96,672 research outputs found
BRST operator for quantum Lie algebras and differential calculus on quantum groups
For a Hopf algebra A, we define the structures of differential complexes on
two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an
exterior extension of the dual algebra A^*. The Heisenberg double of these two
exterior Hopf algebras defines the differential algebra for the Cartan
differential calculus on A. The first differential complex is an analog of the
de Rham complex. In the situation when A^* is a universal enveloping of a Lie
(super)algebra the second complex coincides with the standard complex. The
differential is realized as an (anti)commutator with a BRST- operator Q. A
recurrent relation which defines uniquely the operator Q is given. The BRST and
anti-BRST operators are constructed explicitly and the Hodge decomposition
theorem is formulated for the case of the quantum Lie algebra U_q(gl(N)).Comment: 20 pages, LaTeX, Lecture given at the Workshop on "Classical and
Quantum Integrable Systems", 8 - 11 January, Protvino, Russia; corrected some
typo
Macaulay inverse systems revisited
Since its original publication in 1916 under the title "The Algebraic Theory
of Modular Systems", the book by F. S. Macaulay has attracted a lot of
scientists with a view towards pure mathematics (D. Eisenbud,...) or
applications to control theory (U. Oberst,...).However, a carefull examination
of the quotations clearly shows that people have hardly been looking at the
last chapter dealing with the so-called "inverse systems", unless in very
particular situations. The purpose of this paper is to provide for the first
time the full explanation of this chapter within the framework of the formal
theory of systems of partial differential equations (Spencer operator on
sections, involution,...) and its algebraic counterpart now called "algebraic
analysis" (commutative and homological algebra, differential modules,...). Many
explicit examples are fully treated and hints are given towards the way to work
out computer algebra packages.Comment: From a lecture at the International Conference : Application of
Computer Algebra (ACA 2008) july 2008, RISC, LINZ, AUSTRI
Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multi-dimensions
A direct method for the computation of polynomial conservation laws of
polynomial systems of nonlinear partial differential equations (PDEs) in
multi-dimensions is presented. The method avoids advanced
differential-geometric tools. Instead, it is solely based on calculus,
variational calculus, and linear algebra.
Densities are constructed as linear combinations of scaling homogeneous terms
with undetermined coefficients. The variational derivative (Euler operator) is
used to compute the undetermined coefficients. The homotopy operator is used to
compute the fluxes.
The method is illustrated with nonlinear PDEs describing wave phenomena in
fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters,
the method determines the conditions on the parameters so that a sequence of
conserved densities might exist. The existence of a large number of
conservation laws is a predictor for complete integrability. The method is
algorithmic, applicable to a variety of PDEs, and can be implemented in
computer algebra systems such as Mathematica, Maple, and REDUCE.Comment: To appear in: Thematic Issue on ``Mathematical Methods and Symbolic
Calculation in Chemistry and Chemical Biology'' of the International Journal
of Quantum Chemistry. Eds.: Michael Barnett and Frank Harris (2006
The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces
The Hamiltonian representation for the hierarchy of Lax-type flows on a dual
space to the Lie algebra of shift operators coupled with suitable
eigenfunctions and adjoint eigenfunctions evolutions of associated spectral
problems is found by means of a specially constructed Backlund transformation.
The Hamiltonian description for the corresponding set of squared eigenfunction
symmetry hierarchies is represented. The relation of these hierarchies with Lax
integrable (2+1)-dimensional differential-difference systems and their triple
Lax-type linearizations is analysed. The existence problem of a Hamiltonian
representation for the coupled Lax-type hierarchy on a dual space to the
central extension of the shift operator Lie algebra is solved also
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