Noncommutative Coefficients Research Articles

Differential graded algebras for trivalent plane graphs and their representations

To any trivalent plane graph embedded in the sphere Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the Casals–Murphy dg-algebra to non-commutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank r representations of this dg-algebra, over a field \mathbb{F} , correspond to colorings of the faces of the graph by elements of the Grassmannian \operatorname{Gr}(r,2r;\mathbb{F}) so that bordering faces are transverse, up to the natural action of \operatorname{PGL}_{2r}(\mathbb{F}) . Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that, for Legendrian 2-weaves, rank r representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank r . This is the first explicit such computation of the bijection between the moduli spaces of representations and sheaves for an infinite family of Legendrian surfaces.

The μ-neutral fractional multi-delayed differential equations

The μ-neutral linear fractional multi-delayed differential nonhomogeneous system with noncommutative coefficient matrices is introduced. The novel μ-neutral multi-delayed perturbation of Mittag-Leffler type matrix function is proposed. Based on this, an explicit solution to the system is investigated step by step. The existence uniqueness of solutions to μ-neutral nonlinear fractional multi-delayed differential system is obtained with regard to the supremum norm. The notion of stability analysis in the sense of solutions to the described system is discussed on the grounds of the fixed point approach.

On a study for the neutral Caputo fractional multi-delayed differential equations with noncommutative coefficient matrices

The representation of a solution to a neutral linear fractional multiple-delay differential inhomogeneous system with non-commutative coefficient matrices is studied using a multiple-delay perturbation of a matrix function of the Mittag-Leffler type. Second, the existence and uniqueness of the solution are discussed along with the Ulam-Hyers stability of a semilinear neutral fractional differential with multiple delays. Thirdly, with the help of the Krasnoselskii's fixed point theorem, a sufficient condition for the relative controllability of a semilinear neutral fractional differential system with multiple-delay is obtained. Numerical examples confirm the theoretical conclusions.

New ways to multiply 3 × 3-matrices

It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3×3-matrices. For non-commutative coefficient rings, it is not known whether it can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than 17,000 new and mutually inequivalent schemes for multiplying 3×3-matrices using 23 multiplications. Moreover, we show that the set of all these schemes is a manifold of dimension at least 17.

General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings

ABSTRACT We propose some generalizations of the classical Division Algorithm for polynomials over coefficient rings (possibly non-commutative). These results provide a generalization of the Remainder Theorem that allows calculating the remainder without using the long division method, even if the divisor has degree greater than one. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. Finally, we will refer to some applications of these results for evaluating and dividing on skew polynomial rings. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities.

Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebra

We consider a generalization of (pro)algebraic loops defined on general categories of algebras and the dual notion of a coloop bialgebra suitable to represent them as functors. We prove that the natural loop of formal diffeomorphisms with associative coefficients is proalgebraic, and we give the closed formulas of the codivisions on its coloop bialgebra. This result provides a generalization of the Lagrange inversion formula to series with non-commutative coefficients, and a loop-theoretic explanation to the existence of the non-commutative Faà di Bruno Hopf algebra.

The loop of formal power series with noncommutative coefficients under substitution

The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We study this loop by describing certain Lie and Sabinin algebras related to it. Some examples of Lie algebras satisfying the standard identities of degrees 5 and 6 appear naturally.

A Feynman integral and its recurrences and associators

We determine closed and compact expressions for the ϵ-expansion of certain Gaussian hypergeometric functions expanded around half-integer values by explicitly solving for their recurrence relations. This ϵ-expansion is identified with the normalized solution of the underlying Fuchs system of four regular singular points. We compute its regularized zeta series (giving rise to two independent associators) whose ratio gives the ϵ-expansion at a specific value. Furthermore, we use the well known one-loop massive bubble integral as an example to demonstrate how to obtain all-order ϵ-expansions for Feynman integrals and how to construct representations for Feynman integrals in terms of generalized hypergeometric functions. We use the method of differential equations in combination with the recently established general solution for recurrence relations with non-commutative coefficients.

Noncommutative field with constant background fields and neutral fermions

Introducing constant background fields into the noncommutative gauge theory, we first obtain a Hermitian fermion Lagrangian which involves a Lorentz violation term, then we generalize it to a new deformed canonical noncommutation relations for fermion field. Massless neutrino oscillation in the deformed canonical noncommutation relations is analyzed. The restriction of the noncommutative coefficients is also discussed. By comparing with the existing experimental data of conventional neutrino oscillations, the order of noncommutative deformed coefficients is given from different ways.

Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation

The non-commutative harmonic oscillator (NcHO) is a special type of self-adjoint ordinary differential operator with non-commutative coefficients. In the present note, we aim to provide a reasonable criterion that derives the simplicity of the lowest eigenvalue of NcHO. It actually proves the simplicity of the lowest eigenvalue for a large class of structure parameters. Moreover, this note describes a certain equivalence between the spectral problem of the NcHO (for the even parity) and existence of holomorphic solutions of Heun’s ordinary differential equations in a complex domain. The corresponding Riemann scheme allows us to give another proof to the criterion.

Exact treatment of operator difference equations with nonconstant and noncommutative coefficients

We study a homogeneous linear second-order difference equation with nonconstant and noncommuting operator coefficients in a vector space. We build its exact resolutive formula consisting of the explicit noniterative expression of a generic term of the unknown sequence of vectors. Some nontrivial applications are reported in order to show the usefulness and the broad applicability of the result.

