Related papers: Analogs of Bol operators for $\mathfrak{pgl}(a+1\v…

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the $1|n$-dimensional supermanifold (superstring)…

We define an analogue of the Bol operator on spaces of weakly holomorphic modular forms of half-integral weight. We establish its main properties and relation with other objects.

We distinguish two classifications of bidifferential operators: between (A) spaces of modular forms and (B) spaces of weighted densities. (A) The invariant under the projective action of $\text{SL}(2;\mathbb{Z})$ binary differential…

The modular forms and weighted densities over the 1-dimensional manifold $M$ are transformed ``alike" under the group of linear fractional changes of coordinates, so the classifications of differential operators between spaces of (A)…

Let $M$ be an $n$-dimensional manifold, $V$ the space of a representation $\rho: GL(n)\longrightarrow GL(V)$. Locally, let $T(V)$ be the space of sections of the tensor bundle with fiber $V$ over a sufficiently small open set $U\subset M$,…

Here we continue to list the differential operators invariant with respect to the 15 exceptional simple Lie superalgebras of polynomial vector fields. A part of the list (for operators acting on tensors with finite dimensional fibers) was…

Generalizing the algebra of motion-invariant differential operators on a symmetric space we study invariant operators on equivariant vector bundles. We show that the eigenequation is equivalent to the corresponding eigenequation with…

Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight $\lo$ on a manifold $M$. One can consider a pencil of operators $\hPi(\Delta)=\{\Delta_\l\}$ passing through the operator $\Delta$ such that…

We define the concept of higher order differential operators on a general noncommutative, nonassociative superalgebra A, and show that a vertex operator superalgebra has plenty of them, namely modes of vertex operators. A linear operator…

We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary…

We define a new notion of fiber-wise linear differential operator on the total space of a vector bundle $E$. Our main result is that fiber-wise linear differential operators on $E$ are equivalent to (polynomial) derivations of an…

We study a class of dilatation invariant BPS surface operators in 4-dimensional N=4 Super Yang-Mills theory and their holographic duals in type IIB string theory in AdS_5 x S^5. First we take an example of 1/4 BPS surface operator and study…

The space of smooth sections of an equivariant line bundle over the real projective space $\mathbb{R}{\rm P}^n$ forms a natural representation of the group ${\rm GL}(n+1,\mathbb{R})$. We explicitly construct and classify all intertwining…

We consider the $\mathfrak{aff}(n|1)-$module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify $\mathfrak{aff}(n|1)-$invariant binary differential operators acting on the…

Over n-dimensional manifolds, I classify ternary differential operators acting on the spaces of weighted densities and invariant with respect to the Lie algebra of vector fields. For n=1, some of these operators can be expressed in terms of…

In this paper we study closed subspaces of ultradifferentiable functions which are invariant under the differentiation operator. We propose a version of spectral synthesis which takes into account the presence of non-trivial differentiation…

Over the $(1,n)$-dimensional real superspace, $n>1$, we classify $\mathcal{K}(n)$-invariant binary differential operators acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector…

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with…

We study the properties of nonlinear superalgebras $\mathcal{A}$ and algebras $\mathcal{A}_b$ arising from a one-to-one correspondence between the sets of relations that extract AdS-group irreducible representations $D(E_0,s_1,s_2)$ in…

The space of linear differential operators on a smooth manifold $M$ has a natural one-parameter family of $Diff(M)$ (and $Vect(M)$)-module structures, defined by their action on the space of tensor-densities. It is shown that, in the case…