Abstract. In this paper, we provide a new formulation for the generalized periodic Toda lattice. Since the work of Kostant, Adler and Symes, it has been known that the Toda lattice is related to the structure of simple Lie algebras. Indeed, the non-periodic and the periodic Toda lattices can be expressed as Hamiltonian systems on coadjoint orbits: the former of a simple Lie group and the latter...

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. ...

A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan’s method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras. A number of examples are calculated to illus...

Based on the generalized symmetry group method presented by Lou and Ma [Lou and Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A: Math. Gen. 38 (2005) L129], firstly, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between th...

The Berezin quantization on a simply connected homogeneous Kähler manifold, which is considered as a phase space for a dynam-ical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differentia...

We prove that, general σ-models related by Poisson–Lie T-duality are quantum equivalent under one-loop renormalization group flow. We reveal general properties of the flows, we study the associated generalized coset models and provide explicit examples.

depending on indeterminates x1, . . . , xn. Dyson conjectured the explicit form of this constant term. His conjecture was soon related with the root system of sl(n), generalized to other root systems of simple Lie algebras and proved. The expressions obtained for the Dyson constant are called Macdonald’s identities, see [M]. Let us briefly recall the main results. Let g be a simple (finite dime...