8 research outputs found
Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size
We construct affinization of the algebra of ``complex size''
matrices, that contains the algebras for integral values of the
parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra
results in the quadratic Gelfand--Dickey structure on the
Poisson--Lie group of all pseudodifferential operators of fractional order.
This construction is extended to the simultaneous deformation of orthogonal and
simplectic algebras that produces self-adjoint operators, and it has a
counterpart for the Toda lattices with fractional number of particles.Comment: 29 pages, no figure
Poisson-Lie group of pseudodifferential symbols
We introduce a Lie bialgebra structure on the central extension of the Lie
algebra of differential operators on the line and the circle (with scalar or
matrix coefficients). This defines a Poisson--Lie structure on the dual group
of pseudodifferential symbols of an arbitrary real (or complex) order. We show
that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP
Poisson structures are naturally realized as restrictions of this Poisson
structure to submanifolds of this ``universal'' Poisson--Lie group.
Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in
physical terminology) can be viewed as subspaces of the quotient (or Poisson
reduction) of this Poisson--Lie group by the dressing action of the group of
functions.
Finally, we define an infinite set of functions in involution on the
Poisson--Lie group that give the standard families of Hamiltonians when
restricted to the submanifolds mentioned above. The Poisson structure and
Hamiltonians on the whole group interpolate between the Poisson structures and
Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical
meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes
to 0.Comment: 64 pages, no figure
Representation theory of finite W algebras
In this paper we study the finitely generated algebras underlying
algebras. These so called 'finite algebras' are constructed as Poisson
reductions of Kirillov Poisson structures on simple Lie algebras. The
inequivalent reductions are labeled by the inequivalent embeddings of
into the simple Lie algebra in question. For arbitrary embeddings a coordinate
free formula for the reduced Poisson structure is derived. We also prove that
any finite algebra can be embedded into the Kirillov Poisson algebra of a
(semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that
generalized finite Toda systems are reductions of a system describing a free
particle moving on a group manifold and that they have finite symmetry. In
the second part we BRST quantize the finite algebras. The BRST cohomology
is calculated using a spectral sequence (which is different from the one used
by Feigin and Frenkel). This allows us to quantize all finite algebras in
one stroke. Explicit results for and are given. In the last part
of the paper we study the representation theory of finite algebras. It is
shown, using a quantum version of the generalized Miura transformation, that
the representations of finite algebras can be constructed from the
representations of a certain Lie subalgebra of the original simple Lie algebra.
As a byproduct of this we are able to construct the Fock realizations of
arbitrary finite algebras.Comment: 62 pages, THU-92/32, ITFA-28-9
A bi-Hamiltonian supersymmetric geodesic equation
A supersymmetric extension of the Hunter-Saxton equation is constructed. We
present its bi-Hamiltonian structure and show that it arises geometrically as a
geodesic equation on the space of superdiffeomorphisms of the circle that leave
a point fixed endowed with a right-invariant metric.Comment: 9 pages, no figure