Homology cylinders of higher-order

We study algebraic structures of certain submonoids of the monoid of homology cylinders over a surface and the homology cobordism groups, using Reidemeister torsion with non-commutative coefficients. The submonoids consist of ones whose natural inclusion maps from the boundary surfaces induce isomorphisms on higher solvable quotients of the fundamental groups. We show that for a surface whose first Betti number is positive, the homology cobordism groups are other enlargements of the mapping class group of the surface than that of ordinary homology cylinders. Furthermore we show that for a surface with boundary whose first Betti number is positive, the submonoids consisting of irreducible ones as 3-manifolds trivially acting on the solvable quotients of the surface group are not finitely generated.

Elementary symmetric functions of two solvents of a quadratic matrix equation

Quadratic matrix equations occur in a variety of applications. In this paper we introduce new permutationally invariant functions of two solvents of the n × n quadratic matrix equation X 2 - ℒ 1 X - ℒ 0 = 0, playing the role of the two elementary symmetric functions of the two roots of a quadratic scalar equation. Our results rely on the connection existing between the QME and the theory of linear second-order difference equations with noncommutative coefficients. An application of our results to a simple physical problem is briefly discussed.

Exact treatment of linear difference equations with noncommutative coefficients

AbstractThe exact solution of a Cauchy problem related to a linear second‐order difference equation with constant noncommutative coefficients is reported. Copyright © 2007 John Wiley & Sons, Ltd.

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.

Krull Dimension of Generalized Weyl Algebras with Noncommutative Coefficients

We obtain formulae for the Krull dimension of the generalized Weyl algebra T=R(σ,a), where R is a left noetherian ring.

Exact and approximate solutions of some operator equations based on the Cayley transform

We consider the operator equation SX ≡ Σ j−1 M U j XV j = Y where U j , V j are some communicative sets of operators but in general U j need not compute with V j . Particular cases of this equation are the Sylvester and Ljapunov equations. We give a new representation and an approximation of the solution which is suitable to perform it algorithmically. Error estimates are given which show exponential covergence for bounded operators and polynomial convergence for unbounded ones. Based on these considerations we construct an iterative process and give an existence theorem for the operator equation Z 2 + A 1 Z + A 2 = 0, arising for example when solving an abstract second order differential equation with non-commutative coefficients.

Prime ideals in skew polynomial rings and quantized Weyl algebras

The concern of this paper is to investigate the structure of skew polynomial rings (Ore extensions) of the form T=R[θ; σ, δ] where σ and δ are both nontrivial, and in particular to analyze the prime ideals of T. The main focus is on the case that R is commutative noetherian. In this case, the prime ideals of T are classified, polynomial identities and Artin-Rees separation in prime factor rings are investigated, and cliques of prime ideals are studied. The second layer condition is proved, as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique. Futher, q-skew derivations on noncommutative coefficient rings are introduced, and some preliminary results on contractions of prime ideals of T are obtained in this setting. Finally, prime ideals in quantized Weyl algebras over fields are analyzed.

Solvability of the super KP equation and a generalization of the Birkhoff decomposition

The unique solvability of the initial value problem for the total hierarchy of the super Kadomtsev-Petviashvili system is established. To prove the existence we use a generalization of the Birkhoff decomposition which is obtained by replacing the loop variable and loop groups in the original setting by a super derivation operator and groups of infinite order super micro- (i.e. pseudo-) differential operators. To show the uniqueness we generalize the fact that every flat connection admits horizontal sections to the case of an infinite dimensional super algebra bundle defined over an infinite dimensional super space. The usual KP system with non-commutative coefficients is also studied. The KP system is obtained from the super KP system by reduction modulo odd variables. On the other hand, the first modified KP equation can be obtained from the super KP system by elimination of odd variables. Thus the super KP system is a natural unification of the KP system and the modified KP systems.

Krull dimension of differential operator rings. III. Noncommutative coefficients

This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring S [ θ ; δ ] S[\theta ;\delta ] , where S S is a right noetherian ring with finite Krull dimension n n and δ \delta is a derivation on S S . The main theorem states that S [ θ ; δ ] S[\theta ;\delta ] has Krull dimension n n unless there exists a simple right S S -module A A such that A ⊗ S S [ θ ; δ ] A{ \otimes _S}S[\theta ;\delta ] is not simple (as an S [ θ ; δ ] S[\theta ;\delta ] -module) and A A has height n n in the sense that there exist critical right S S -modules A = A 0 , A 1 , … , A n A = {A_0},{A_1},\ldots ,{A_n} such that each A i ⊗ s S [ θ ; δ ] {A_i} \otimes _s S[\theta ;\delta ] is a critical S [ θ ; δ ] S[\theta ;\delta ] -module, each A i {A_i} is a minor subfactor of A i + 1 {A_{i + 1}} and A n {A_n} is a subfactor of S S . If such an A A does exist, then S [ θ ; δ ] S[\theta ;\delta ] has Krull dimension n + 1 n + 1 . This criterion is simplified when S S is fully bounded, in which case it is shown that S [ θ ; δ ] S[\theta ;\delta ] has Krull dimension n n unless S S has a maximal ideal M M of height n n such that either char( S / M ) > 0 {\text {char(}}S/M) > 0 or δ ( M ) ⊆ M \delta (M) \subseteq M , and in these cases S [ θ ; δ ] S[\theta ;\delta ] has Krull dimension n + 1 n + 1 .