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U q ( gl ( m | 1 )) and canonical bases Sean Clark Northeastern University / Max Planck Institute for Mathematics Algebraic Groups, Quantum Groups and Geometry Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 1 / 12 Q UANTUM

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SLIDE 1

Uq(gl(m|1)) and canonical bases

Sean Clark Northeastern University / Max Planck Institute for Mathematics Algebraic Groups, Quantum Groups and Geometry

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 1 / 12

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QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM ALGEBRAS AND CANONICAL BASES

Uq(g): algebra over Q(q) coming from root data of simple Lie algebra g.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 2 / 12

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QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM ALGEBRAS AND CANONICAL BASES

Uq(g): algebra over Q(q) coming from root data of simple Lie algebra g. (∼1990) Lusztig and Kashiwara: miraculous bases for U−

q (g) = Uq(n−)

(CB1) B is a Z[q, q−1]-basis of the integral form AU−

q (g);

(CB2) For any b ∈ B, b = b (· is natural involution q → q−1); (CB3) PBW → B is qZ[q]-unitriangular for any PBW (CB4) B induces a basis on simple modules

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 2 / 12

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QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM SUPERALGEBRAS

Question: what if g is a Lie superalgebra?

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 3 / 12

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SLIDE 5

QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM SUPERALGEBRAS

Question: what if g is a Lie superalgebra? Few examples of CBs known:

◮ osp(1|2n) and anisotropic Kac-Moody super [C-Hill-Wang, ’13] ◮ Partial results for gl(m|n), osp(2|2n) [C-Hill-Wang, ’13; Du-Gu ’14]

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 3 / 12

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SLIDE 6

QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM SUPERALGEBRAS

Question: what if g is a Lie superalgebra? Few examples of CBs known:

◮ osp(1|2n) and anisotropic Kac-Moody super [C-Hill-Wang, ’13] ◮ Partial results for gl(m|n), osp(2|2n) [C-Hill-Wang, ’13; Du-Gu ’14]

Why partial? Problems:

◮ Different simple roots= Different U− (in general)

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 3 / 12

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SLIDE 7

QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM SUPERALGEBRAS

Question: what if g is a Lie superalgebra? Few examples of CBs known:

◮ osp(1|2n) and anisotropic Kac-Moody super [C-Hill-Wang, ’13] ◮ Partial results for gl(m|n), osp(2|2n) [C-Hill-Wang, ’13; Du-Gu ’14]

Why partial? Problems:

◮ Different simple roots= Different U− (in general)

Workaround: in standard case, PBW basis → B = (CB1), (CB2)

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 3 / 12

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SLIDE 8

QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM SUPERALGEBRAS

Question: what if g is a Lie superalgebra? Few examples of CBs known:

◮ osp(1|2n) and anisotropic Kac-Moody super [C-Hill-Wang, ’13] ◮ Partial results for gl(m|n), osp(2|2n) [C-Hill-Wang, ’13; Du-Gu ’14]

Why partial? Problems:

◮ Different simple roots= Different U− (in general)

Workaround: in standard case, PBW basis → B = (CB1), (CB2)

◮ if m > 1 and n > 1, this basis is not canonical!

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 3 / 12

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SLIDE 9

QUANTUM ENVELOPING gl(m|1) Canonical bases

QUANTUM SUPERALGEBRAS

Question: what if g is a Lie superalgebra? Few examples of CBs known:

◮ osp(1|2n) and anisotropic Kac-Moody super [C-Hill-Wang, ’13] ◮ Partial results for gl(m|n), osp(2|2n) [C-Hill-Wang, ’13; Du-Gu ’14]

Why partial? Problems:

◮ Different simple roots= Different U− (in general)

Workaround: in standard case, PBW basis → B = (CB1), (CB2)

◮ if m > 1 and n > 1, this basis is not canonical!

