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The (secret?) homological algebra of the BatalinVilkovisky approach
"... Stasheff ..."
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Another Look On Recursion Operators
"... . Recursion operators of partial differential equations are identified with Backlund autotransformations of linearized diffieties. Relations to the classical concept and its recent Guthrie's generalization are discussed. Traditionally, a recursion operator of a PDE is a linear operator L actin ..."
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. Recursion operators of partial differential equations are identified with Backlund autotransformations of linearized diffieties. Relations to the classical concept and its recent Guthrie's generalization are discussed. Traditionally, a recursion operator of a PDE is a linear operator L acting on symmetries: if f is a symmetry then so is Lf . This provides a convenient way to generate infinite families of symmetries (Olver [13]). Standard reference is [14]. The presence of a recursion operator has been soon recognized as one of the attributes of the complete integrability. An exact and rich theory has been developed within the class of evolution equations (see Fokas [2] and references therein). Recursion operators are, as a rule, nonlocal and so may turn out to be the symmetries they generate (giving a powerful source of nonlocal symmetries [10, 5]). A remarkable problem of inverting a recursion operator motivated Guthrie [3] to a generalization, with consequences for generation of n...
On the spectral parameter problem
"... We consider the problem whether a nonparametric zerocurvature representation can be embedded into a oneparameter family within the same Lie algebra. After introducing a computable cohomological obstruction, a method using the recursion operator to incorporate the parameter is discussed. 1 ..."
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We consider the problem whether a nonparametric zerocurvature representation can be embedded into a oneparameter family within the same Lie algebra. After introducing a computable cohomological obstruction, a method using the recursion operator to incorporate the parameter is discussed. 1
Secondary Calculus and the Covariant Phase Space
 J. Geom. Phys
, 2009
"... Abstract. The covariant phase space of a lagrangian ¦eld theory is the solution space of the associated EulerLagrange equations. It is, in principle, a nice environment for covariant quantization of a lagrangian ¦eld theory. Indeed, it is manifestly covariant and possesses a canonical (functional) ..."
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Abstract. The covariant phase space of a lagrangian ¦eld theory is the solution space of the associated EulerLagrange equations. It is, in principle, a nice environment for covariant quantization of a lagrangian ¦eld theory. Indeed, it is manifestly covariant and possesses a canonical (functional) ¤presymplectic structure ¥ ω (as ¦rst noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of jet spaces and (A. M. Vinogradov£s) secondary calculus. In particular, we describe the degeneracy distribution of ω. As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.
Homological Perturbation Theory And Computability Of Hochschild And Cyclic Homologies Of Cdgas
, 1997
"... . We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some ..."
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. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the plocal homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...