Papers by Magnot Jean-Pierre
book published in Lambert Academic publishing
We consider two principal bundles of embeddings with total space Emb(M, N), with structure groups... more We consider two principal bundles of embeddings with total space Emb(M, N), with structure groups Dif f (M) and Dif f + (M), where Dif f + (M) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B(M, N) = Emb(M, N)/Dif f (M) and B + (M, N) = Emb(M, N)/Dif f + (M). From the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. This is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in [38]. We show that these groups are regular, and develop the necessary properties for applications to the geometry of B(M, N). A case of particular interest is M = S 1 , where connected components of B + (S 1 , N) are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space T B + (S 1 , N) is a subgroup of some group GLres, following the classical notations of [45]. These constructions suggest some approaches in the spirit of [27] that could lead to knot invariants through a theory of Chern-Weil forms. MSC (2010) : 47G30, 47N50, 22E67
In this paper, we start from an extension of the notion of ho-lonomy on diffeological bundles, re... more In this paper, we start from an extension of the notion of ho-lonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in [15], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q−series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable in the sense of Frobenius. Here, we get an integration of the Zakharov-Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group some groups of q−deformed operators. The integration obtained by Mulase [16], and the key tools he developped, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose-Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modelled on a complete locally convex topological vector space is also recovered and the work developped in [14, 15] is completed.
Using the procedure initiated in \cite{Ma2013}, we deform Lax-type equations
though a scaling of ... more Using the procedure initiated in \cite{Ma2013}, we deform Lax-type equations
though a scaling of the time parameter. This gives an equivalent (deformed)
equation which is integrable in terms of power series of the scaling parameter.
We then describe a regular Fr\"olicher Lie group of symmetries of this deformed
equation
One of the goals of this article is to define a an unified setting adapted to
the description of ... more One of the goals of this article is to define a an unified setting adapted to
the description of means (normalized integrals or invariant means) on an
infinite product of measured spaces with infinite measure. We first remark that
some known examples coming from the theory of metric measured spaces and also
from oscillatory integrals are obtained as limits of means with respect to
finite measures. Then, we explore in a systematic way the limit of means of the
type $$ \lim \frac{1}{\mu(U_n)} \int_{U_n} f d\mu$$ where $\mu$ is a a
$\sigma-$finite Radon measure $\mu.$ In some cases, we get a linear extension
of the limit at infinity. Then, the mean value on an infinite product is
defined, first for cylindrical functions and secondly taking the uniform limit.
Finally, the mean value for the heuristic Lebesgue measure on a separable
infinite dimensional topological vector space (but principally on a Hilbert
space) is defined. This last object is shown to be invariant by translation,
scaling and restriction.
unpublished preliminary version of a paper published in 2013
In this article, we describe a class of algebras with unbounded operators on which the Schwinger ... more In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E→M. We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin–Kravchenko–Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M=S1, we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S1-pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111–127]. When dim(M)>1, we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.
We state a Chern-Weil type theorem which is a generalization of a Chern-Weil type theorem for Fre... more We state a Chern-Weil type theorem which is a generalization of a Chern-Weil type theorem for Fredholm structures stated by Freed in [4]. Using this result, we investigate Chern forms on based manifold of maps M ap b (M, N) following two approaches, the first one using the Wodzicki residue, and the second one using renormalized traces of pseudo-differential operators along the lines of [1], [19], [20], [14]. We specialize to the case N = G to study current groups. Finally, we apply these results to a class of holomorphic connections on the loop group H
Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus o... more Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit
three cocycles on the Lie algebra of classical pseudo-differential operators
$$Cl(S^1,\mathbb{C}^n)$$ acting on
$$L^2(S^1,\mathbb{C}^n)$$. We first show that the Schwinger functional
$$c_S^D$$ associated to the Dirac operator is a cocycle on
$$Cl(S^1,\mathbb{C}^n)$$, and not only on a restricted algebra
$$Cl(S^1,\mathbb{C}^n)^D_{\rm res}.$$ Then, we investigate two bilinear functionals
$$c_{+}^D$$ and
$$c_{++}^D$$, which satisfies
$${1 \over 2}c_S^D = c_+^D - c_{++}^D.$$
We show that
$$c_{+}^D$$ and
$$c_{++}^D$$ are two cocycles in
$$Cl(S^1,\mathbb{C}^n)$$, and
$${1 \over 2}c_S^D$$ and
$$c_+^D$$ have the same nonvanishing cohomology class. We finaly calculate
$$c_{+}^D$$ on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols.
By this last computation, we recover the Virasoro cocyle and the Kähler form of the loop group.
We show that the holonomy group and the holonomy bundle of a connection of an infinite dimensiona... more We show that the holonomy group and the holonomy bundle of a connection of an infinite dimensional principal bundle can be equipped with a natural diffeology.
We give a theorem of reduction of the structure group of a principal bundle P with regular struct... more We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820–839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose–Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.