Workaround: when m = 1 or n = 1: canonical signed basis.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 3 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

ROOT DATA FOR gl(m|1)

P =  m

i=1 Z ǫi

  • even

  ⊕ Z ǫm+1

  • dd

with (ǫi, ǫi) = (−1)p(ǫi), P∨, ·, · as usual. Φ =

  • ǫi − ǫj | 1 ≤ i = j ≤ m + 1
  • with simple roots Π = {αi | i ∈ I}

Standard choice: Πstd = {ǫ1 − ǫ2, . . . , ǫm − ǫm+1}

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 4 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

ROOT DATA FOR gl(m|1)

P =  m

i=1 Z ǫi

  • even

  ⊕ Z ǫm+1

  • dd

with (ǫi, ǫi) = (−1)p(ǫi), P∨, ·, · as usual. Φ =

  • ǫi − ǫj | 1 ≤ i = j ≤ m + 1
  • with simple roots Π = {αi | i ∈ I}

Standard choice: Πstd = {ǫ1 − ǫ2, . . . , ǫm − ǫm+1} Note:

◮ odd roots are isotropic; e.g. (ǫm − ǫm+1, ǫm − ǫm+1) = 1 + −1 = 0;

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 4 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

ROOT DATA FOR gl(m|1)

P =  m

i=1 Z ǫi

  • even

  ⊕ Z ǫm+1

  • dd

with (ǫi, ǫi) = (−1)p(ǫi), P∨, ·, · as usual. Φ =

  • ǫi − ǫj | 1 ≤ i = j ≤ m + 1
  • with simple roots Π = {αi | i ∈ I}

Standard choice: Πstd = {ǫ1 − ǫ2, . . . , ǫm − ǫm+1} Note:

◮ odd roots are isotropic; e.g. (ǫm − ǫm+1, ǫm − ǫm+1) = 1 + −1 = 0; ◮ no odd simple reflection! (Still obvious Sm+1 action: Weyl groupoid)

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 4 / 12

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SLIDE 13

QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

ROOT DATA FOR gl(m|1)

P =  m

i=1 Z ǫi

  • even

  ⊕ Z ǫm+1

  • dd

with (ǫi, ǫi) = (−1)p(ǫi), P∨, ·, · as usual. Φ =

  • ǫi − ǫj | 1 ≤ i = j ≤ m + 1
  • with simple roots Π = {αi | i ∈ I}

Standard choice: Πstd = {ǫ1 − ǫ2, . . . , ǫm − ǫm+1} Note:

◮ odd roots are isotropic; e.g. (ǫm − ǫm+1, ǫm − ǫm+1) = 1 + −1 = 0; ◮ no odd simple reflection! (Still obvious Sm+1 action: Weyl groupoid) ◮ different choices of simple roots may have different GCMs!

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 4 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

ROOT DATA FOR gl(m|1)

P =  m

i=1 Z ǫi

  • even

  ⊕ Z ǫm+1

  • dd

with (ǫi, ǫi) = (−1)p(ǫi), P∨, ·, · as usual. Φ =

  • ǫi − ǫj | 1 ≤ i = j ≤ m + 1
  • with simple roots Π = {αi | i ∈ I}

Standard choice: Πstd = {ǫ1 − ǫ2, . . . , ǫm − ǫm+1} Note:

◮ odd roots are isotropic; e.g. (ǫm − ǫm+1, ǫm − ǫm+1) = 1 + −1 = 0; ◮ no odd simple reflection! (Still obvious Sm+1 action: Weyl groupoid) ◮ different choices of simple roots may have different GCMs!

GCMs for m = 3:   2 −1 −1 2 −1 −1  

  • standard choice

,   2 −1 −1 1 1  

  • Π={ǫ1−ǫ2,ǫ2−ǫ4,ǫ4−ǫ3}

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 4 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

Uq(gl(m|1))

Fix a choice of Π. We define U = Uq(Π) = Q(q)

  • Ei, Fi, qh | i ∈ I, h ∈ P∨

subject to usual relations: e.g. qhEiq−h = qh,αiEi, [Ei, Fj]

super commutator

= EiFj − (−1)p(i)p(j)FjEi = δij qhi − q−hi q − q−1

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 5 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

Uq(gl(m|1))

Fix a choice of Π. We define U = Uq(Π) = Q(q)

