Integrable Hamiltonian systems generated by antisymmetric matrices

Transcription

1 Journal of Physics: Conference Series OPEN ACCESS Integrable Hamiltonian systems generated by antisymmetric matrices To cite this article: Alina Dobrogowska 013 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - Kinematics of semiclassical spin and spin fiber bundle associated with son Lie- Poisson manifold A A Deriglazov - Hamiltonian structure of an operator valued extension of Super KdV equations Alvaro Restuccia and Adrián Sotomayor - Poisson structure and stability analysis of a coupled system arising from the supersymmetric breaking of Super KdV Adrián Sotomayor and Alvaro Restuccia This content was downloaded from IP address on 08/06/018 at 01:49

2 XXIst International Conference on Integrable Systems and Quantum Symmetries ISQS1 IOP Publishing Journal of Physics: Conference Series doi: / /474/1/01015 Integrable Hamiltonian systems generated by antisymmetric matrices Alina Dobrogowska Institute of Mathematics, University of Białystok Akademicka, Białystok, Poland Abstract. We construct a family of integrable systems generated by the Casimir functions of Lie algebra of skew-symmetric matrices, where the Lie bracket is deformed by a symmetric matrix. 1. Introduction Let An be the vector space of antisymmetric n n matrices and Symn be the vector space of symmetric n n matrices. The An, [, ] S is a Lie algebra with the S bracket defined by a deformeded commutator [X, Y ] S = XSY Y SX 1 for fixed S Symn and X, Y An, see [4, 5]. In this paper we construct the family of integrable systems a hierarchy generated by the Casimir functions on the dual of Lie algebra An. We prove that the integrals of this family of Hamiltonian systems are in involution. The idea of considering these systems comes from [1]. In this paper we present more general case, which reduces to the case considering in [1] if we put that the matrix S = 1. Also in [3] the authors studied similar systems in the complex setting and for matrices with a different internal structure.. Hierarchy generated by Casimir functions We identify An with its dual A n = An using natural non-degenerate pairing by trace of the product X, ρ = TrρX, ρ A n, X An. We shall write a general element An as X = A B B C, 3 where A A, C An and B Mat n R. Having Lie algebra An, [, ] S one defines the Lie-Poisson bracket on C An by [ {f, g} S = Tr X X, g ], f, g C An, 4 X Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 S

3 XXIst International Conference on Integrable Systems and Quantum Symmetries ISQS1 IOP Publishing Journal of Physics: Conference Series doi: / /474/1/01015 where X = A, 5 In order to obtain the second Poisson bracket "frozen" Poisson bracket we fix the element X 0 An and put [ {f, g} F S = Tr X 0 X, g ], f, g C An. 6 X S It is a general fact that the Lie-Poisson bracket and frozen bracket are compatible in the sense that their linear combination α{, } S β{, } F S 7 is also a Poisson bracket. We shall choose X 0 = 1 where A 0 is matrix defined by A A 0 := S1 S, S = 3 S3 S 0 1 0, 8. 9 After simple calculation we show that the Poisson bracket 6 can be written in the form g {f, g} F S = Tr A A 0 A A g 0 S3 10 Tr A g 0 S. Basic assumption. From now we put the block S 3 equal to zero S 3 0. After reducing to this case we obtain {f, g} F S = Tr A g 0 S. 11 Thus, we can think of this bracket as being defined on C Mat n R thus An Mat n R is injective smooth Poisson map. In the case when det S 0 the Casimir functions for the Lie Poisson bracket 4 are given by C k X = 1 k TrXS k, k = 1,,... 1 see [5]. For the degenerate case when S 1 0 we know only some Casimir functions of the following form C k X = 1 k k Tr B BS, k = 1,,..., 13 see [] for the case S = 1. In this case the Lie-Poisson bracket 4 can be rewritten in the form {f, g} S = Tr A g S g C S 14 Tr B g g B S.

4 XXIst International Conference on Integrable Systems and Quantum Symmetries ISQS1 IOP Publishing Journal of Physics: Conference Series doi: / /474/1/01015 Now, we show that the functions given by 13 are Casimir functions for the bracket 14. Since the derivative of C k is k k = BS B BS, 15 k k = S B BS B, 16 k = 0, 17 we have k {C k, C l } S = Tr A l S = k l = 4 Tr S B BS B ABS B BS S = 18 kl = 4 Tr B ABS B BS = 0 = {C k, C l } F S, because the matrix B AB is antisymmetric and S we have the following proposition. B BS kl is symmetric. Moreover Proposition 1 The Casimir functions C k defined by 1 or 13 for the Lie-Poisson bracket 4 considered as functions of B are in involution with respect to the frozen bracket 11. Proof 1 Since the derivative of C k given by 1 is k = P XS k P, 19 k = P S X k P, 0 where P, P are the orthogonal projectors given, in block matrix notation, by P =, P 0 0 = After a direct calculation we obtain k {C k, C l } F S = Tr A l S = P S X k P A 0 P XS l P S = P XS k XP A 0 P XS l P = XS k XP A 0 P XS l 4 Tr P XS k XP A 0 P XS l P = XS kl XP A 0 P X S X kl S = 0. Above vanishes because in the first term we have a product of three antisymmetric matrices which is also antisymmetric and in the second term we have a product of an antisymmetric matrix XS kl XP A 0 P X S X kl and symmetric matrix S. The proof of the involution of the functions 13 with respect to the frozen Poisson bracket 11, was given before this proposition. 3