Using {\it weighted traces} which are linear functionals of the type $$A\to tr^Q(A):=(tr(A Q^{-z}... more Using {\it weighted traces} which are linear functionals of the type $$A\to tr^Q(A):=(tr(A Q^{-z})-z^{-1} tr(A Q^{-z}))_{z=0}$$ defined on the whole algebra of (classical) pseudo-differential operators (P.D.O.s) and where $Q$ is some positive invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces
In this study, a pairwise comparison matrix is generalized to the
case when coefficients create L... more In this study, a pairwise comparison matrix is generalized to the
case when coefficients create Lie group G, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons
matrix (extended to the class of group G) are proposed. A geometric interpretation of pairwise comparisons matrices in terms of connections to a simplex is given.
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Papers by Magnot Jean-Pierre
though a scaling of the time parameter. This gives an equivalent (deformed)
equation which is integrable in terms of power series of the scaling parameter.
We then describe a regular Fr\"olicher Lie group of symmetries of this deformed
equation
the description of means (normalized integrals or invariant means) on an
infinite product of measured spaces with infinite measure. We first remark that
some known examples coming from the theory of metric measured spaces and also
from oscillatory integrals are obtained as limits of means with respect to
finite measures. Then, we explore in a systematic way the limit of means of the
type $$ \lim \frac{1}{\mu(U_n)} \int_{U_n} f d\mu$$ where $\mu$ is a a
$\sigma-$finite Radon measure $\mu.$ In some cases, we get a linear extension
of the limit at infinity. Then, the mean value on an infinite product is
defined, first for cylindrical functions and secondly taking the uniform limit.
Finally, the mean value for the heuristic Lebesgue measure on a separable
infinite dimensional topological vector space (but principally on a Hilbert
space) is defined. This last object is shown to be invariant by translation,
scaling and restriction.
three cocycles on the Lie algebra of classical pseudo-differential operators
$$Cl(S^1,\mathbb{C}^n)$$ acting on
$$L^2(S^1,\mathbb{C}^n)$$. We first show that the Schwinger functional
$$c_S^D$$ associated to the Dirac operator is a cocycle on
$$Cl(S^1,\mathbb{C}^n)$$, and not only on a restricted algebra
$$Cl(S^1,\mathbb{C}^n)^D_{\rm res}.$$ Then, we investigate two bilinear functionals
$$c_{+}^D$$ and
$$c_{++}^D$$, which satisfies
$${1 \over 2}c_S^D = c_+^D - c_{++}^D.$$
We show that
$$c_{+}^D$$ and
$$c_{++}^D$$ are two cocycles in
$$Cl(S^1,\mathbb{C}^n)$$, and
$${1 \over 2}c_S^D$$ and
$$c_+^D$$ have the same nonvanishing cohomology class. We finaly calculate
$$c_{+}^D$$ on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols.
By this last computation, we recover the Virasoro cocyle and the Kähler form of the loop group.
case when coefficients create Lie group G, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons
matrix (extended to the class of group G) are proposed. A geometric interpretation of pairwise comparisons matrices in terms of connections to a simplex is given.
though a scaling of the time parameter. This gives an equivalent (deformed)
equation which is integrable in terms of power series of the scaling parameter.
We then describe a regular Fr\"olicher Lie group of symmetries of this deformed
equation
the description of means (normalized integrals or invariant means) on an
infinite product of measured spaces with infinite measure. We first remark that
some known examples coming from the theory of metric measured spaces and also
from oscillatory integrals are obtained as limits of means with respect to
finite measures. Then, we explore in a systematic way the limit of means of the
type $$ \lim \frac{1}{\mu(U_n)} \int_{U_n} f d\mu$$ where $\mu$ is a a
$\sigma-$finite Radon measure $\mu.$ In some cases, we get a linear extension
of the limit at infinity. Then, the mean value on an infinite product is
defined, first for cylindrical functions and secondly taking the uniform limit.
Finally, the mean value for the heuristic Lebesgue measure on a separable
infinite dimensional topological vector space (but principally on a Hilbert
space) is defined. This last object is shown to be invariant by translation,
scaling and restriction.
three cocycles on the Lie algebra of classical pseudo-differential operators
$$Cl(S^1,\mathbb{C}^n)$$ acting on
$$L^2(S^1,\mathbb{C}^n)$$. We first show that the Schwinger functional
$$c_S^D$$ associated to the Dirac operator is a cocycle on
$$Cl(S^1,\mathbb{C}^n)$$, and not only on a restricted algebra
$$Cl(S^1,\mathbb{C}^n)^D_{\rm res}.$$ Then, we investigate two bilinear functionals
$$c_{+}^D$$ and
$$c_{++}^D$$, which satisfies
$${1 \over 2}c_S^D = c_+^D - c_{++}^D.$$
We show that
$$c_{+}^D$$ and
$$c_{++}^D$$ are two cocycles in
$$Cl(S^1,\mathbb{C}^n)$$, and
$${1 \over 2}c_S^D$$ and
$$c_+^D$$ have the same nonvanishing cohomology class. We finaly calculate
$$c_{+}^D$$ on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols.
By this last computation, we recover the Virasoro cocyle and the Kähler form of the loop group.
case when coefficients create Lie group G, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons
matrix (extended to the class of group G) are proposed. A geometric interpretation of pairwise comparisons matrices in terms of connections to a simplex is given.