Related papers: $\lambda$-Differential operators and $\lambda$-dif…
We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under suitable…
Let $\operatorname{Witt}$ be the Lie algebra generated by the set $\{L_i\,\vert\, i \in {\mathbb Z}\}$ and $\operatorname{Vir}$ its universal central extension. Let $\operatorname{Diff}(V)$ be the Lie algebra of differential operators on…
In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the…
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…
In this paper, we study the structure of the differential operator algebra \( \mathcal{D}(W) \) and its associated eigenvalue algebra \( \Lambda(W) \) for matrix-valued orthogonal polynomials. While \( \Lambda(W) \) is isomorphic to \(…
The Virasoro Lie algebra is a one-dimensional central extension of the Witt algebra, which can be realized as the Lie algebra of derivations on the algebra $\cc [t^{\pm}]$ of Laurent polynomials. Using this fact, we define a natural family…
The $\lambda$-differential operators and modified $\lambda$-differential operators are generalizations of classical differential operators. This paper introduces the notions of $\lambda$-differential Poisson ($\lambda$-DP for short)…
In this paper, extensions of nonunitary rational Virasoro vertex operator algebras corresponding to some exceptional modular invariants are constructed. The uniqueness of these extensions is also established.
We construct differential operators for families of overconvergent Hilbert modular forms by interpolating the Gauss--Manin connection on strict neighborhoods of the ordinary locus. This is related to work done by Harron and Xiao and by…
A difference operator on an associative algebra is an algebraic abstraction of the forward and backward difference operators. In this paper, we first introduce difference operators on associative $2$-algebras and consider the category of…
In earlier work, Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to Heisenberg parabolic subalgebras in simple Lie algebras. The construction was systematic, but the…
In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential…
We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semi-linear representations of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie…
The Lie algebra gl(\lambda) with \lambda \in \mathbb C, introduced by B.L.Feigin, can be embedded into the Lie algebra of differential operators on the real line. We give an explicit formula of the embedding of gl(\lambda) into the algebra…
We study relationships between spinor representations of certain Lie algebras and Lie superalgebras of differential operators on the circle and values of $\zeta$--functions at the negative integers. By using formal calculus techniques we…
In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
In contrast with differential operators on modules over commutative and graded commutative rings, there is no satisfactory notion of a differential operator in noncommutative geometry.
Structures of dual Lie bialgebras on the one sided Witt algebra, the Witt algebra and the Virasoro algebra are investigated. As a result, we obtain some infinite dimensional Lie algebras.
For a Hopf algebra A, we define the structures of differential complexes on two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an exterior extension of the dual algebra A^*. The Heisenberg double of these two exterior…