi-R-matrix); ◮ N(λ, λ′) admits a CB (bar involution + CB on simples); ◮ The CB of N(λ, λ′) is compatible with N(λ + λ′′, λ′′ + λ′);

These facts allow us to build a basis for ˙ U.

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

CANONICAL BASIS

Theorem (C)

˙ U admits a π-signed canonical basis generalizing the basis for U−. This basis is π-almost orthonormal under the bilinear form. For π = 1, this specializes to Lusztig’s canonical basis for ˙ U|π=1.

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

SOME 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, to appear in Trans. Groups. [CHW2] Quantum supergroups II. Canonical Basis, arXiv:1304.7837. [C] Quantum supergroups IV. Modified form, forthcoming.

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

INTRODUCTION MOTIVATION COVERING QUANTUM GROUPS Modified Form

Thank you for your attention!