Quadratic Poisson structures on representations and double brackets - PowerPoint PPT Presentation

Quadratic Poisson structures on representations and double brackets Vladimir Rubtsov, ITEP, Moscow and LAREMA, Universit´ e d’Angers, (joint work with A.Odesskii and V. Sokolov , Theor. Math. Phys. 2012, 171, 442447 , arXiv:1208.2935v1 and work in progress) Talk at "Quantum Integrable Systems and Geometry". Faro-Olh˜ ao, Portugal, Septembre 3, 2012

Plan: ◮ Motivations; ◮ Representation functor, double and H − Poisson structures; ◮ Reminder of matrix case; ◮ AYBE and Compatibility of non-abelian Poisson brackets; ◮ Double Poisson brackets depending on a parameter ◮ Perspectives and output;

◮ A , an associative unital algebra over k V − a finite-dimensional k -vector space. ◮ ◮ The space Rep V ( A ) of all representations of A in V is an affine k -scheme; it comes equipped with a natural GL ( V ) -action whose orbits correspond to the isomorphism classes of representations. ◮ Many important varieties arise as (quotients of) representation spaces. For example, the moduli spaces of flat connections on a closed manifold M can be interpreted as the space of isomorphisms classes of representations of the group algebra A = C [ π 1 ( M )] for the fundamental group of M .

Noncommutatitve Algebraic Geometry ◮ Varying A (while keeping V fixed) defines a contravariant functor Rep V : Alg k → Sch k ◮ Heuristic Principle ( Kontsevich ): A structure on a noncommutative algebra A has a geometric meaning if it naturally induces standard geometric structures on all representation spaces { Rep V ( A ) } . ◮ Applications: Cuntz-Quillen (1995), Kontsevich-Rosenberg (2000), Ginzburg, Etingof, Schedler (2001-12), Van den Bergh (2004-2012), Le Bruyn (2008), Crawley-Boevey (2011), Berest with coauthors (2012) etc‘; ◮ In practice, the above principle works well only when A is a formally smooth (= quasi-free) algebra (since in that case all Rep V ( A ) ’s are nonsingular).

◮ Example Consider A / [ A , A ] . There is a canonical trace map Tr V ( A ) : A / [ A , A ] → k [ Rep V ( A )] which transforms elements of A / [ A , A ] to functions on Rep V ( A ) for all V . ◮ Thus, by Kontsevich , A / [ A , A ] should be thought of as a space of functions on the noncommutative (and probably non-existent!) ‘ Spec ( A ) ’. ◮ The schemes { Rep V ( A ) } are supposed to give a good approximation to Spec ( A ) that becomes ‘better and better’ as dim k V → ∞ . ◮ Remark There are interesting algebras for which all Rep V ( A ) are trivial (e.g. the Weyl algebras).

Procesi Theorem ◮ To put a multiplicative structure in the game , we extend Sym Tr V ( A ) : Sym k ( A / [ A , A ]) → k [ Rep V ( A )] . ◮ Theorem ( Procesi ). The image of Sym Tr V ( A ) is k [ Rep V ( A )] GL ( V ) .

Van den Bergh’s Double Poisson Structures-1 ◮ Let A be a f.g. associative C − algebra; ◮ Rep m ( A ) := Hom ( A , Mat m ( C )); ◮ GL m ( C ) acts by conjugation on Mat m ( C ); ◮ Question ( M.Van den Bergh ): "What kind of structures we need on A in order that Rep m ( A ) and Rep m ( A ) GL m ( C ) possess a Poisson structure?"

