We construct a class of integrable generalization of Toda mechanics withlong-range interactions. These systems are associated with the loop algebras L(C_r) and L(D_r) inthe sense that their Lax matrices can he realize...We construct a class of integrable generalization of Toda mechanics withlong-range interactions. These systems are associated with the loop algebras L(C_r) and L(D_r) inthe sense that their Lax matrices can he realized in terms of the c = 0 representations of theaffine Lie algebras C_r~((1)) and D_r~((1)) and the interactions pattern involved bears the typicalcharacters of the corresponding root systems. We present the equations of motion and the Hamiltoninnstructure. These generalized systems can be identified unambiguously by specifying the underlyingloop algebra together with an ordered pair of integers (n, m). It turns out that different systemsassociated with the same underlying loop algebra but with different pairs of integers (n_1, m_1) and(n_2, m_2) with n_2 【 n_1 and m_2 【 m_2 can be related by a nested Hamiltonian reduction procedure.For all nontrivial generalizations, the extra coordinates besides the standard Toda variables arePoisson non-commute, and when either n or m ≥ 3, the Poisson structure for the extra coordinatevariables becomes some Lie algebra (i.e. the extra variables appear linearly on the right-hand sideof the Poisson brackets). In the quantum case, such generalizations will become systems withnoncommutative variables without spoiling the integrability.展开更多
文摘We construct a class of integrable generalization of Toda mechanics withlong-range interactions. These systems are associated with the loop algebras L(C_r) and L(D_r) inthe sense that their Lax matrices can he realized in terms of the c = 0 representations of theaffine Lie algebras C_r~((1)) and D_r~((1)) and the interactions pattern involved bears the typicalcharacters of the corresponding root systems. We present the equations of motion and the Hamiltoninnstructure. These generalized systems can be identified unambiguously by specifying the underlyingloop algebra together with an ordered pair of integers (n, m). It turns out that different systemsassociated with the same underlying loop algebra but with different pairs of integers (n_1, m_1) and(n_2, m_2) with n_2 【 n_1 and m_2 【 m_2 can be related by a nested Hamiltonian reduction procedure.For all nontrivial generalizations, the extra coordinates besides the standard Toda variables arePoisson non-commute, and when either n or m ≥ 3, the Poisson structure for the extra coordinatevariables becomes some Lie algebra (i.e. the extra variables appear linearly on the right-hand sideof the Poisson brackets). In the quantum case, such generalizations will become systems withnoncommutative variables without spoiling the integrability.