  • Ei, Fi, qh | i ∈ I, h ∈ P∨

subject to usual relations: e.g. qhEiq−h = qh,αiEi, [Ei, Fj]

super commutator

= EiFj − (−1)p(i)p(j)FjEi = δij qhi − q−hi q − q−1 and both usual and unusual Serre relations: if p(i) = 1, E2

i = F2 i = 0,

[Ei−1, [Ei, [Ei+1, Ei]q]q]q

  • super q−commutators

= 0

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 5 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

Uq(gl(m|1))

Fix a choice of Π. We define U = Uq(Π) = Q(q)

  • Ei, Fi, qh | i ∈ I, h ∈ P∨

subject to usual relations: e.g. qhEiq−h = qh,αiEi, [Ei, Fj]

super commutator

= EiFj − (−1)p(i)p(j)FjEi = δij qhi − q−hi q − q−1 and both usual and unusual Serre relations: if p(i) = 1, E2

i = F2 i = 0,

[Ei−1, [Ei, [Ei+1, Ei]q]q]q

  • super q−commutators

= 0 This has standard structural features (integral form, triangular decomposition, bar-involution, . . . ) NOTE: Different Π yield different U− = Q(q) Fi | i ∈ I in general!

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 5 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

QUANTUM SUPERALGEBRAS

Theorem (C)

U−(Πstd) has a canonical basis B; that is, a basis satisfying (CB1)-(CB3), and (CB4) for most (but not all∗) finite-dimensional simple modules. Main strokes of proof:

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 6 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

QUANTUM SUPERALGEBRAS

Theorem (C)

U−(Πstd) has a canonical basis B; that is, a basis satisfying (CB1)-(CB3), and (CB4) for most (but not all∗) finite-dimensional simple modules. Main strokes of proof:

  • 1. Construct crystal on U− using Benkart-Kang-Kashiwara;

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 6 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

QUANTUM SUPERALGEBRAS

Theorem (C)

U−(Πstd) has a canonical basis B; that is, a basis satisfying (CB1)-(CB3), and (CB4) for most (but not all∗) finite-dimensional simple modules. Main strokes of proof:

  • 1. Construct crystal on U− using Benkart-Kang-Kashiwara;
  • 2. Globalize using pseudo-canonical basis

⇒ get canonical B satisfying (CB1), (CB2), and (CB4) in many cases

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 6 / 12

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QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

QUANTUM SUPERALGEBRAS

Theorem (C)

U−(Πstd) has a canonical basis B; that is, a basis satisfying (CB1)-(CB3), and (CB4) for most (but not all∗) finite-dimensional simple modules. Main strokes of proof:

  • 1. Construct crystal on U− using Benkart-Kang-Kashiwara;
  • 2. Globalize using pseudo-canonical basis

⇒ get canonical B satisfying (CB1), (CB2), and (CB4) in many cases

  • 3. Prove (CB3) using braid group action.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 6 / 12

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SLIDE 22

QUANTUM ENVELOPING gl(m|1) Quantum gl(m|1)

QUANTUM SUPERALGEBRAS

Theorem (C)

U−(Πstd) has a canonical basis B; that is, a basis satisfying (CB1)-(CB3), and (CB4) for most (but not all∗) finite-dimensional simple modules. Main strokes of proof:

  • 1. Construct crystal on U− using Benkart-Kang-Kashiwara;
  • 2. Globalize using pseudo-canonical basis

⇒ get canonical B satisfying (CB1), (CB2), and (CB4) in many cases

  • 3. Prove (CB3) using braid group action.