Van den Bergh’s Double Poisson Structures-2 Definition A double Poisson bracket on A is a C − bilinear map { {− , −} } : A × A �→ A ⊗ A such that: ◮ } ◦ , ( a ⊗ b ) ◦ = b ⊗ a ; { { a , b } } = −{ { b , a } (1) ◮ { {− , −} } is a derivation in its second argument (wrt the outer bimodule structure on A): { { a , bc } } = ( b ⊗ 1 ) { { a , c } } + { { a , b } } ( 1 ⊗ c ); (2) ◮ { {− , −} } satisfies a sort of Jacobi identity: } + σ 2 { { { a , { { b , c } }} } + σ { { b , { { c , a } }} { c , { { a , c } }} } = 0 , (3) where { { a , b ⊗ c } } := { { a , b } } ⊗ c and σ ( a ⊗ b ⊗ c ) := c ⊗ a ⊗ b .

M. Van den Bergh’s Double Poisson Structures-3 Define {− , −} : A × A �→ A by } ′ { {− , −} := µ ( { { a , b } } ) = { { a , b } { a , b } } ” Theorem ( M.Van den Bergh ) Let A , { {− , −} } be a double Poisson algebra. Then, 1. {− , −} is a derivation in its second argument and vanishes on commutators in its first argument; 2. {− , −} induces an anti-symmetric bracket on A / [ A , A ] ; 3. {− , −} makes A into a left Loday algebra ( {− , −} satisfies the following version of the Jacobi identity: { a , { b , c }} = {{ a , b } , c } + { b , { a , c }} ); 4. {− , −} makes A / [ A , A ] into a Lie algebra.

Crawley-Boevey H − structure-1 ◮ Let A be a f.g. associative C − algebra, Der ( A ) its derivation algebra and HH 0 ( A ) = A / [ A , A ] ; ◮ Any ∂ ∈ Der ( A ) - "descends" under the projection p : A �→ HH 0 ( A ) to the map p ( ∂ ) : HH 0 ( A ) �→ HH 0 ( A ) such that p ( ∂ )( p ( a )) = p ( ∂ ( a )) ; ◮ Definition H − Poisson structure on A is a Lie bracket [ − , − ] on HH 0 ( A ) such that the map [ p ( a ) , − ] ∈ End HH 0 ( A ) is induced by some derivation ∂ a : p ( ∂ a ) = [ p ( a ) , − ] .

Crawley-Boevey H − structure-2 ◮ Example Any double Poisson structure on A induces on H − structure on A via the multiplication µ : [ p ( a ) , p ( b )] := p ( µ ( { { a , b } } )) . ◮ Theorem ( W.Crawley-Boevey ) Each H − Poisson structure on A defines a unique Poisson structure on C [ Rep m ( A ) GL m ] such that { Tr ( a ) , Tr ( b ) } = Tr ([ p ( a ) , p ( b )]) . .

The coordinate ring k [ Rep m ( A )] is generated by symbols x j i where 1 ≤ i , j ≤ m for all x ∈ A with the relations ( xy ) j � i y j 1 j i = δ j x l i = l , i . l Example Let A = k < x 1 , . . . , x N > then k [ Rep m ( A )] = k [( x j i ,α )] where 1 ≤ α ≤ N i a i The trace Tr ( a ) := � i defines GL m − invariant polynomial on k [ Rep m ( A )]

Non-abelian Poisson brackets : matrix case. Suppose now k = C ◮ Mat m ( C ) - matrix algebra . ◮ Take N exemplairs of Mat m ( C ) and let x j i ,α be a matrix element of the matrix with a number 1 ≤ α ≤ N and 1 ≤ i , j ≤ m . ◮ A Poisson bracket on Mat m ( C ) × . . . × Mat m ( C ) is called a non-abelian Poisson bracket iff the bracket between traces of any two matrix polynomials P i ( x 1 , ..., x N ) , i = 1 , 2 is the trace of a matrix polynomial P 3 .

