Download Poisson algebras of block-upper-triangular bilinear forms and braid

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
hx, yi := xT A 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,l ak,j +
+ sign(j − k) + 1 aj,l ai,k + sign(i − l) − 1 al,j ak,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,l ak,j +
+ sign(j − k) + 1 aj,l ai,k + sign(i − l) − 1 al,j ak,i
{aN,1, ak,l } = (−1+1)aN,l ak,1+(−1+1)a1,l aN,k +(1−1)al,1ak,N ,
{aN,1, a1,l } = a1,l aN,1
for
{aN,1, aN,l } = −aN,1aN,l
for
l 6= 1, N,
for
l 6= 1, N,
In general:
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k, l 6= 1, N,
Case m = 1, Dubrovin–Ugaglia bracket
{ai,j , ak,l } = (sign(i − l) − sign(j − l))(al,j ak,i − al,iak,j ) +
+ sign(i − k) − sign(j − k) (ak,j ai,l − ak,iaj,l )
Notation:



A=

1 a1,2 . . . a1,n
0
1 . . . a2,n
... ... ... ...
0 ... 0
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 7→ BAB T .
• This action of GL(Cn) does not preserve A.
• For every A ∈ A we take a subset:
n
T
MA = B ∈ GL(C ) | A 7→ BAB ∈ 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, BAB T ∈ A}
Partial multiplication:
m
(B1AB1T , B2), (A, B1)
Identity morphism:
= (A, B2B1).
e = (A, 11),
Inverse:
i : (A, B) → (BAB T , B −1).
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Case m = 1: algebroid structure.
Infinitesimal version of the condition BAB T ∈ A.
Lie algebroid (A, g):
g := ∪A∈AgA
T
gA := g ∈ gln(C), | A + Ag + g A ∈ A .
Natural isomorphism anchor map:
DA : gA →
TA A
g 7→ Ag + g T A.
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
T A ∼ {strictly upper triangular matrices}
T ∗A ∼ {strictly lower triangular matrices}
⇒ Lemma: The following map
gA
PA : TA∗ A →
w 7→ P−,1/2(wA) − P+,1/2(wT AT ),
(1)
where P±,1/2 are the projection operators:
P±,1/2ai,j
1 ± sign(j − i)
:=
ai,j ,
2
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:
C ∞(A)
Π : TA∗ A × TA∗ A →
(ω1, ω2)
Tr (ω1DAPA(ω2))
The Poisson structure is thus automatically invariant under groupoid
action.
The braid group elements are:
βi,i+1A =
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T
Bi,i+1ABi,i+1
,
Bi,i+1 =
...
i
i+1
...







1
...
1
ai,i+1
1
−1
0
1
...
1
11


.


Case m = 1: Central elements
T
T
Since βi,i+1A = Bi,i+1ABi,i+1
and βi,i+1AT = Bi,i+1AT Bi,i+1
⇒ the central elements are generated by
T
det A + λA
⇒
n
2
central elements so that the symplectic leaves have dimension
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n(n − 1) h n i
−
2
2
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General case: We keep the same Poisson bi–vector:
Π : TA∗ A × TA∗ A →
(ω1, ω2)
C ∞(A)
Tr (ω1DAPA(ω2))
⇒ we keep the structure
TA∗ An,m
w
PA
DA
→
gA
→ TAAn,m
7→ P−,1/2(wA) − P+,1/2(wT AT ) 7→ A g + g T A
where gA := Im(PA)
T A ∼ {upper block triangular matrices s.t. Tr(A−1
J,J δAJ,J ) = 0}
T ∗A ∼ {lower block triangular matrices s.t. Tr(A−1
J,J wJ,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 .
N
• dim(M) = dim g = N 2 − 2
• M ⊂ ∪A∈AN {B| BAB T ∈ AN }
How to find the groupoid? Idea: it must preserve central
elements.
N +2
T
det A + λA
generates
independent central elements.
2
N +2
2
is not always even
N −
2
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⇒ We need more central elements. Insight: freedom of block upper
triangular reduction.
⇒ The bottom left minors:


AN −d+1,1 . . . AN +d−1,d


.
.
Md := det 
.
...
.
,
AN,1
...
AN,d
must play a role.
bd := det MN −d/ det Md,
for
N +1
d = 1, . . . ,
,
2
are central elements.This leads to symplectic leaves of the dimension
N2 − N
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always even
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General case
Theorem The Lie groupoid M is
M := UA∈An,m MA,
where
MA :=
B ∈ GLN | BAB T ∈ An,m and
o
m
(I)
(I)
T
bd (BAB ) = bd (A), ∀d = 0, . . . , [ ], I = 1, . . . , n ,
2
The braid group generators are found as elementary elements of
this groupoid.
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General case: braid group action
T
βI,I+1[A] = BI,I+1ABI,I+1
,
I = 1, . . . , n − 1

E

...

.. 
E


−T
T
A
A
I 
I,I+1 I,I −E
BI,I+1 =

I +1 
O
AI,I A−T
I,I
.. 

E

...

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
E






,





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Affinisation
[L. Chekhov, M.M. Advances 2010]
Generating function:
Gi,j (λ) :=
(0)
Gi,j
+
∞
X
(0)
Gi,j
(p) −p
Gi,j λ ,
= ai,j ,
p=1
the matrices G(p) are arbitrary full-size matrices.
k
k
j
l
(1)
Gij
j
l
k
j
l
(2)
Gij
(0)
Gji
i
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i
i
18
λ+µ
{Gi,j (λ), Gk,l (µ)} = sign(i − k) −
Gk,j (λ)Gi,l (µ) +
λ−µ
λ+µ
+ sign(j − l) +
Gk,j (µ)Gi,l (λ) +
λ−µ
1 + λµ
Gi,k (λ)Gj,l (µ) +
+ sign(j − k) −
1 − λµ
1 + λµ
+ sign(i − l) +
Gl,j (λ)Gk,i(µ).
1 − λµ
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:
T
βn,1[G(λ)] = Bn,1(λ)G(λ) Bn,1(λ ) ,
−1
where




Bn,1(λ) = 


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0
0
..
0
−λ−1
0
1
0
..
0
... 0
λ
0 ... 0
..
... ...
... 1
0
(1)
. . . 0 Gn,1
(3)




.


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Affine case with arbitrary m




Bn,1(λ) = 


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λAn,nA−T
n,n
O
E
...
E
−1
−λ E
(1) T −T
Gn,1 An,n







(4)
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Quantisation:
1
T1 2
−1
2
−1
T1 1
R(λ, µ)G (λ)R(λ , µ) G (µ) = G (µ)R(λ , µ) G (λ)R(λ, µ)
R(λ, µ) = (λ − µ)
X
Eii ⊗ Ejj + (q −1λ − qµ)
+ (q −1 − q)λ
Eii ⊗ Eii +
i
i6=j
X
X
Eij ⊗ Eji + (q −1 − q)µ
i<j
X
Eij ⊗ Eji
i>j
• For m = 1: twisted q–Yangian Yq′(on)
• Forf m = 2: twisted q–Yangian Yq′(sp2n).
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Quantisation of braid group action, m = 2:
We need a quantum inverse:
1
a11 a12
a22
(q − 1/q)a21 − a12
−1
forA =
A =
a21 a22
a11
qdet −q 2a21



.. 


I 
BI,I+1 =

I +1 
.. 



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
E
...
E
2
−q
E
qATI,I+1A−T
I,I
O
AI,I A−T
I,I
E
...
E






,





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