200 research outputs found
On the nature of the Virasoro algebra
The multiplication in the Virasoro algebra comes from the commutator in a quasiassociative algebra with the multiplication
\renewcommand{\theequation}{} \be \ba{l} \ds e_p * e_q = - {q (1 + \epsilon
q) \over 1 + \epsilon (p + q)} e_{p+q} + {1 \over 2} \theta \left[p^3 - p +
\left(\epsilon - \epsilon^{-1} \right) p^2 \right] \delta^0_{p+q},
\vspace{3mm}\\ \ds e_p * \theta = \theta* e_p = 0. \ea \ee The multiplication
in a quasiassociative algebra satisfies the property
\renewcommand{\theequation}{} \be a * (b * c) - (a * b) * c = b * (a * c) -
(b * a) * c, \qquad a, b, c \in {\cal R}. \ee This property is necessary and
sufficient for the Lie algebra {\it Lie} to have a phase space. The
above formulae are put into a cohomological framework, with the relevant
complex being different from the Hochschild one even when the relevant
quasiassociative algebra becomes associative. Formula above
also has a differential-variational counterpart
Coadjoint Poisson actions of Poisson-Lie groups
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie
algebra induces on it a non-trivial class of quadratic Poisson structures
extending the linear Poisson bracket on the coadjoint orbits
Quadratic Poisson brackets compatible with an algebra structure
Quadratic Poisson brackets on a vector space equipped with a bilinear
multiplication are studied. A notion of a bracket compatible with the
multiplication is introduced and an effective criterion of such compatibility
is given. Among compatible brackets, a subclass of coboundary brackets is
described, and such brackets are enumerated in a number of examples.Comment: 6 page
r-matrices for relativistic deformations of integrable systems
We include the relativistic lattice KP hierarchy, introduced by Gibbons and
Kupershmidt, into the -matrix framework. An -matrix account of the
nonrelativistic lattice KP hierarchy is also provided for the reader's
convenience. All relativistic constructions are regular one-parameter
perturbations of the nonrelativistic ones. We derive in a simple way the linear
Hamiltonian structure of the relativistic lattice KP, and find for the first
time its quadratic Hamiltonian structure. Amasingly, the latter turns out to
coincide with its nonrelativistic counterpart (a phenomenon, known previously
only for the simplest case of the relativistic Toda lattice)
On the Miura map between the dispersionless KP and dispersionless modified KP hierarchies
We investigate the Miura map between the dispersionless KP and dispersionless
modified KP hierarchies. We show that the Miura map is canonical with respect
to their bi-Hamiltonian structures. Moreover, inspired by the works of Takasaki
and Takebe, the twistor construction of solution structure for the
dispersionless modified KP hierarchy is given.Comment: 19 pages, Latex, no figure
2D Quantum Gravity and The Miura Map
We study the sL(3,C) mKDV string theories. We obtain the flows and the string
equations. Using the generalized Miura map, we show that we have an unification
of these models with the [P,Q]=Q sL(3,C) KDV ones in the framework of
open-closed string theories in minimal models backgrounds.Comment: 12 pages, phyzz
On the B\"acklund Transformation for the Moyal Korteweg-de Vries Hierarchy
We study the B\"acklund symmetry for the Moyal Korteweg-de Vries (KdV)
hierarchy based on the Kuperschmidt-Wilson Theorem associated with second
Gelfand-Dickey structure with respect to the Moyal bracket, which generalizes
the result of Adler for the ordinary KdV.Comment: 9 pages, Revte
Dispersionful analogues of Benney's equations and -wave systems
We recall Krichever's construction of additional flows to Benney's hierarchy,
attached to poles at finite distance of the Lax operator. Then we construct a
``dispersionful'' analogue of this hierarchy, in which the role of poles at
finite distance is played by Miura fields. We connect this hierarchy with
-wave systems, and prove several facts about the latter (Lax representation,
Chern-Simons-type Lagrangian, connection with Liouville equation,
-functions).Comment: 12 pages, latex, no figure
Nonstandard Drinfeld-Sokolov reduction
Subject to some conditions, the input data for the Drinfeld-Sokolov
construction of KdV type hierarchies is a quadruplet (\A,\Lambda, d_1, d_0),
where the are -gradations of a loop algebra \A and \Lambda\in \A
is a semisimple element of nonzero -grade. A new sufficient condition on
the quadruplet under which the construction works is proposed and examples are
presented. The proposal relies on splitting the -grade zero part of \A
into a vector space direct sum of two subalgebras. This permits one to
interpret certain Gelfand-Dickey type systems associated with a nonstandard
splitting of the algebra of pseudo-differential operators in the
Drinfeld-Sokolov framework.Comment: 19 pages, LaTeX fil
Conformal Covariantization of Moyal-Lax Operators
A covariant approach to the conformal property associated with Moyal-Lax
operators is given. By identifying the conformal covariance with the second
Gelfand-Dickey flow, we covariantize Moyal-Lax operators to construct the
primary fields of one-parameter deformation of classical -algebras.Comment: 13 pages, Revtex, no figures, v.2: typos corrected, references added
and conclusion modifie
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