Today, focus on 3.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 6 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PERSPECTIVE ON BRAIDS: NON-SUPER

Usual definition of braid action: automorphism Ti : Uq(g) → Uq(g) “lifting si action on weights”, e.g. for Uq(sl(3)) T2(F1) = F2F1 − qF1F2 ∈ U−

q (sl(3))s2·α1

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 7 / 12

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SLIDE 24

QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PERSPECTIVE ON BRAIDS: NON-SUPER

Usual definition of braid action: automorphism Ti : Uq(g) → Uq(g) “lifting si action on weights”, e.g. for Uq(sl(3)) T2(F1) = F2F1 − qF1F2 ∈ U−

q (sl(3))s2·α1

Interpretation: Ti is weight-preserving translation between choices of simples Conjugacy of Borels ↔ presentation is “unique”; e.g.      Fǫ1−ǫ2, Fǫ2−ǫ3, . . . Π = {ǫ1 − ǫ2, ǫ2 − ǫ3} Fǫ1−ǫ3, Fǫ3−ǫ2, . . . Π = {ǫ1 − ǫ3, ǫ3 − ǫ2} F1, F2, . . . Π = {α1, α2} all essentially the same Braid operator is automorphism sending e.g. T2(F1) = F2F1 − qF1F2 ⇒ T2(Fǫ1−ǫ2) = Fǫ3−ǫ2Fǫ1−ǫ3 − qFǫ1−ǫ3Fǫ3−ǫ2

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 7 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PERSPECTIVE ON BRAIDS: SUPER

Borels are non-conjugate in general → can’t just identify different presentations! Example: for standard Uq(gl(2|1)), F2

1 = 0 yet

“T2(F1)”2 = (F2F1 − qF1F2)2 = 0.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 8 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PERSPECTIVE ON BRAIDS: SUPER

Borels are non-conjugate in general → can’t just identify different presentations! Example: for standard Uq(gl(2|1)), F2

1 = 0 yet

“T2(F1)”2 = (F2F1 − qF1F2)2 = 0. Solution: really, T2 : U(s2 · Πstd) → U(Πstd) with T2(Fǫ1−ǫ3) = Fǫ2−ǫ3

F2

Fǫ1−ǫ2

F1

−qFǫ1−ǫ2Fǫ2−ǫ3 where Fǫ1−ǫ3 = F1 ∈ U(s2 · Πstd); note F2

ǫ1−ǫ3 = 0.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 8 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

BRAID ISOMORPHISMS

Theorem (C)

If Π and Π′ are related via the “reflection” si, then there is an isomorphism Ti : Uq(Π) → Uq(Π′). These maps satisfy braid relations:

Uq(Π) Uq(Πi) Uq(Πj) Uq(Πij)

  • Ti

Tj Tj Ti

Uq(Π) Uq(Πi) Uq(Πj) Uq(Πij) Uq(Πji) Uq(Π′)

  • Ti

Tj Tj Ti Ti Tj Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 9 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PBW

w0 = si1 . . . sit: reduced expression for the longest element of Sm+1 → root vectors and a PBW basis.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 10 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PBW

w0 = si1 . . . sit: reduced expression for the longest element of Sm+1 → root vectors and a PBW basis. To show the Z[q]-span of the bases are the same:

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 10 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PBW

w0 = si1 . . . sit: reduced expression for the longest element of Sm+1 → root vectors and a PBW basis. To show the Z[q]-span of the bases are the same:

◮ Reduce to braid relations, hence rank 2 computations;

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 10 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PBW

w0 = si1 . . . sit: reduced expression for the longest element of Sm+1 → root vectors and a PBW basis. To show the Z[q]-span of the bases are the same:

◮ Reduce to braid relations, hence rank 2 computations; ◮ New phenomena: three rank 2 cases:

◮

2 −1 −1 2

  • : sl(3)-case, which is known.

◮

2 −1 −1

  • : standard gl(2|1)-case, which is easy.

◮

1 1

  • : non-standard gl(2|1)-case, which is false, but doesn’t occur.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 10 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

PBW

w0 = si1 . . . sit: reduced expression for the longest element of Sm+1 → root vectors and a PBW basis. To show the Z[q]-span of the bases are the same:

◮ Reduce to braid relations, hence rank 2 computations; ◮ New phenomena: three rank 2 cases:

◮

2 −1 −1 2

  • : sl(3)-case, which is known.

◮

2 −1 −1

  • : standard gl(2|1)-case, which is easy.

◮

1 1

  • : non-standard gl(2|1)-case, which is false, but doesn’t occur.

