Poisson algebras of block-upper-triangular bilinear forms and braid - - PowerPoint PPT Presentation

Poisson algebras of block-upper-triangular bilinear forms and braid group action Marta Mazzocco, Loughborough University

Work in collaboration with Leonid Chekhov

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Bilinear forms on CN x, y := xTA y, ∀ x, y ∈ CN. “Block–upper–triangular bilinear form” = “A block–upper– triangular” A =      A1,1 A1,2 . . . A1,n O A2,2 . . . A2,n . . . . . . . . . . . . O O . . . An,n      , AI,J ∈ GL(m), det(AI,I) = 1 An,m ⊂ GL(nm) is the set of such bilinear forms. ak,l denote the entries of A.

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Homogeneous quadratic Poisson bracket on GLN(C), N = nm: {ai,j, ak,l} =

• sign(j − l) + sign(i − k)
• ai,lak,j +

+

• sign(j − k) + 1
• aj,lai,k +
• sign(i − l) − 1
• al,jak,i
• This bracket admits a Poisson reduction to An,m for any n, m

such that N = nm.

• Admits a suitable action of Braid grorup preserving it.
• Affine version and quantisation.

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Why do we care?

• Case m = 1: Dubrovin–Ugaglia bracket appearing in Frobenius

Manifold theory. Its quantization is also known as Nelson–Regge algebra in 2 + 1-dimensional quantum gravity and as Fock–Rosly bracket in Chern–Simons theory.

• Case m = 2: its quantization is a Twisted Yangian associated to

the Lie algebra sp2n.

• Algebraic geometers are interested in the vanishing locus of

quadratic Poisson algebras on Projective spaces (Hitchin 2011).

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Poisson reductions: {ai,j, ak,l} =

• sign(j − l) + sign(i − k)
• ai,lak,j +

+

• sign(j − k) + 1
• aj,lai,k +
• sign(i − l) − 1
• al,jak,i

{aN,1, ak,l} = (−1+1)aN,lak,1+(−1+1)a1,laN,k+(1−1)al,1ak,N,

for k, l = 1, N,

{aN,1, a1,l} = a1,laN,1

for

l = 1, N, {aN,1, aN,l} = −aN,1aN,l

for

l = 1, N, In general:

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Case m = 1, Dubrovin–Ugaglia bracket {ai,j, ak,l} = (sign(i − l) − sign(j − l))(al,jak,i − al,iak,j) + +

• sign(i − k) − sign(j − k)
• (ak,jai,l − ak,iaj,l)

Notation: A =      1 a1,2 . . . a1,n 1 . . . a2,n . . . . . . . . . . . . . . . 1      ∈ A Braid group action in the context of Frobenius manifolds due to Dubrovin.

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Bondal’s approach for m = 1.

• GLn(C) acts on bilinear forms as

∀A, B ∈ GLn(C), A → BABT.

• This action of GL(Cn) does not preserve A.
• For every A ∈ A we take a subset:

MA =

• B ∈ GL(Cn) | A → BABT ∈ A
• .
• We define a groupoid

(A, M) = {(A, B) such that A ∈ A, B ∈ MA} M = ∪A∈AMA

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Case m = 1: groupoid structure. (A, M) = {(A, B) such that A ∈ A, BABT ∈ A} Partial multiplication: m

• (B1ABT

1 , B2), (A, B1)

• = (A, B2B1).

Identity morphism: e = (A, 1 1), Inverse: i : (A, B) → (BABT, B−1).

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Case m = 1: algebroid structure. Infinitesimal version of the condition BABT ∈ A. Lie algebroid (A, g): g := ∪A∈AgA gA :=

• g ∈ gln(C), | A + Ag + gTA ∈ A
• .

Natural isomorphism anchor map: DA : gA → TAA g → Ag + gTA. Bondal’s main idea: give a parameterization of all g ∈ gA.

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Case m = 1: Bondal’s parameterization of all g ∈ gA TA ∼ {strictly upper triangular matrices} T ∗A ∼ {strictly lower triangular matrices} ⇒ Lemma: The following map PA : T ∗

AA →

gA w → P−,1/2(wA) − P+,1/2(wTAT), (1) where P±,1/2 are the projection operators: P±,1/2ai,j := 1 ± sign(j − i) 2 ai,j, i, j = 1, . . . , n, (2) defines an isomorphism between the Lie algebroid (g, DA) and the Lie algebroid (T ∗A, DAPA).

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Case m = 1: Poisson structure The Lie algebroid (T ∗A, DAPA) defines the Poisson bi-vector: Π : T ∗

AA × T ∗ AA →

C∞(A) (ω1, ω2) Tr (ω1DAPA(ω2)) The Poisson structure is thus automatically invariant under groupoid action. The braid group elements are: βi,i+1A = Bi,i+1ABT

i,i+1,

Bi,i+1 =

. . . i i + 1 . . .

     

1 ... 1 ai,i+1 −1 1 1 ... 1

      .

