Related papers: Derivations of the Lie Algebras of Differential Op…

We establish a H\"{o}rmander type theorem for the multilinear pseudo-differential operators, which is also a generalization of the results in \cite{MR4322619} to symbols depending on the spatial variable. Most known results for multilinear…

We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_\delta^n=\bigoplus_{j=0}^n \mathcal{F}_{\delta-j}$. We study deformations of this…

Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms.…

Let $\BB$ be the Lie algebra of Block type with basis $\{L_{\a,i}|\,\a,i\in\Z, i\geq0\}$ and relations $[L_{\a,i},L_{\b,j}]=\left((\a-1)(j+1)-(\b-1)(i+1)\right)L_{\a+\b,i+j}$. In the present paper, the derivation algebra and the…

E.B. Vinberg's theory of quasi-derivations of algebras is extended to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. Although basic results apply to arbitrary algebras, our…

Let M\"ob be the biholomorphic automorphism group of the unit disc of the complex plane, $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{U}(\mathcal{H})$ be the group of all unitary operators. Suppose $\mathcal{H}$ is a…

In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions.…

Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifold, fractals, graphs ...). Boundedness on $L^p$ for pseudodifferential operators of…

In this paper we extend to the difference case the notion of Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators. A local scalar difference Hamiltonian…

This paper surveys recent work on Lie algebras of differential operators and their application to the construction of quasi-exactly solvable Schroedinger operators.

We show that the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type…

Let M be a compact Riemannian manifold equipped with a parallel differential form \omega. We prove a version of Kaehler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquotient…

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie…

This thesis is devoted to the study of joint spectral multipliers for a system of pairwise commuting, self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G. Under the assumption that the algebra generated…

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-K\"ahler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces…

We prove a Lie-algebraic characterization of vector bundle for the Lie algebra $\mathcal{D}(E,M),$ seen as ${\rm C}^\infty(M)-$module, of all linear operators acting on sections of a vector bundle $E\to M$. We obtain similar result for its…

We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group $U(\mathcal{H})$ in a Hilbert space $\mathcal{H}$ with $U(\mathcal{H})$ equipped with the strong operator…

We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{\lambda,\mu})$ ($\lambda, \mu\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace $\mathfrak{D}_{\lambda,\mu}$ of…

We review the definition of a Lie manifold $(M, \VV)$ and the construction of the algebra $\Psi\sp{\infty}\sb{\VV}(M)$ of pseudodifferential operators on a Lie manifold $(M, \VV)$. We give some concrete Fredholmness conditions for…

A class of nongraded Hamiltonian Lie algebras was earlier introduced by Xu. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a ``sandwich'' method…