423 research outputs found
Dirac reduction revisited
The procedure of Dirac reduction of Poisson operators on submanifolds is
discussed within a particularly useful special realization of the general
Marsden-Ratiu reduction procedure. The Dirac classification of constraints on
'first-class' constraints and 'second-class' constraints is reexamined.Comment: This is a revised version of an article published in J. Nonlinear
Math. Phys. vol. 10, No. 4, (2003), 451-46
Algebraic properties of Manin matrices 1
We study a class of matrices with noncommutative entries, which were first
considered by Yu. I. Manin in 1988 in relation with quantum group theory. They
are defined as "noncommutative endomorphisms" of a polynomial algebra. More
explicitly their defining conditions read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). The basic claim
is that despite noncommutativity many theorems of linear algebra hold true for
Manin matrices in a form identical to that of the commutative case. Moreover in
some examples the converse is also true. The present paper gives a complete
list and detailed proofs of algebraic properties of Manin matrices known up to
the moment; many of them are new. In particular we present the formulation in
terms of matrix (Leningrad) notations; provide complete proofs that an inverse
to a M.m. is again a M.m. and for the Schur formula for the determinant of a
block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered
recently [arXiv:0809.3516], which includes the classical Capelli and related
identities. We also discuss many other properties, such as the Cramer formula
for the inverse matrix, the Cayley-Hamilton theorem, Newton and
MacMahon-Wronski identities, Plucker relations, Sylvester's theorem, the
Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, some
multiplicativity properties for the determinant, relations with
quasideterminants, calculation of the determinant via Gauss decomposition,
conjugation to the second normal (Frobenius) form, and so on and so forth. We
refer to [arXiv:0711.2236] for some applications.Comment: 80 page
Algebraic properties of Manin matrices II: q-analogues and integrable systems
We study a natural q-analogue of a class of matrices with noncommutative
entries, which were first considered by Yu. I. Manin in 1988 in relation with
quantum group theory, (called Manin Matrices in [5]) . These matrices we shall
call q-Manin matrices(qMMs). They are defined, in the 2x2 case, by the
relations M_21 M_12 = q M_12 M_21; M_22 M_12 = q M_12 M_22; [M_11;M_22] = 1/q
M_21 M_12 - q M_12 M_21: They were already considered in the literature,
especially in connection with the q-Mac Mahon master theorem [16], and the
q-Sylvester identities [25]. The main aim of the present paper is to give a
full list and detailed proofs of algebraic properties of qMMs known up to the
moment and, in particular, to show that most of the basic theorems of linear
algebras (e.g., Jacobi ratio theorems, Schhur complement, the Cayley-Hamilton
theorem and so on and so forth) have a straightforward counterpart for q-Manin
matrices. We also show how this classs of matrices ?ts within the theory of
quasi-determninants of Gel'fand-Retakh and collaborators (see, e.g., [17]). In
the last sections of the paper, we frame our definitions within the tensorial
approach to non-commutative matrices of the Leningrad school, and we show how
the notion of q-Manin matrix is related to theory of Quantum Integrable
Systems.Comment: 62 pages, v.2 cosmetic changes, typos fixe
Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
Gaudin algebras form a family of maximal commutative subalgebras in the
tensor product of copies of the universal enveloping algebra U(\g) of a
semisimple Lie algebra \g. This family is parameterized by collections of
pairwise distinct complex numbers . We obtain some new commutative
subalgebras in U(\g)^{\otimes n} as limit cases of Gaudin subalgebras. These
commutative subalgebras turn to be related to the hamiltonians of bending flows
and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of
spectrum in the Gaudin model for some new cases.Comment: 11 pages, references adde
A Note on Fractional KdV Hierarchies
We introduce a hierarchy of mutually commuting dynamical systems on a finite
number of Laurent series. This hierarchy can be seen as a prolongation of the
KP hierarchy, or a ``reduction'' in which the space coordinate is identified
with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV
hierarchies are gotten by means of further reductions, obtained by constraining
the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are
discussed in detail.Comment: Final version to appear in J. Math. Phys. Some changes in the order
of presentation, with more emphasis on the geometrical picture. One figure
added (using epsf.sty). 30 pages, Late
Algebraic properties of Manin matrices II: q-analogues and integrable systems
We study a natural q-analogue of a class of matrices with non-commutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory, (called Manin matrices in [5]). We call these q-analogues q-Manin matrices . These matrices are defined, in the 2×22×2 case by the following relations among their matrix entries:
M21M12=qM12M21, M22M12 = qM12M22
[M11,M22]=q-1M21M12-qM12M21
They were already considered in the literature, especially in connection with the q-MacMahon master theorem [10], and the q-Sylvester identities [22]. The main aim of the present paper is to give a full list and detailed proofs of the algebraic properties of q-Manin matrices known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schur complement, the Cayley–Hamilton theorem and so on and so forth) have a straightforward counterpart for such a class of matrices. We also show how q-Manin matrices fit within the theory of quasideterminants of Gelfand–Retakh and collaborators (see, e.g., [11]). We frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school in the last sections. We finally discuss how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems
Gaudin Models and Bending Flows: a Geometrical Point of View
In this paper we discuss the bihamiltonian formulation of the (rational XXX)
Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued
spins. In particular, we focus on the homogeneous models. We find a pencil of
Poisson brackets that recursively define a complete set of integrals of the
motion, alternative to the set of integrals associated with the 'standard' Lax
representation of the Gaudin model. These integrals, in the case of su(2),
coincide wih the Hamiltonians of the 'bending flows' in the moduli space of
polygons in Euclidean space introduced by Kapovich and Millson. We finally
address the problem of separability of these flows and explicitly find
separation coordinates and separation relations for the r=2 case.Comment: 27 pages, LaTeX with amsmath and amssym
The Sato Grassmannian and the CH hierarchy
We discuss how the Camassa-Holm hierarchy can be framed within the geometry
of the Sato Grassmannian.Comment: 10 pages, no figure
On Separation of Variables for Integrable Equations of Soliton Type
We propose a general scheme for separation of variables in the integrable
Hamiltonian systems on orbits of the loop algebra
. In
particular, we illustrate the scheme by application to modified Korteweg--de
Vries (MKdV), sin(sinh)-Gordon, nonlinear Schr\"odinger, and Heisenberg
magnetic equations.Comment: 22 page
An original growth mode of MWCNTs on alumina supported iron catalysts
Multi-walled carbon nanotubes (MWCNTs) have been produced from ethylene by Fluidized Bed Catalytic Chemical Vapor Deposition (FB-CCVD) on alumina supported iron catalyst powders. Both catalysts and MWCNTs-catalyst composites have been characterized by XRD, SEM-EDX, TEM, Mössbauer Spectroscopy, TGA and nitrogen adsorption measurements at different stages of the process. The fresh catalyst is composed of amorphous iron (III) oxide nanoparticles located inside the porosity of the support and of a micrometric crystalline &-iron (III) oxide surface film. The beginning of the CVD process provokes a brutal reconstruction and simultaneous carburization of the surface film that allows MWCNT nucleation and growth. These MWCNTs grow aligned between the support and the surface catalytic film, leading to a uniform consumption and uprising of the film. When the catalytic film has been consumed, the catalytic particles located inside the alumina porosity are slowly reduced and activated leading to a secondary MWCNT growth regime, which produces a generalized grain explosion and entangled MWCNT growth. Based on experimental observations and characterizations, this original two-stage growth mode is discussed and a general growth mechanism is proposed
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