72 research outputs found
Dequantized Differential Operators between Tensor Densities as Modules over the Lie Algebra of Contact Vector Fields
In recent years, algebras and modules of differential operators have been
extensively studied. Equivariant quantization and dequantization establish a
tight link between invariant operators connecting modules of differential
operators on tensor densities, and module morphisms that connect the
corresponding dequantized spaces. In this paper, we investigate dequantized
differential operators as modules over a Lie subalgebra of vector fields that
preserve an additional structure. More precisely, we take an interest in
invariant operators between dequantized spaces, viewed as modules over the Lie
subalgebra of infinitesimal contact or projective contact transformations. The
principal symbols of these invariant operators are invariant tensor fields. We
first provide full description of the algebras of such affine-contact- and
contact-invariant tensor fields. These characterizations allow showing that the
algebra of projective-contact-invariant operators between dequantized spaces
implemented by the same density weight, is generated by the vertical cotangent
lift of the contact form and a generalized contact Hamiltonian. As an
application, we prove a second key-result, which asserts that the Casimir
operator of the Lie algebra of infinitesimal projective contact
transformations, is diagonal. Eventually, this upshot entails that invariant
operators between spaces induced by different density weights, are made up by a
small number of building bricks that force the parameters of the source and
target spaces to verify Diophantine-type equations.Comment: 22 page
Simultaneous deformations and Poisson geometry
We consider the problem of deforming simultaneously a pair of given
structures. We show that such deformations are governed by an L-infinity
algebra, which we construct explicitly. Our machinery is based on Th. Voronov's
derived bracket construction.
In this paper we consider only geometric applications, including deformations
of coisotropic submanifolds in Poisson manifolds, of twisted Poisson
structures, and of complex structures within generalized complex geometry.
These applications can not be, to our knowledge, obtained by other methods such
as operad theory.Comment: 32 pages. Results in Section 2 improved (Lemma 2.6 and Corollaries
2.20, 2.22). Corollary 2.5 and Corollary 2.11 added. Final version, accepted
for publicatio
Optimized Crystallographic Graph Generation for Material Science
Graph neural networks are widely used in machine learning applied to
chemistry, and in particular for material science discovery. For crystalline
materials, however, generating graph-based representation from geometrical
information for neural networks is not a trivial task. The periodicity of
crystalline needs efficient implementations to be processed in real-time under
a massively parallel environment. With the aim of training graph-based
generative models of new material discovery, we propose an efficient tool to
generate cutoff graphs and k-nearest-neighbours graphs of periodic structures
within GPU optimization. We provide pyMatGraph a Pytorch-compatible framework
to generate graphs in real-time during the training of neural network
architecture. Our tool can update a graph of a structure, making generative
models able to update the geometry and process the updated graph during the
forward propagation on the GPU side. Our code is publicly available at
https://github.com/aklipf/mat-graph
L-infinity algebra actions
We define the notion of action of an L-infinity algebra on a graded
manifold , and show that such an action corresponds to a homological vector
field on of a specific form. This generalizes the
correspondence between Lie algebra actions on manifolds and transformation Lie
algebroids. In particular, we consider actions of on a second L-infinity
algebra, leading to a notion of "semidirect product" of L-infinity algebras
more general than those we found in the literature.Comment: v2: Final version, to appear in Diff. Geom. App
Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras
In this paper we construct ternary -Virasoro-Witt algebras which
-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie
and Zachos using enveloping algebra techniques. The ternary
Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a
parameter and are not Nambu-Lie algebras for all but finitely many values of
this parameter. For the parameter values for which the ternary Virasoro-Witt
algebras are Nambu-Lie, the corresponding ternary -Virasoro-Witt algebras
constructed in this article are also Hom-Nambu-Lie because they are obtained
from the ternary Nambu-Lie algebras using the composition method. For other
parameter values this composition method does not yield Hom-Nambu Lie algebra
structure for -Virasoro-Witt algebras. We show however, using a different
construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and
Zachos, as well as the general ternary -Virasoro-Witt algebras we construct,
carry a structure of ternary Hom-Nambu-Lie algebra for all values of the
involved parameters
The homotopy theory of simplicial props
The category of (colored) props is an enhancement of the category of colored
operads, and thus of the category of small categories. In this paper, the
second in a series on "higher props," we show that the category of all small
colored simplicial props admits a cofibrantly generated model category
structure. With this model structure, the forgetful functor from props to
operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat
Derived coisotropic structures I: affine case
We define and study coisotropic structures on morphisms of commutative dg
algebras in the context of shifted Poisson geometry, i.e. -algebras.
Roughly speaking, a coisotropic morphism is given by a -algebra acting
on a -algebra. One of our main results is an identification of the space
of such coisotropic structures with the space of Maurer--Cartan elements in a
certain dg Lie algebra of relative polyvector fields. To achieve this goal, we
construct a cofibrant replacement of the operad controlling coisotropic
morphisms by analogy with the Swiss-cheese operad which can be of independent
interest. Finally, we show that morphisms of shifted Poisson algebras are
identified with coisotropic structures on their graph.Comment: 49 pages. v2: many proofs rewritten and the paper is split into two
part
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