"Polynomial and Elliptic Algebras of "small - PowerPoint PPT Presentation

Introduction Examples of regular leave algebras, n = 3 Heisenberg invariancy, Cremona and all that... Cremona transformations and Poisson morphisms of P 4 "Polynomial and Elliptic Algebras of "small dimensions." Vladimir Roubtsov 1 1 LAREMA, U.M.R. 6093 associé au CNRS Université d’Angers and Theory Division, ITEP, Moscow August, 3, 2009 - "SQS’09", Dubna Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Heisenberg invariancy, Cremona and all that... Cremona transformations and Poisson morphisms of P 4 Plan 1 Introduction Poisson algebras associated to elliptic curves. 2 Examples of regular leave algebras, n = 3 Elliptic algebras "Mirror transformation" 3 Heisenberg invariancy, Cremona and all that... 4 Cremona transformations and Poisson morphisms of P 4 Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 A Poisson structure on a manifold M (smooth or algebraic) is given by a bivector antisymmetric tensor field π ∈ Λ 2 ( TM ) defining on the corresponded algebra of functions on M a structure of (infinite dimensional) Lie algebra by means of the Poisson brackets { f , g } = � π, df ∧ dg � . The Jacobi identity for this brackets is equivalent to an analogue of (classical) Yang-Baxter equation namely to the "Poisson Master Equation": [ π, π ] = 0, where the brackets [ , ] : Λ p ( TM ) × Λ q ( TM ) �→ Λ p + q − 1 ( TM ) are the only Lie super-algebra structure on Λ . ( TM ) . Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Nambu-Poisson Let us consider n − 2 polynomials Q i in C n with coordinates x i , i = 1 , ..., n . For any polynomial λ ∈ C [ x 1 , ..., x n ] we can define a bilinear differential operation { , } : C [ x 1 , ..., x n ] ⊗ C [ x 1 , ..., x n ] �→ C [ x 1 , ..., x n ] by the formula { f , g } = λ df ∧ dg ∧ dQ 1 ∧ ... ∧ dQ n − 2 , f , g ∈ C [ x 1 , ..., x n ] . (1) dx l ∧ dx 2 ∧ ... ∧ dx n Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Sklyanin algebra The case n = 4 in (1) corresponds to the classical (generalized) Sklyanin quadratic Poisson algebra. The very Sklyanin algebra is associated with the following two quadrics in C 4 : Q 1 = x 2 1 + x 2 2 + x 2 3 , (2) Q 2 = x 2 4 + J 1 x 2 1 + J 2 x 2 2 + J 3 x 2 3 . (3) Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 The Poisson brackets (1) with λ = 1 between the affine coordinates looks as follows � ∂ Q k � { x i , x j } = ( − 1 ) i + j det , l � = i , j , i > j . (4) ∂ x l Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 A wide class of the polynomial Poisson algebras arises as a quasi-classical limit q n , k ( E ) of the associative quadratic algebras Q n , k ( E , η ) . Here E is an elliptic curve and n , k are integer numbers without common divisors ,such that 1 ≤ k < n while η is a complex number and Q n , k ( E , 0 ) = C [ x 1 , ..., x n ] . Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Feigin-Odesskii-Sklyanin algebras Let E = C / Γ be an elliptic curve defined by a lattice Γ = Z ⊕ τ Z , τ ∈ C , ℑ τ > 0. The algebra Q n , k ( E , η ) has generators x i , i ∈ Z / n Z subjected to the relations θ j − i + r ( k − 1 ) ( 0 ) � θ j − i − r ( − η ) θ kr ( η ) x j − r x i + r = 0 r ∈ Z / n Z and have the following properties: Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Basic properties Q n , k ( E , η ) = C ⊕ Q 1 ⊕ Q 2 ⊕ ... such that Q α ∗ Q β = Q α + β , here ∗ denotes the algebra multiplication. The algebras Q n , k ( E , η ) are Z - graded; The Hilbert function of Q n , k ( E , η ) is α ≥ 0 dim Q α t α = 1 � ( 1 − t ) n . Q n , k ( E , η ) ≃ Q n , k ′ ( E , η ) , if kk ′ ≡ 1 (mod n ); The maps x i �→ x i + 1 et x i �→ ε i x i , where ε n = 1 , define automorphisms of the algebra Q n , k ( E , η ); We see that the algebra Q n , k ( E , η ) for fixed E is a flat deformation of the polynomial ring C [ x 1 , ..., x n ] . Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Basic properties Q n , k ( E , η ) = C ⊕ Q 1 ⊕ Q 2 ⊕ ... such that Q α ∗ Q β = Q α + β , here ∗ denotes the algebra multiplication. The algebras Q n , k ( E , η ) are Z - graded; The Hilbert function of Q n , k ( E , η ) is α ≥ 0 dim Q α t α = 1 � ( 1 − t ) n . Q n , k ( E , η ) ≃ Q n , k ′ ( E , η ) , if kk ′ ≡ 1 (mod n ); The maps x i �→ x i + 1 et x i �→ ε i x i , where ε n = 1 , define automorphisms of the algebra Q n , k ( E , η ); We see that the algebra Q n , k ( E , η ) for fixed E is a flat deformation of the polynomial ring C [ x 1 , ..., x n ] . Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Basic properties Q n , k ( E , η ) = C ⊕ Q 1 ⊕ Q 2 ⊕ ... such that Q α ∗ Q β = Q α + β , here ∗ denotes the algebra multiplication. The algebras Q n , k ( E , η ) are Z - graded; The Hilbert function of Q n , k ( E , η ) is α ≥ 0 dim Q α t α = 1 � ( 1 − t ) n . Q n , k ( E , η ) ≃ Q n , k ′ ( E , η ) , if kk ′ ≡ 1 (mod n ); The maps x i �→ x i + 1 et x i �→ ε i x i , where ε n = 1 , define automorphisms of the algebra Q n , k ( E , η ); We see that the algebra Q n , k ( E , η ) for fixed E is a flat deformation of the polynomial ring C [ x 1 , ..., x n ] . Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Basic properties Q n , k ( E , η ) = C ⊕ Q 1 ⊕ Q 2 ⊕ ... such that Q α ∗ Q β = Q α + β , here ∗ denotes the algebra multiplication. The algebras Q n , k ( E , η ) are Z - graded; The Hilbert function of Q n , k ( E , η ) is α ≥ 0 dim Q α t α = 1 � ( 1 − t ) n . Q n , k ( E , η ) ≃ Q n , k ′ ( E , η ) , if kk ′ ≡ 1 (mod n ); The maps x i �→ x i + 1 et x i �→ ε i x i , where ε n = 1 , define automorphisms of the algebra Q n , k ( E , η ); We see that the algebra Q n , k ( E , η ) for fixed E is a flat deformation of the polynomial ring C [ x 1 , ..., x n ] . Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,

Introduction Examples of regular leave algebras, n = 3 Polynomial Poisson structures Heisenberg invariancy, Cremona and all that... Poisson algebras associated to elliptic curves. Cremona transformations and Poisson morphisms of P 4 Basic properties Q n , k ( E , η ) = C ⊕ Q 1 ⊕ Q 2 ⊕ ... such that Q α ∗ Q β = Q α + β , here ∗ denotes the algebra multiplication. The algebras Q n , k ( E , η ) are Z - graded; The Hilbert function of Q n , k ( E , η ) is α ≥ 0 dim Q α t α = 1 � ( 1 − t ) n . Q n , k ( E , η ) ≃ Q n , k ′ ( E , η ) , if kk ′ ≡ 1 (mod n ); The maps x i �→ x i + 1 et x i �→ ε i x i , where ε n = 1 , define automorphisms of the algebra Q n , k ( E , η ); We see that the algebra Q n , k ( E , η ) for fixed E is a flat deformation of the polynomial ring C [ x 1 , ..., x n ] . Vladimir Roubtsov "SQS’09", Dubna Workshop August, 3, 2009,