Theorem Theorem ( Odesskii, R., Sokolov ) Any linear non-abelian Poisson bracket has the form { x j i ,α , x j ′ i ′ ,β } = b γ α,β x j ′ i ,γ δ j i ′ − b γ β,α x j i ′ ,γ δ j ′ i ; (4) Any quadratic non-abelian Poisson bracket is given by i ,α , x j ′ αβ x j ′ i ,γ x j ′ k ,ǫ δ j ′ { x j i ′ ,β } = r γǫ i ,γ x j i ′ ,ǫ + a γǫ αβ x k k ,ǫ δ j i ′ − a γǫ βα x k i ′ ,γ x j i , (5) Formula (4) defines a Poisson bracket iff b µ αµ b µ αβ b σ µγ = b σ βγ ; (6)

Quadratic Non-Abelian brackets conditions Formula (5) defines a Poisson bracket iff the following relations hold: r σǫ αβ = − r ǫσ βα , (7) στ + r µσ τα r λµ r λσ αβ r µν βτ r νλ σα + r νσ σβ = 0 , (8) a σλ αβ a µν τσ = a µσ τα a νλ σβ , (9) στ = a µσ a σλ αβ a µν αβ r λν τσ + a µν ασ r σλ (10) βτ and στ + a µν a λσ αβ a µν τσ = a σν αβ r λµ σβ r σλ τα . (11)

Poisson conditions: three ways of presentation We may regard the tensors r and a as: ◮ operators on V ⊗ V , where V is an m -dimensional C − vector space; ◮ elements of Mat m ( C ) ⊗ Mat m ( C ) ; ◮ operators on Mat m ( C ) .

First interpretation: AYBE ◮ The identities (7-11) can be written as r 12 = − r 21 , r 23 r 12 + r 31 r 23 + r 12 r 31 = 0 , a 12 a 31 = a 31 a 12 , (12) σ 23 a 13 a 12 = a 12 r 23 − r 23 a 12 , a 32 a 12 = r 13 a 12 − a 32 r 13 . ◮ Here all operators act in V ⊗ V ⊗ V , σ ij means the transposition of i -th and j -th components of the tensor product, and a ij , r ij mean operators a , r acting in the product of the i -th and j -th components.

Second interpretation ◮ In the second interpretation we consider the following elements from Mat m ( C ) ⊗ Mat m ( C ) : k ⊗ e j k ⊗ e j r = r km ij e i a = a km ij e i m , where e i m , j are the matrix unities: ◮ e j k = δ j i e m k e m i . Then (7), (8)-(11) are equivalent to (12), where tensors belong to Mat m ( C ) ⊗ Mat m ( C ) ⊗ Mat m ( C ) . ◮ Namely, r 12 = r mk k ⊗ e j e i m ⊗ 1 and so on. ij ◮ The element σ is given by σ = e j i ⊗ e i j . .

Third interpretation: Rota-Baxter operators In the third interpretation one can define operators r , a ∗ : Mat m ( C ) → Mat m ( C ) by r , a , ¯ r ( x ) p q = r lp nq x n a ( x ) p q = a lp nq x n l , l , r ( x ) p q = r pl a ∗ ( x ) p q = a pl nq x n qn x n ¯ l , l . Then for any x , y ∈ Mat m ( C ) a ( x ) a ∗ ( y ) = a ∗ ( y ) a ( x ) , r ( x ) = − r ∗ ( x ) , r ( x ) r ( y ) = r ( xr ( y )) + r ( x ) y ) , r ∗ ( x ) , ¯ r ( x ) = − ¯ ¯ r ( x )¯ r ( y ) = ¯ r ( x ¯ r ( y )) + ¯ r ( x ) y ) , a ∗ ( ya ( x )) = r ( xa ∗ ( y )) − r ( x ) a ∗ ( y ) , a ( x ) a ( y ) = − a ( r ( y ) x ) − a ( yr ( x )) , a ∗ ( a ( x ) y ) = r ( a ∗ ( y ) x ) − a ∗ ( y ) r ( x ) , a ( ya ∗ ( x )) = − ¯ r ( xa ( y )) + ¯ r ( x ) a ( y ) , a ∗ ( x ) a ∗ ( y ) = a ∗ (¯ r ( y ) x ) + a ∗ ( y ¯ r ( x )) , a ( a ∗ ( x ) y ) = − ¯ r ( a ( y ) x ) + a ( y )¯ r ( x )