Theorem (C)

B ≡q PBW(i1,...,it)

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 10 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

FURTHER CASES

◮ The other simple modules

◮ V(λ): highest wt. simple mod with hwv vλ ◮ {bvλ | b ∈ B, bvλ = 0} is not always a basis :-( ◮ Conjecture (supported by some low rank examples): CB is

B(λ) = {bvλ | bvλ / ∈ qABvλ}

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 11 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

FURTHER CASES

◮ The other simple modules

◮ V(λ): highest wt. simple mod with hwv vλ ◮ {bvλ | b ∈ B, bvλ = 0} is not always a basis :-( ◮ Conjecture (supported by some low rank examples): CB is

B(λ) = {bvλ | bvλ / ∈ qABvλ}

◮ CB for non-standard half-quantum groups:

◮ Easy CB for rank 2 case (though module compatibility is worse) ◮ Can get signed CB for hqg associated to the Dynkin

X . . . X

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 11 / 12

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SLIDE 35

QUANTUM ENVELOPING gl(m|1) Proving (CB3)

FURTHER CASES

◮ The other simple modules

◮ V(λ): highest wt. simple mod with hwv vλ ◮ {bvλ | b ∈ B, bvλ = 0} is not always a basis :-( ◮ Conjecture (supported by some low rank examples): CB is

B(λ) = {bvλ | bvλ / ∈ qABvλ}

◮ CB for non-standard half-quantum groups:

◮ Easy CB for rank 2 case (though module compatibility is worse) ◮ Can get signed CB for hqg associated to the Dynkin

X . . . X

◮ CB for gl(2|2) and beyond

◮ “chirality” is key difficulty; Uq(gl(m|n))0 ∼

= Uq(gl(m)) ⊗ Uq−1(gl(n))

◮ seems (CB3) must be modified! Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 11 / 12

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QUANTUM ENVELOPING gl(m|1) Proving (CB3)

THANKS FOR YOUR ATTENTION!

Some references:

◮ S. C., Canonical bases for the quantum enveloping algebra of gl(m|1) and its modules,

arXiv:1605.04266

◮ S. C., D. Hill and W. Wang, Quantum shuffles and quantum supergroups of basic type,

  • Quant. Top. (to appear), arXiv:1310.7523.

◮ J. Du and H. Gu, Canonical bases for the quantum supergroups U(glm|n), Math. Zeit.

281, 631-660

◮ G. Benkart, S.-J. Kang and M. Kashiwara, Crystal bases for the quantum superalgebra

Uq(gl(m, n)), Journal of Amer. Math. Soc. 13 (2000), 295–331.

◮ M. Khovanov, How to categorify one-half of quantum gl(1|2), arXiv:1007.3517.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 12 / 12

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SLIDE 37

Extra Slides

WHY NOT CANONICAL?

The standard GCM for gl(2|2) is   2 −1 −1 1 1 −2  . Let U− be the half-quantum group associated to this GCM. The α2 + α3 root vectors are either: F23 = F3F2 − q−1F2F3 (if α2 < α3); F32 = F2F3 − q−1F3F2 (if α3 < α2). So the respective bar-invariant basis elements of weight α2 + α3 are Bα2<α3 =

  • F2F3, F3F2 − (q + q−1)F2F3 = F23 − qF2F3
  • Bα3<α2 =
  • F3F2, F2F3 − (q + q−1)F3F2 = F32 − qF3F2
  • Return

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 1 / 8

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SLIDE 38

Extra Slides

EXAMPLES FOR m = 2

Let A = 2 −1 −1

  • ,

B = 1 1

  • ,

both GCMs for gl(2|1). U−(A) has generators F1, F2 subject to the relation F2

1F2 + F2F2 1 = (q + q−1)F1F2F1;

U−(B) has generators F1, F2 subject to the relations F2

1 = F2 2 = 0.

Different GCMs typically yield non-isomorphic half-quantum groups!

