Poisson algebras of block-upper-triangular bilinear forms and braid group action Marta Mazzocco, Loughborough University Work in collaboration with Leonid Chekhov – Typeset by Foil T EX – 1
Bilinear forms on C N � x, y � := x T A y, ∀ x, y ∈ C N . = “ A block–upper– “Block–upper–triangular bilinear form” triangular” A 1 , 1 A 1 , 2 . . . A 1 ,n A 2 , 2 . . . O A 2 ,n A = , A I,J ∈ GL ( m ) , det( A I,I ) = 1 . . . . . . . . . . . . . . . O O A n,n A n,m ⊂ GL ( nm ) is the set of such bilinear forms. a k,l denote the entries of A . – Typeset by Foil T EX – 2
Homogeneous quadratic Poisson bracket on GL N ( C ) , N = nm : � � { a i,j , a k,l } = sign( j − l ) + sign( i − k ) a i,l a k,j + � � � � + sign( j − k ) + 1 a j,l a i,k + sign( i − l ) − 1 a l,j a k,i • This bracket admits a Poisson reduction to A n,m for any n , m such that N = nm . • Admits a suitable action of Braid grorup preserving it. • Affine version and quantisation. – Typeset by Foil T EX – 3
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 sp 2 n . • Algebraic geometers are interested in the vanishing locus of quadratic Poisson algebras on Projective spaces (Hitchin 2011). – Typeset by Foil T EX – 4
Poisson reductions: � � { a i,j , a k,l } = sign( j − l ) + sign( i − k ) a i,l a k,j + � � � � + sign( j − k ) + 1 a j,l a i,k + sign( i − l ) − 1 a l,j a k,i { a N, 1 , a k,l } = ( − 1+1) a N,l a k, 1 +( − 1+1) a 1 ,l a N,k +(1 − 1) a l, 1 a k,N , for k, l � = 1 , N, { a N, 1 , a 1 ,l } = a 1 ,l a N, 1 l � = 1 , N, for { a N, 1 , a N,l } = − a N, 1 a N,l l � = 1 , N, for In general: – Typeset by Foil T EX – 5
Case m = 1 , Dubrovin–Ugaglia bracket { a i,j , a k,l } = (sign( i − l ) − sign( j − l ))( a l,j a k,i − a l,i a k,j ) + � � + sign( i − k ) − sign( j − k ) ( a k,j a i,l − a k,i a j,l ) Notation: 1 a 1 , 2 . . . a 1 ,n 0 1 . . . a 2 ,n A = ∈ A . . . . . . . . . . . . 0 . . . 0 1 Braid group action in the context of Frobenius manifolds due to Dubrovin. – Typeset by Foil T EX – 6
Bondal’s approach for m = 1 . • GL n ( C ) acts on bilinear forms as A �→ BAB T . ∀ A, B ∈ GL n ( C ) , • This action of GL ( C n ) does not preserve A . • For every A ∈ A we take a subset: B ∈ GL ( C n ) | A �→ BAB T ∈ A � � M A = . • We define a groupoid ( A , M ) = { ( A, B ) such that A ∈ A , B ∈ M A } M = ∪ A ∈A M A – Typeset by Foil T EX – 7
Case m = 1 : groupoid structure. ( A , M ) = { ( A, B ) such that A ∈ A , BAB T ∈ A} Partial multiplication: ( B 1 AB T � � m 1 , B 2 ) , ( A, B 1 ) = ( A, B 2 B 1 ) . Identity morphism: e = ( A, 1 1) , Inverse: i : ( A, B ) → ( BAB T , B − 1 ) . – Typeset by Foil T EX – 8
Case m = 1 : algebroid structure. Infinitesimal version of the condition BAB T ∈ A . Lie algebroid ( A , g ) : g ∈ gl n ( C ) , | A + Ag + g T A ∈ A � � g := ∪ A ∈A g A g A := . Natural isomorphism anchor map: D A : g A → T A A �→ Ag + g T A. g Bondal’s main idea: give a parameterization of all g ∈ g A . – Typeset by Foil T EX – 9
Case m = 1 : Bondal’s parameterization of all g ∈ g A T A ∼ { strictly upper triangular matrices } T ∗ A ∼ { strictly lower triangular matrices } ⇒ Lemma: The following map P A : T ∗ A A → g A (1) �→ P − , 1 / 2 ( wA ) − P + , 1 / 2 ( w T A T ) , w where P ± , 1 / 2 are the projection operators: P ± , 1 / 2 a i,j := 1 ± sign( j − i ) a i,j , i, j = 1 , . . . , n, (2) 2 defines an isomorphism between the Lie algebroid ( g , D A ) and the Lie algebroid ( T ∗ A , D A P A ) . – Typeset by Foil T EX – 10
Case m = 1 : Poisson structure The Lie algebroid ( T ∗ A , D A P A ) defines the Poisson bi-vector: C ∞ ( A ) Π : T ∗ A A × T ∗ A A → ( ω 1 , ω 2 ) Tr ( ω 1 D A P A ( ω 2 )) The Poisson structure is thus automatically invariant under groupoid action. The braid group elements are: 1 ... . . . 1 β i,i +1 A = B i,i +1 AB T i a i,i +1 − 1 i,i +1 , B i,i +1 = . i + 1 1 0 . . . 1 ... 1 – Typeset by Foil T EX – 11
