Differential Operators on the free algebras (original) (raw)
NON-COMMUTATIVE FIRST ORDER DIFFERENTIAL CALCULUS OVER FINITELY GENERATED ASSOCIATIVE ALGEBRAS
arxive(Cornell University), 2019
In this review article the construction of first order coordinate differential calculi on finitely generated and finitely related associative algebras are considered and explicit construction of the bimodule of one form over such algebras is presented. The concept of optimal algebras for such caculi are also discussed. Detailed computations presented will make this note particularly useful for physicists.
2009
In this paper, we establish the Composition-Diamond lemma for lambda\lambdalambda-differential associative algebras over a field KK K with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free lambda\lambdalambda-differential Rota-Baxter algebras. In particular, linear bases of free lambda\lambdalambda-differential Rota-Baxter algebras are obtained and consequently, the free lambda\lambdalambda-differential Rota-Baxter algebras are constructed by words.
Proceedings of the International Geometry Center
We review main differential-algebraic structures \ lying in background of \ analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and \ Leibniz type algebraic structures are derived, a new nonassociative "Riemann" algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. \ As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed inves...
Differential operators and W-algebra
Physics Letters B, 1992
The connection between W-algebras and the algebra of differential operators is conjectured. The bosonized representation of the differential operator algebra with c=-2n and all the subalgehras are examined. The degenerate representations and null-state classifications for c=-2 are presented.
Differential Operators on Azumaya algebra and Heisenberg algebra
Arxiv preprint math/0002014, 2000
2. If An denotes the n-th Weyl algebra over a field of characteristic 0, then Dk(An) = A2n (Corollary 4.1.8). In the case of Azumaya algebras, we show that there is a one-to-one correspondence between ideals of Dk(A) and Dk(R) (section 3.3). If Hn denotes the nth-Heisenberg ...
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
ph ] 2 0 Ju l 2 01 0 Noncommutative ε-graded connections ∗
2012
We introduce the new notion of ε-graded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of ε-derivation-based differential calculus, which generalizes the derivation-based differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of ε-graded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also to interpret mathematically a noncommutative gauge field theory.