5 XXIst International Conference on Integrable Systems and Quantum Symmetries ISQS1 IOP Publishing Journal of Physics: Conference Series doi: / /474/1/01015 Proposition The smooth functions δ k : Mat n R R defined by δ k B = Tr BS k B A 0 3 are in involution with respect to the frozen Poisson bracket 11 Proof Since the derivative of δ k is {δ k, δ l } F S = 0. 4 we have δk {δ k, δ l } F S = Tr A 0 = 4 Tr δ k = A 0BS δ k = S S k, 5 k B A 0, 6 δ l S = 7 k B A 0 A 0 A 0 BS l S = B A 0 BS kl = 0, because the matrix B A 0 B is antisymmetric and S kl is symmetric. Proposition 3 Assume that S 1 = 1. Then the functions δ k and C l given by 1, k, l = 1,,..., are in involution with respect to the frozen Poisson bracket 11 {δ k, C l } F S = 0. 8 Proof 3 First, we show that δ 1 commutes with C k given by 1 δ1 {δ 1, C k } F S = Tr A k 0 = 9 S B A 0 A 0 P XS k P S = = 4 Tr B P XS k P = P XS P XP XS k P = XS P XP XS k 4 Tr P XS P XP XS k P = P XP XS k 4 Tr P XP XS k P XP S = XP XS k 4 Tr P XP XS k P = P XS k XP = 0, 4

6 XXIst International Conference on Integrable Systems and Quantum Symmetries ISQS1 IOP Publishing Journal of Physics: Conference Series doi: / /474/1/01015 because P XS k XP is antisymmetric, P XP is antisymmetric and P XS k P is symmetric. Second, the functions δ k and C l satisfy the following recursion formula Thus the relation 8 is valid for any k. {δ k, C l } F S = {δ k, C l1 } F S. 30 Proposition 4 The functions δ k and C l given by 13, k, l = 1,,..., are in involution with respect to the frozen Poisson bracket 11 Proof 4 For the functions C k given by 13 we have δk {δ k, C l } F S = Tr = 4 Tr A 0 S {δ k, C l } F S = l = 3 l S k B A 0 A 0 BS l k B BS B BS = = 0, 33 because the matrix k C is antisymmetric and S B BS l is symmetric. We obtain a hierarchy of Hamilton s equations generated by Hamiltonians C k given by 1 or 13 with respect the frozen Poisson bracket 11 t k = A 0 k S, k = 1,, Example 1 In this example we consider the case when X is 5 5-matrix which we denote 0 a p 1 p p 3 a 0 q 1 q q 3 X = p 1 q 1 0 c 3 c 35 p q c 3 0 c 1 p 3 q 3 c c 1 0 and matrix S is degenerate, that mean S 1 = 0 and S = e e e 3 The frozen Poisson bracket in this case is g {f, g} F S p i, q i =e 1 g g e g p 1 q 1 q 1 p 1 p q q p g e 3 g p 3 q 3 q p

7 XXIst International Conference on Integrable Systems and Quantum Symmetries ISQS1 IOP Publishing Journal of Physics: Conference Series doi: / /474/1/01015 The integrals in involution are Hamilton s equations for the Hamiltonian C 1 are Hamilton s equations for the Hamiltonian C are C 1 = S p p S q q, 38 C = 1 C 1 S q S p q p, 39 δ 1 = C S q S p. 40 p = q, t 41 q = p. t 4 p t = C 1 q S p q, 43 q t = C 1 p S p q. 44 References [1] A. Dobrogowska, T.S. Ratiu, Integrable systems of Neumann type, J. Dyn. Diff. Equat., DOI : /s , 013. [] A. Odzijewicz, A. Dobrogowska, Integrable Hamiltonian systems related to the Hilbert Schmidt ideal, J. Geom. Phys., 61, [3] A. Odzijewicz, T. Goliński, Hierarchy of integrable Hamiltonians describing the nonlinear n-wave interaction, J. Phys. A Math. Theor., 45, no [4] A.B. Yanovski, Linear bundles of lie algebras and their applications, J. Math. Phys., 41, No. 1, 000. [5] V.V. Trofimov, A.T. Fomenko, Algebra and geometry of integrable Hamiltonian differential equations, Faktorial, Moscow, Russian