For each of the Lie algebras gln and gl～n., we construct a family of integrable generalizations of the Toda chains characterized by two integers m+ and m-. The Lax matrices and the equations of motion are given expli...For each of the Lie algebras gln and gl～n., we construct a family of integrable generalizations of the Toda chains characterized by two integers m+ and m-. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m+ = m-,we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic (m+,m-)-Toda chains, which turns out to be very different for different pairs of m+ and m-. Finally we also obtain thenonabelian generalizations of the (m+, m-)-Toda chains in an explicit form.展开更多
A class of generalization of Toda mechanics with long range interactions isconstructed in this paper. These systems are associated with the loop algebras L(B_r) in the sensethat their Lax matrices can be realized in t...A class of generalization of Toda mechanics with long range interactions isconstructed in this paper. These systems are associated with the loop algebras L(B_r) in the sensethat their Lax matrices can be realized in terms of the c = 0 representations of the affine Liealgebras B_r~((1)) . We adopt a pair of ordered integers (m, n) to describe the Toda mechanicssystem when we present the equations of motion and the Hamiltonian structure. We also extract theclassical r matrix which satisfy the classical Yang-Baxter relation. Such generalizations willbecome systems with noncommutative variables in the quantum case.展开更多
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For each of the Lie algebras gln and gl～n., we construct a family of integrable generalizations of the Toda chains characterized by two integers m+ and m-. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m+ = m-,we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic (m+,m-)-Toda chains, which turns out to be very different for different pairs of m+ and m-. Finally we also obtain thenonabelian generalizations of the (m+, m-)-Toda chains in an explicit form.
文摘A class of generalization of Toda mechanics with long range interactions isconstructed in this paper. These systems are associated with the loop algebras L(B_r) in the sensethat their Lax matrices can be realized in terms of the c = 0 representations of the affine Liealgebras B_r~((1)) . We adopt a pair of ordered integers (m, n) to describe the Toda mechanicssystem when we present the equations of motion and the Hamiltonian structure. We also extract theclassical r matrix which satisfy the classical Yang-Baxter relation. Such generalizations willbecome systems with noncommutative variables in the quantum case.
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.
