240 research outputs found
Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies
Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade
one regular semisimple elements from non-equivalent Heisenberg subalgebras of a
loop algebra \G\otimes{\bf C}[\lambda,\lambda^{-1}] are studied. The graded
Heisenberg subalgebras containing such elements are labelled by the regular
conjugacy classes in the Weyl group {\bf W}(\G) of the simple Lie algebra
\G. A representative w\in {\bf W}(\G) of a regular conjugacy class can be
lifted to an inner automorphism of \G given by , where is the defining vector of an subalgebra
of \G.The grading is then defined by the operator and any grade one regular element from the
Heisenberg subalgebra associated to takes the form , where and is included in an
subalgebra containing . The largest eigenvalue of is
except for some cases in , . We explain how these Lie
algebraic results follow from known results and apply them to construct
integrable systems.If the largest eigenvalue is , then
using any grade one regular element from the Heisenberg subalgebra associated
to we can construct a KdV system possessing the standard \W-algebra
defined by as its second Poisson bracket algebra. For \G a classical
Lie algebra, we derive pseudo-differential Lax operators for those
non-principal KdV systems that can be obtained as discrete reductions of KdV
systems related to . Non-abelian Toda systems are also considered.Comment: 44 pages, ENSLAPP-L-493/94, substantial revision, SWAT-95-77. (use
OLATEX (preferred) or LATEX
Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction
The matrix version of the -KdV hierarchy has been recently
treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian
symmetry reduction applied to a Poisson submanifold in the dual of the Lie
algebra . Here a
series of extensions of this matrix Gelfand-Dickey system is derived by means
of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra
using the natural
embedding for any positive integer. The
hierarchies obtained admit a description in terms of a matrix
pseudo-differential operator comprising an -KdV type positive part and a
non-trivial negative part. This system has been investigated previously in the
case as a constrained KP system. In this paper the previous results are
considerably extended and a systematic study is presented on the basis of the
Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson
brackets and makes clear the conformal (-algebra) structures related to
the KdV type hierarchies. Discrete reductions and modified versions of the
extended -KdV hierarchies are also discussed.Comment: 60 pages, plain TE
On Matrix KP and Super-KP Hierarchies in the Homogeneous Grading
Constrained KP and super-KP hierarchies of integrable equations (generalized
NLS hierarchies) are systematically produced through a Lie algebraic AKS-matrix
framework associated to the homogeneous grading. The role played by different
regular elements to define the corresponding hierarchies is analyzed as well as
the symmetry properties under the Weyl group transformations. The coset
structure of higher order hamiltonian densities is proven.\par For a generic
Lie algebra the hierarchies here considered are integrable and essentially
dependent on continuous free parameters. The bosonic hierarchies studied in
\cite{{FK},{AGZ}} are obtained as special limit restrictions on hermitian
symmetric-spaces.\par In the supersymmetric case the homogeneous grading is
introduced consistently by using alternating sums of bosons and fermions in the
spectral parameter power series.\par The bosonic hierarchies obtained from
and the supersymmetric ones derived from the
affinization of , and are explicitly constructed.
\par An unexpected result is found: only a restricted subclass of the
bosonic hierarchies can be supersymmetrically extended while preserving
integrability.Comment: 36 pages, LaTe
The Shapovalov determinant for the Poisson superalgebras
Among simple Z-graded Lie superalgebras of polynomial growth, there are
several which have no Cartan matrix but, nevertheless, have a quadratic even
Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields
on the (1|6)-dimensional supercircle preserving the contact form, and the
series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian
fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using
C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for
the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov
described irreducible finite dimensional representations of po(0|n) and
sh(0|n); we generalize his result for Verma modules: give criteria for
irreducibility of the Verma modules over po(0|2k) and sh(0|2k)
Polar orthogonal representations of real reductive algebraic groups
We prove that a polar orthogonal representation of a real reductive algebraic
group has the same closed orbits as the isotropy representation of a
pseudo-Riemannian symmetric space. We also develop a partial structural theory
of polar orthogonal representations of real reductive algebraic groups which
slightly generalizes some results of the structural theory of real reductive
Lie algebras.Comment: 23 pages, Late
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