422 research outputs found
Algebraic theories of brackets and related (co)homologies
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets
in the category of modules over a commutative algebra is described. Some
related structures and (co)homology invariants are discussed, as well as
applications to geometry.Comment: 14 pages; v2: minor correction
Generalized Lenard Chains, Separation of Variables and Superintegrability
We show that the notion of generalized Lenard chains naturally allows
formulation of the theory of multi-separable and superintegrable systems in the
context of bi-Hamiltonian geometry. We prove that the existence of generalized
Lenard chains generated by a Hamiltonian function defined on a four-dimensional
\omega N manifold guarantees the separation of variables. As an application, we
construct such chains for the H\'enon-Heiles systems and for the classical
Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler
potential are found.Comment: 14 pages Revte
Kodaira-Spencer formality of products of complex manifolds
We shall say that a complex manifold is emph{Kodaira-Spencer formal} if its Kodaira-Spencer differential graded Lie algebra
is formal; if this happen, then the deformation theory of
is completely determined by the graded Lie algebra and the base space of the semiuniversal deformation is a quadratic singularity..
Determine when a complex manifold is Kodaira-Spencer formal is generally difficult and
we actually know only a limited class of cases where this happen. Among such examples we have
Riemann surfaces, projective spaces, holomorphic Poisson manifolds with surjective anchor map
and every compact K"{a}hler manifold with trivial or torsion canonical
bundle.
In this short note we investigate the behavior of this property under finite products. Let be compact complex manifolds; we prove that whenever and are
K"{a}hler, then is Kodaira-Spencer formal if and only if the same
holds for and . A revisit of a classical example by Douady shows that the above result fails if the K"{a}hler assumption is droppe
Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets
In this paper we derive the most general first-order symmetry operator
commuting with the Dirac operator in all dimensions and signatures. Such an
operator splits into Clifford even and Clifford odd parts which are given in
terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous
forms respectively. We study commutators of these symmetry operators and give
necessary and sufficient conditions under which they remain of the first-order.
In this specific setting we can introduce a Killing-Yano bracket, a bilinear
operation acting on odd Killing-Yano and even closed conformal Killing-Yano
forms, and demonstrate that it is closely related to the Schouten-Nijenhuis
bracket. An important non-trivial example of vanishing Killing-Yano brackets is
given by Dirac symmetry operators generated from the principal conformal
Killing-Yano tensor [hep-th/0612029]. We show that among these operators one
can find a complete subset of mutually commuting operators. These operators
underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all
dimensions [arXiv:0711.0078].Comment: 37 pages, no figure
Do Killing-Yano tensors form a Lie Algebra?
Killing-Yano tensors are natural generalizations of Killing vectors. We
investigate whether Killing-Yano tensors form a graded Lie algebra with respect
to the Schouten-Nijenhuis bracket. We find that this proposition does not hold
in general, but that it does hold for constant curvature spacetimes. We also
show that Minkowski and (anti)-deSitter spacetimes have the maximal number of
Killing-Yano tensors of each rank and that the algebras of these tensors under
the SN bracket are relatively simple extensions of the Poincare and (A)dS
symmetry algebras.Comment: 17 page
Кон'юнктурний аналіз розвитку ринку рекреаційних послуг АР Крим
Метою дослідження є кон’юнктурний аналіз розвитку ринку рекреаційних послуг АР Крим та порівняльна оцінка функціонування конкурентоспроможних рекреаційних районів
Combined use of ISCR and biostimulation techniques in incomplete processes of reductive dehalogenation of chlorinated solvents
Pools of chloroethenes are more recalcitrant in the transition zone between aquifers and basal aquitards than those elsewhere in the aquifer. Although biodegradation of chloroethenes occur in this zone, it is a slow process and a remediation strategy is needed. The aim of this study was to demonstrate that combined strategy of biostimulation and in situ chemical reduction (ISCR) is more efficient than the two separated strategies. Four different microcosm experiments with sediment and groundwater of a selected field site where an aged perchloroethene (PCE)-pool exists at the bottom of a transition zone, were designed under i) natural conditions, ii) biostimulation with lactic acid, iii) in situ chemical reduction (ISCR)with zero valent iron (ZVI) and under iv) a combined strategy with lactic acid and ZVI. Biotic and abiotic dehalogenation, terminal electron acceptor processes and evolution of microbial communities were investigated for each experiment. The main results where: i) limited reductive dehalogenation of PCE occurs under sulfate-reducing conditions; ii) biostimulation with lactic acid promotes a more pronounced reductive dehalogenation of PCE in comparison under natural conditions, but resulted in an accumulation of cis-dichloroethene (cDCE); iii) ISCR with zero-valent iron (ZVI) facilitates a sustained dehalogenation of PCE and its metabolites to non-halogenated products, however, the iv) combined strategy results in the fastest and sustained dehalogenation of PCE to non-halogenated products in comparison of all four set-ups. These findings suggest that biostimulation and ISCRwith ZVI are the most suitable strategy for a complete reductive dehalogenation of PCE-pools in the transition zone
Jacobi structures revisited
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra
associated with a vector bundle which satisfy a property similar to that of the
Jacobi brackets, are introduced. They turn out to be equivalent to generalized
Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as
odd Jacobi brackets on the supermanifolds associated with the vector bundles.
Jacobi bialgebroids are defined in the same manner. A lifting procedure of
elements of this Grassmann algebra to multivector fields on the total space of
the vector bundle which preserves the corresponding brackets is developed. This
gives the possibility of associating canonically a Lie algebroid with any local
Lie algebra in the sense of Kirillov.Comment: 20 page
Cohomology of the Lie Superalgebra of Contact Vector Fields on and Deformations of the Superspace of Symbols
Following Feigin and Fuchs, we compute the first cohomology of the Lie
superalgebra of contact vector fields on the (1,1)-dimensional
real superspace with coefficients in the superspace of linear differential
operators acting on the superspaces of weighted densities. We also compute the
same, but -relative, cohomology. We explicitly give
1-cocycles spanning these cohomology. We classify generic formal
-trivial deformations of the -module
structure on the superspaces of symbols of differential operators. We prove
that any generic formal -trivial deformation of this
-module is equivalent to a polynomial one of degree .
This work is the simplest superization of a result by Bouarroudj [On
(2)-relative cohomology of the Lie algebra of vector fields and
differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112--127].
Further superizations correspond to -relative cohomology
of the Lie superalgebras of contact vector fields on -dimensional
superspace
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