Case m = 1 : Central elements i,i +1 and β i,i +1 A T = B i,i +1 A T B T Since β i,i +1 A = B i,i +1 AB T i,i +1 ⇒ the central elements are generated by A + λA T � � det � n � ⇒ central elements so that the symplectic leaves have dimension 2 n ( n − 1) � n � − 2 2 – Typeset by Foil T EX – 12
General case: We keep the same Poisson bi–vector: C ∞ ( A ) Π : T ∗ A A × T ∗ A A → ( ω 1 , ω 2 ) Tr ( ω 1 D A P A ( ω 2 )) ⇒ we keep the structure P A D A T ∗ A A n,m → → T A A n,m g A �→ P − , 1 / 2 ( w A ) − P + , 1 / 2 ( w T A T ) A g + g T A w �→ where g A := Im( P A ) T A ∼ { upper block triangular matrices s.t. Tr( A − 1 J,J δA J,J ) = 0 } T ∗ A ∼ { lower block triangular matrices s.t. Tr( A − 1 J,J w J,J ) = 0 } dim (ker P A ) > 0 . – Typeset by Foil T EX – 13
To find the braid group generators we need to find the groupoid ( A n,m , M ) which integrates ( A n,m , g ) . We deal with the case of full size matrices N × N . • dim( M ) = dim g = N 2 − � N � 2 • M ⊂ ∪ A ∈A N { B | B A B T ∈ A N } How to find the groupoid? Idea: it must preserve central elements. � N + 2 � A + λA T � � det generates independent central elements. 2 � N + 2 � N 2 − is not always even 2 – Typeset by Foil T EX – 14
⇒ We need more central elements. Insight: freedom of block upper triangular reduction. ⇒ The bottom left minors: A N − d +1 , 1 . . . A N + d − 1 ,d . . . . M d := det . . . . . , . . . A N, 1 A N,d must play a role. � N + 1 � b d := det M N − d / det M d , for d = 1 , . . . , , 2 are central elements.This leads to symplectic leaves of the dimension N 2 − N always even – Typeset by Foil T EX – 15
General case Theorem The Lie groupoid M is M := U A ∈A n,m M A , where B ∈ GL N | B A B T ∈ A n,m and � M A := d ( A ) , ∀ d = 0 , . . . , [ m � b ( I ) d ( B A B T ) = b ( I ) 2 ] , I = 1 , . . . , n , The braid group generators are found as elementary elements of this groupoid. – Typeset by Foil T EX – 16
General case: braid group action β I,I +1 [ A ] = B I,I +1 A B T I,I +1 , I = 1 , . . . , n − 1 E ... . . . E I,I +1 A − T A T − E I I,I B I,I +1 = , A I,I A − T I + 1 O I,I . . . E ... E – Typeset by Foil T EX – 17
Affinisation [L. Chekhov, M.M. Advances 2010] Generating function: ∞ G i,j ( λ ) := G (0) G ( p ) G (0) � i,j λ − p , i,j + i,j = a i,j , p =1 the matrices G ( p ) are arbitrary full-size matrices. k k k G (1) ij j j j l l l G (2) G (0) ij ji i i i – Typeset by Foil T EX – 18
� � sign( i − k ) − λ + µ {G i,j ( λ ) , G k,l ( µ ) } = G k,j ( λ ) G i,l ( µ ) + λ − µ � � sign( j − l ) + λ + µ + G k,j ( µ ) G i,l ( λ ) + λ − µ � � sign( j − k ) − 1 + λµ + G i,k ( λ ) G j,l ( µ ) + 1 − λµ � � sign( i − l ) + 1 + λµ + G l,j ( λ ) G k,i ( µ ) . 1 − λµ This is an abstract Poisson algebra whatever zero level we pick. – Typeset by Foil T EX – 19
Affine case with m = 1 A further braid group generator: � T , B n, 1 ( λ − 1 ) � β n, 1 [ G ( λ )] = B n, 1 ( λ ) G ( λ ) (3) where 0 0 . . . 0 λ 0 1 0 . . . 0 . . ... ... . . . 0 . B n, 1 ( λ ) = . . ... . 0 . 1 0 − λ − 1 0 . . . G (1) 0 n, 1 – Typeset by Foil T EX – 20
Affine case with arbitrary m λ A n,n A − T O n,n E ... B n, 1 ( λ ) = (4) E � T A − T G (1) − λ − 1 E � n, 1 n,n – Typeset by Foil T EX – 21
Quantisation: 1 ( λ ) R ( λ − 1 , µ ) T 1 G 2 ( µ ) = G 2 ( µ ) R ( λ − 1 , µ ) T 1 G 1 ( λ ) R ( λ, µ ) R ( λ, µ ) G � � E ii ⊗ E jj + ( q − 1 λ − qµ ) R ( λ, µ ) = ( λ − µ ) E ii ⊗ E ii + i � = j i + ( q − 1 − q ) λ E ij ⊗ E ji + ( q − 1 − q ) µ � � E ij ⊗ E ji i<j i>j • For m = 1 : twisted q –Yangian Y ′ q ( o n ) • Forf m = 2 : twisted q –Yangian Y ′ q ( sp 2 n ) . – Typeset by Foil T EX – 22
Quantisation of braid group action, m = 2 : We need a quantum inverse: � a 11 a 12 � � � 1 a 22 ( q − 1 /q ) a 21 − a 12 A − 1 = for A = − q 2 a 21 a 21 a 22 a 11 qdet E ... . . . E I,I +1 A − T q A T − q 2 E I I,I B I,I +1 = , A I,I A − T I + 1 O I,I . . . E ... E – Typeset by Foil T EX – 23
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