Return Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 2 / 8

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SLIDE 39

Extra Slides

BRAID DEFINITION

Ti(EX,j) =      −(−1)pY(i)K−e

Y,iFY,i

if j = i; EY,jEY,i − (−1)pY(i)pY(j)qeYijEY,iEY,j if j = i ± 1; EY,j

  • therwise;

Ti(FX,j) =      −(−1)pY(i)EY,iKe

Y,i

if j = i; FY,iFY,j − (−1)pY(i)pY(j)q−eYijFY,jFY,i if j = i ± 1; FY,j

  • therwise;

Ti(KX,j) =      (−1)pY(i)K−1

Y,i

if j = i; (−1)pY(i)pY(j)KY,iKY,j if j = i ± 1; KY,j

  • therwise;

Return Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 3 / 8

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Extra Slides

NON-STANDARD RANK 2

The non-standard GCM for gl(2|1) is 1 1

  • .

The PBW bases are given by Fx

1(F2F1 − q−1F1F2)(y)Fz 2,

Fx

2(F1F2 − q−1F2F1)(y)Fz 1.

Clearly not the same Z[q]-span; in fact, also not the same Z[q−1]-span! (Nevertheless, there is a sort of CB: Fx

1(F2F1)yFz 2.)

However, this case doesn’t bother the theorem: e.g. w0 = s3s2s1s3s2s3 = s3s2s1s2s3s2

2

−1 −1 2 −1 −1

  • 2

−1 0 −1 1 1

  • 1

1 −1 0 −1 2

  • −1

−1 2 −1 −1 2

  • s3

s2 s1

s1, s2 s1 s3 s2, s3

Return Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 4 / 8

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Extra Slides

U-MODULES AND CRYSTALS

As usual, can consider finite-dimensional weight representations. λ ∈ P+: weights in P which are gl(m)-dominant K(λ): induced module from Uq(gl(m))-rep Vgl(m)(λ) V(λ): simple quotient Crystal basis is a pair (L, B) where

◮ L is a lattice over A ⊂ Q(q) (no poles at 0) ◮ B is a basis of L/qL

Theorem (Benkart-Kang-Kashiwara, Kwon)

Let λ ∈ P+. Then K(λ) has a crystal basis (LK(λ), BK(λ)). If additionally λ is a polynomial, V(λ) admits a crystal basis (L(λ), B(λ)), combinatorially realized by super semistandard Young tableaux.

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 5 / 8

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Extra Slides

CRYSTAL FOR U−

Theorem (C)

U− admits a crystal basis B which is compatible with those on modules. Moreover, these crystals “globalize” to a basis satisfying (CB1)-(CB3) and (CB5).

  • Crystal for U−

q (gl(2|1))

  • Crystal for V(3ǫ1 + ǫ2)

Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 6 / 8

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Extra Slides

IDEAS IN PROOF

◮ Simplify [BKK] results: for the m ≥ n = 1 case,

◮ no upper crystals; ◮ no fake highest weights; ◮ no super signs.

◮ Construct crystal inductively using “grand loop” and [BKK]. ◮ Characterize lattice and (signed) basis with bilinear form. ◮ Pseudo-canonical basis mod q is crystal basis (up to sign). ◮ Existence of globalizations follow easily.

Return Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 7 / 8

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Extra Slides

CANONICAL BASIS OF gl(3|1)

Strategy for constructing B from canonical basis for gl(m).:

  • 1. Inductively build B for weights with larger αm multiplicity
  • 2. Multiplicity ≤ 2m, so done in finite number of steps.

Theorem (C)

For x, y, z ∈ {0, 1} and a, b, c ∈ Z≥0, let u = u(x, y, z, a, b, c) ∈ B be the unique element equal to the PBW vector Fx

3Fy 32Fz 321F(a) 2 F(b) 21 F(c) 1

modulo q. Then u =                      Fx

3 F(a+y) 2

F(b+c+z)

1

Fy

3 F(b+z) 2

Fz

3

if c ≥ a, Fx

3Fy 2F(b+z) 1

Fy

3F(a+b+z) 2

Fz

3F(c) 1

if a > c and y ≤ x,

b

  • t=0

(−1)t

  • a − c − 1 + t

t

  • F(a+1+t)

2

F(b+c+z)

1

F3F(b+z−t)

2

Fz

3

  • therwise.

Return Sean Clark Uq(gl(m|1)) and canonical bases May 25, 2016 8 / 8