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

Equivariant quantizations and Cartan connections

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Approximations of Lovasz extensions and their induced interaction index

The Lovasz extension of a pseudo-Boolean function $f : \{0,1\}^n \to R$ is defined on each simplex of the standard triangulation of $[0,1]^n$ as the unique affine function $\hat f : [0,1]^n \to R$ that interpolates $f$ at the $n+1$ vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses $f$. In this paper we investigate the least squares approximation problem of an arbitrary Lovasz extension $\hat f$ by Lovasz extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of $\hat f$ and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page

A classification of polynomial functions satisfying the Jacobi identity over integral domains

The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes

Measuring the interactions among variables of functions over the unit hypercube

By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of $f$. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of $f$ or, under certain natural conditions on $f$, as an expected value of the derivatives of $f$. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a few applications of the interaction index

A classification of barycentrically associative polynomial functions

We describe the class of polynomial functions which are barycentrically associative over an infinite commutative integral domain

Existence of natural and conformally invariant quantizations of arbitrary symbols

peer reviewedA quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection. In general, there is no natural quantization procedure, that is, spaces of symbols and of differential operators are not equivalent, if the action of local diffeomorphisms is taken into account. However, considering manifolds endowed with additional structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account. The existence of such a quantization was proved recently in a series of papers in the context of projective geometry. Here, we show that the construction of the quantization based on Cartan connections can be adapted from projective to pseudo-conformal geometry to yield the natural and conformally invariant quantization for arbitrary symbols, outside some critical situations

On the structure of symmetric nn-ary bands

We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\ldots,x)=x$ for all $x\in X$. We first provide a structure theorem for these symmetric $n$-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong $n$-ary semilattice of $n$-ary semigroups and we show that the symmetric $n$-ary bands are exactly the strong $n$-ary semilattices of $n$-ary extensions of Abelian groups whose exponents divide $n-1$. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric $n$-ary band to be reducible to a semigroup