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Case m = 1: Central elements Since βi,i+1A = Bi,i+1ABT

i,i+1 and βi,i+1AT = Bi,i+1ATBT i,i+1

⇒ the central elements are generated by det

• A + λAT

⇒ n

2

• central elements so that the symplectic leaves have dimension

n(n − 1) 2 − n 2

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EX – 12

General case: We keep the same Poisson bi–vector: Π : T ∗

AA × T ∗ AA →

C∞(A) (ω1, ω2) Tr (ω1DAPA(ω2)) ⇒ we keep the structure PA DA T ∗

AAn,m →

gA → TAAn,m w → P−,1/2(wA) − P+,1/2(wTAT) → A g + gTA where gA := Im(PA) TA ∼ {upper block triangular matrices s.t. Tr(A−1

J,JδAJ,J) = 0}

T ∗A ∼ {lower block triangular matrices s.t. Tr(A−1

J,JwJ,J) = 0}

dim (ker PA) > 0.

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To find the braid group generators we need to find the groupoid (An,m, M) which integrates (An,m, g). We deal with the case of full size matrices N × N.

• dim(M) = dim g = N 2 −

N

2

• M ⊂ ∪A∈AN{B| BABT ∈ AN}

How to find the groupoid? Idea: it must preserve central elements. det

• A + λAT

generates N + 2 2

• independent central elements.

N 2 − N + 2 2

• is not always even

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⇒ We need more central elements. Insight: freedom of block upper triangular reduction. ⇒ The bottom left minors: Md := det    AN−d+1,1 . . . AN+d−1,d . . . . . . . . . AN,1 . . . AN,d    , must play a role. bd := det MN−d/ det Md, for d = 1, . . . , N + 1 2

• ,

are central elements.This leads to symplectic leaves of the dimension N 2 − N always even

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General case Theorem The Lie groupoid M is M := UA∈An,mMA, where MA :=

• B ∈ GLN | BABT ∈ An,m and

b(I)

d (BABT) = b(I) d (A), ∀d = 0, . . . , [m

2 ], I = 1, . . . , n

• ,

The braid group generators are found as elementary elements of this groupoid.

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General case: braid group action βI,I+1[A] = BI,I+1ABT

I,I+1,

I = 1, . . . , n − 1 BI,I+1 = . . . I I + 1 . . .              E ... E AT

I,I+1A−T I,I

−E AI,IA−T

I,I

O E ... E              ,

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Affinisation [L. Chekhov, M.M. Advances 2010] Generating function: Gi,j(λ) := G(0)

i,j + ∞

• p=1

G(p)

i,j λ−p,

G(0)

i,j = ai,j,

the matrices G(p) are arbitrary full-size matrices. G(0)

ji

j k l i G(1)

ij

j k l i G(2)

ij

j k l i

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{Gi,j(λ), Gk,l(µ)} =

• sign(i − k) − λ + µ

λ − µ

• Gk,j(λ)Gi,l(µ) +

+

• sign(j − l) + λ + µ

λ − µ

• Gk,j(µ)Gi,l(λ) +

+

• sign(j − k) − 1 + λµ

1 − λµ

• Gi,k(λ)Gj,l(µ) +

+

• sign(i − l) + 1 + λµ

1 − λµ

• Gl,j(λ)Gk,i(µ).

This is an abstract Poisson algebra whatever zero level we pick.

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Affine case with m = 1 A further braid group generator: βn,1[G(λ)] = Bn,1(λ)G(λ)

• Bn,1(λ−1)

T, (3) where Bn,1(λ) =        0 . . . λ 1 . . . . . . ... ... . . . . . . ... 1 −λ−1 0 . . . G(1)

n,1

       .

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Affine case with arbitrary m Bn,1(λ) =        O λAn,nA−T

n,n

E ... E −λ−1E

• G(1)

n,1

TA−T

n,n

       (4)

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Quantisation: R(λ, µ)G

1 (λ)R(λ−1, µ)T1G 2 (µ) = G 2 (µ)R(λ−1, µ)T1G 1 (λ)R(λ, µ)

R(λ, µ) = (λ − µ)

• i=j

Eii ⊗ Ejj + (q−1λ − qµ)

• i

Eii ⊗ Eii + + (q−1 − q)λ

• i<j

Eij ⊗ Eji + (q−1 − q)µ

• i>j

Eij ⊗ Eji

• For m = 1: twisted q–Yangian Y ′

q(on)

• Forf m = 2: twisted q–Yangian Y ′

q(sp2n).

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Quantisation of braid group action, m = 2: We need a quantum inverse: forA = a11 a12 a21 a22

• A−1 =

1 qdet

• a22

(q − 1/q)a21 − a12 −q2a21 a11

• BI,I+1 =

. . . I I + 1 . . .              E ... E qAT

I,I+1A−T I,I

−q2E AI,IA−T

I,I

O E ... E              ,

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