Differential Operators on Azumaya algebra and Heisenberg algebra (original) (raw)

Some Algebras in Terms of Differential Operators

International Electronic Journal of Algebra, 2021

Let C be a commutative ring and C[x 1 , x 2 ,. . .] the polynomial ring in a countable number of variables x i of degree 1. Suppose that the differential operator d 1 = i x i ∂ i acts on C[x 1 , x 2 ,. . .]. Let Zp be the p-adic integers, K the extension field of the p-adic numbers Qp, and F 2 the 2-element filed. In this article, first, the C-algebra A 1 (C) of differential operators is constructed by the divided differential operators (d 1) ∨k /k! as its generators, where ∨ stands for the wedge product. Then, the free Baxter algebra of weight 1 over ∅, the λ-divided power Hopf algebra A λ , the algebra C(Zp, K) of continuous functions from Zp to K, and the algebra of all F 2-valued continuous functions on the ternary Cantor set are represented in terms of the differential operators algebra A 1 (C).

Non-Noetherian generalized Heisenberg algebras

Journal of Algebra and Its Applications, 2016

In this note, we classify the non-Noetherian generalized Heisenberg algebras [Formula: see text] introduced in [R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291]. In case deg [Formula: see text] > 1, we determine all locally finite and also all locally nilpotent derivations of [Formula: see text] and describe the automorphism group of these algebras.

Volichenko Algebras as Algebras of Differential Operators

Journal of Nonlinear Mathematical Physics, 2006

Throughout this paper, k denotes a field of characteristic 0 and all tensor products are over k. Further, k[X; n] is the polynomial algebra in n commuting indeterminates X = (x 1 , x 2 , • • • , x n) and Λ[Y ; m] is the Grassmann algebra in n anti-commuting indeterminates Y = (y 1 , y 2 , • • • , y m). Supersymmetries are symmetries of supervarieties, i.e., objects, functions on which depend on both usual commuting (even) variables and on anticommuting (odd) ones. For numerous applications of supersymmetry and for basics, see [3], [2] and [10]. Supersymmetries widened the notion of group in order to be able to mix Bose and Fermi particles. However, the collection of morphisms of supervarieties (locally, of its superalgebra of functions F)-supersymmetries-is not the largest possible group of automorphisms of the algebra F , with superstructure ignored. Besides, not every subalgebra or a quotient of a supercommutative superalgebra is supercommutative, whereas they are metaabelian and the notion of superscheme was first given ([7]) in terms of such, not necessarily homogeneous, subalgebras and quotients of supercommutative superalgebras. Recall that a ring M is said to be metaabelian if [a, [b, c]] = 0 for all a, b, c ∈ M , where [a, b] = ab − ba. Volichenko showed (see [8]) that every metaabelian algebra can be realized as a nonhomogeneous subalgebra of a universal supercommutative superalgebra, called its supercommutative envelope. The purpose of this note it to construct an appropriate analog of differential operators on metaabelian algebras, more precisely, viewing a metaabelian algebra M as an analog of the algebra of functions, construct the corresponding algebra of vector fields. Lunts and Rosenberg ([9]) constructed algebras of differential operators on (graded) noncommutative algebras. In particular, one can study differential operators on superalgebras. Superderivations of a superalgebra, which are first order differential operators, form a Lie superalgebra. In a work aborted by his death, Volichenko gave a conjectural intrinsic description of nongraded subalgebras of Lie superalgebras. In his memory then, Leites and Serganova ([8]) called such subalgebras Volichenko algebras and (under a technical assumption) listed simple Volichenko algebras (finite dimensional and of vector fields). Like the list of simple

On the 3-generated commutative rings of differential operators

Journal of Mathematical Sciences

The theory of commutative rings of differential operators relating the completely integrable systems with the geometry of algebraic curves was constructed several decades ago. It was especially complete in the case of rings, generated by two operators of the coprime order, usually of order 2 and of some odd order; the theory of such rings turned out to be equivalent to the theory of KdV hierarchy. However, the corresponding algebraic curves were always hyperelliptic. In order to handle the general (canonical curves), one should consider the rings, generated by more than two operators. In the previous paper of 1980, the author considered the simplest possible case of this kind-that of generators of orders 3, 4, 5. The goal of the present paper is to give the details of the calculations in that paper and to explain the conjectural geometry underlying some enigmatic phenomena that were used in 1980 to complete the calculations and give some algebro-geometric applications.

O ct 2 00 6 On N-differential graded algebras

2008

We introduce the concept of N -differential graded algebras (N-dga), and study the moduli space of deformations of the differential of an N-dga. We prove that it is controlled by what we call the (M,N)-Maurer-Cartan equation. Introduction The goal of this paper is to take the first step towards finding a generalization of Homological Mirror Symmetry (HMS) [11] to the context of N -homological algebra [5]. In [7] Fukaya introduced HMS as the equivalence of the deformation functor of the differential of a differential graded algebra associated with the holomorphic structure, with the deformation functor of an A∞-algebra associated with the symplectic structure of a Calabi-Yau variety. This idea motivated us to define deformation functors of the differential of an N -differential graded algebra. An N -dga is a graded associative algebra A, provided with an operator d : A → A of degree 1 such that d(ab) = d(a)b + (−1)ad(b) and d = 0. A nilpotent differential graded algebra (Nil-dga) wil...

A Parametric Family of Subalgebras of the Weyl Algebra III. Derivations

arXiv: Rings and Algebras, 2014

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A_h generated by elements x,y, which satisfy yx-xy = h, where h is in F[x]. When h is nonzero, these algebras are subalgebras of the Weyl algebra A_1 and can be viewed as differential operators with polynomial coefficients. In previous work, we investigated the structure of A_h, determined its automorphisms and their invariants, and studied the irreducible A_h-modules. Here we determine the derivations of A_h over an arbitrary field.

Differential Operators on the free algebras

Selecta Mathematica, New Series, 2011

Following the definitions of the algebras of differential operators, β-differential operators, and the quantum differential operators on a noncommutative (graded) algebra given in [6], we describe these operators on the free associative algebra. We further study their properties.