191 research outputs found
Dualities between Poisson brackets and antibrackets
Recently it has been shown that antibrackets may be expressed in terms of
Poisson brackets and vice versa for commuting functions in the original
bracket. Here we also introduce generalized brackets involving higher
antibrackets or higher Poisson brackets where the latter are of a new type. We
give generating functions for these brackets for functions in arbitrary
involutions in the original bracket. We also give master equations for
generalized Maurer-Cartan equations. The presentation is completely symmetric
with respect to Poisson brackets and antibrackets.Comment: 24 pages,Latexfile,corrected (2.7-8) and removed text between (2.9)
and (2.10
Cohomology of Lie superalgebras and of their generalizations
The cohomology groups of Lie superalgebras and, more generally, of color Lie
algebras, are introduced and investigated. The main emphasis is on the case
where the module of coefficients is non-trivial. Two general propositions are
proved, which help to calculate the cohomology groups. Several examples are
included to show the peculiarities of the super case. For L = sl(1|2), the
cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple
graded L-module, are determined, and the result is used to show that
H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies
that the superalgebra U(L) does not admit of any non-trivial formal
deformations (in the sense of Gerstenhaber). Garland's theory of universal
central extensions of Lie algebras is generalized to the case of color Lie
algebras.Comment: 50 pages, Latex, no figures. In the revised version the proof of
Lemma 5.1 is greatly simplified, some references are added, and a pertinent
result on sl(m|1) is announced. To appear in the Journal of Mathematical
Physic
Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices
This paper is mainly a review of the multi--Hamiltonian nature of Toda and
generalized Toda lattices corresponding to the classical simple Lie groups but
it includes also some new results. The areas investigated include master
symmetries, recursion operators, higher Poisson brackets, invariants and group
symmetries for the systems. In addition to the positive hierarchy we also
consider the negative hierarchy which is crucial in establishing the
bi--Hamiltonian structure for each particular simple Lie group. Finally, we
include some results on point and Noether symmetries and an interesting
connection with the exponents of simple Lie groups. The case of exceptional
simple Lie groups is still an open problem.Comment: 65 pages, 67 reference
Linear connections on matrix geometries
A general definition of a linear connection in noncommutative geometry has
been recently proposed. Two examples are given of linear connections in
noncommutative geometries which are based on matrix algebras. They both possess
a unique metric connection.Comment: 14p, LPTHE-ORSAY 94/9
Super-Poincare' algebras, space-times and supergravities (I)
A new formulation of theories of supergravity as theories satisfying a
generalized Principle of General Covariance is given. It is a generalization of
the superspace formulation of simple 4D-supergravity of Wess and Zumino and it
is designed to obtain geometric descriptions for the supergravities that
correspond to the super Poincare' algebras of Alekseevsky and Cortes'
classification.Comment: 29 pages, v2: minor improvements at the end of Section 5.
A Unified Algebraic Approach to Classical Yang-Baxter Equation
In this paper, the different operator forms of classical Yang-Baxter equation
are given in the tensor expression through a unified algebraic method. It is
closely related to left-symmetric algebras which play an important role in many
fields in mathematics and mathematical physics. By studying the relations
between left-symmetric algebras and classical Yang-Baxter equation, we can
construct left-symmetric algebras from certain classical r-matrices and
conversely, there is a natural classical r-matrix constructed from a
left-symmetric algebra which corresponds to a parak\"ahler structure in
geometry. Moreover, the former in a special case gives an algebraic
interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a
classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica
Lagrange structure and quantization
A path-integral quantization method is proposed for dynamical systems whose
classical equations of motion do \textit{not} necessarily follow from the
action principle. The key new notion behind this quantization scheme is the
Lagrange structure which is more general than the Lagrangian formalism in the
same sense as Poisson geometry is more general than the symplectic one. The
Lagrange structure is shown to admit a natural BRST description which is used
to construct an AKSZ-type topological sigma-model. The dynamics of this
sigma-model in dimensions, being localized on the boundary, are proved to
be equivalent to the original theory in dimensions. As the topological
sigma-model has a well defined action, it is path-integral quantized in the
usual way that results in quantization of the original (not necessarily
Lagrangian) theory. When the original equations of motion come from the action
principle, the standard BV path-integral is explicitly deduced from the
proposed quantization scheme. The general quantization scheme is exemplified by
several models including the ones whose classical dynamics are not variational.Comment: Minor corrections, format changed, 40 page
Poisson brackets with prescribed Casimirs
We consider the problem of constructing Poisson brackets on smooth manifolds
with prescribed Casimir functions. If is of even dimension, we achieve
our construction by considering a suitable almost symplectic structure on ,
while, in the case where is of odd dimension, our objective is achieved by
using a convenient almost cosymplectic structure. Several examples and
applications are presented.Comment: 24 page
Nambu-Poisson manifolds and associated n-ary Lie algebroids
We introduce an n-ary Lie algebroid canonically associated with a
Nambu-Poisson manifold. We also prove that every Nambu-Poisson bracket defined
on functions is induced by some differential operator on the exterior algebra,
and characterize such operators. Some physical examples are presented
Linear Connections on Fuzzy Manifolds
Linear connections are introduced on a series of noncommutative geometries
which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23
- …