arXiv subjects
Publications and source records attributed to Michael Forger.
We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.
A new Poisson bracket for Hamiltonian forms on the full multisymplectic phase space is defined. At least for forms of degree n-1, where n is the dimension of space-time, Jacobi's identity is fulfilled.
We develop a new, systematic approach towards studying the integrability of the ordinary Calogero-Moser-Sutherland models as well as the elliptic Calogero models associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras. It is based on the introduction of a function F, defined on the relevant root system and with values in the respective Cartan subalgebra, satisfying a certain set of combinatoric identities that ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, given by completely explicit formulas. It is shown that among the simple Lie algebras, only those belonging to the A-series admit such a function F, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p,q)/S(U(p) x U(q)), allows non-trivial solutions when |p-q| <= 1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our approach provides new ones, namely for the ordinary models when |p-q| = 1 and for the elliptic models when |p-q| = 1 or p = q.
Continuing our attempt to explain the degeneracy of the genetic code using basic classical Lie superalgebras, we present the branching schemes for the typical codon representations (typical 64-dimensional irreducible representations) of basic classical Lie superalgebras and find three schemes that do reproduce the degeneracies of the standard code, based on the orthosymplectic algebra osp(5|2) and differing only in details of the symmetry breaking pattern during the last step.