Commutative Product Research Articles

Connected sum for modular operads and Beilinson–Drinfeld algebras

Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. Motivated by this example, we introduce a notion of connected sum for general modular operads. We show that a connected sum induces a commutative product on the space of functions associated to the modular operad. Moreover, we combine this product with Barannikov’s non-commutative Batalin–Vilkovisky structure present on this space of functions, obtaining a Beilinson–Drinfeld algebra. Finally, we study the quantum master equation using the exponential defined using this commutative product.

Decomposition of matrices into products of commutators of quadratic matrices

Let F be a field, p ( x ) be a quadratic polynomial in F [ x ] with a non-zero constant term, and n ≥ 2 be a positive integer. We say that T ∈ M n ( F ) is p ( x ) -quadratic if p ( T ) = 0 . In this paper, given A ∈ SL n ( F ) , we show that A can be decomposed into a product of at most two commutators of p ( x ) -quadratic matrices if either p ( x ) has two distinct roots in F and A is non-scalar, or if F is algebraically closed and has characteristic different from 2. A similar result for matrices over real quaternion division rings is also provided.

A note on products of commutators of quadratic matrices

A note on products of commutators of quadratic matrices

Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

Rings and C*-algebras generated by commutators

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number N such that every element is a sum of N products of pairs of commutators. We show that one can take N≤2 for matrix rings, and that one may choose N≤3 for rings that contain a direct sum of matrix rings – this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that N≤6 can be arranged.For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. Using that a C*-algebra cannot be a radical extension over a proper ideal, we deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

Products of commutators of unipotent matrices of index $2$ in $\mathrm{GL}_n(\mathbb H)$

The aim of this paper is to show that if $\mathbb{H}$ is the real quaternion division ring and $n$ is an integer greater than $1,$ then every matrix in the special linear group $\mathrm{SL}_n(\mathbb{H})$ can be expressed as a product of at most three commutators of unipotent matrices of index $2$.

Two Results on the Commutative Product of Distributions and Functions

Two Results on the Commutative Product of Distributions and Functions

Correction to: Derivable maps at commutative products on Banach algebras

Correction to: Derivable maps at commutative products on Banach algebras

On algebras embeddable into bicommutative algebras

We consider algebras embeddable into free bicommutative algebras with respect to commutator and anti-commutator products over a field of characteristic zero. We show that every metabelian Lie algebra can be embedded into a bicommutative algebra with respect to the commutator product. Furthermore, we prove that the class of commutative algebras embeddable into bicommutative algebras with respect to the anti-commutator product forms a variety. As a consequence, we obtain that every metabelian Lie algebra can be embedded into a Novikov algebra with respect to the commutator product.

Products of Commutators of Involutions in Skew Linear Groups

Products of Commutators of Involutions in Skew Linear Groups

Transposed Poisson structures on Lie incidence algebras

Let X be a finite connected poset, K a field of characteristic zero and I(X,K) the incidence algebra of X over K seen as a Lie algebra under the commutator product. In the first part of the paper we show that any 12-derivation of I(X,K) decomposes into the sum of a central-valued 12-derivation, an inner 12-derivation and a 12-derivation associated with a map σ:X<2→K that is constant on chains and cycles in X. In the second part of the paper we use this result to prove that any transposed Poisson structure on I(X,K) is the sum of a structure of Poisson type, a mutational structure and a structure determined by λ:Xe2→K, where Xe2 is the set of (x,y)∈X2 such that x<y is a maximal chain not contained in a cycle.

Products of commutators in certain rings

Let [Formula: see text] be a ring. For each positive integer [Formula: see text], [Formula: see text] (respectively, [Formula: see text]) is denoted by the set of all products of (respectively, at most) [Formula: see text] additive commutators of [Formula: see text]. If [Formula: see text] is an algebra over a field [Formula: see text] such that [Formula: see text] for some [Formula: see text], then [Formula: see text]. If [Formula: see text] is a noncommutative ring such that [Formula: see text] for some [Formula: see text], then the necessary and sufficient condition for [Formula: see text] to be a division ring is that every nonzero element of [Formula: see text] is invertible. If [Formula: see text] is the matrix ring over a division ring [Formula: see text], then [Formula: see text], and in particular, if [Formula: see text] is a field, then [Formula: see text].

Some decompositions of matrices over local rings

Let [Formula: see text] be a local ring with maximal ideal [Formula: see text], let [Formula: see text] be a natural number greater than [Formula: see text] and let [Formula: see text] be a matrix in the general linear group [Formula: see text] of degree [Formula: see text] over [Formula: see text]. We firstly show that if the matrix [Formula: see text] is nonscalar in [Formula: see text] and [Formula: see text] are invertible elements in [Formula: see text], then there exists an invertible element [Formula: see text] such that [Formula: see text] is similar to the product [Formula: see text] in which [Formula: see text] is a lower uni-triangular matrix and [Formula: see text] is an upper triangular matrix whose diagonal entries are [Formula: see text]. We then present some applications of this factorization to find decompositions of matrices in [Formula: see text] into product of commutators and involutions.

The Eliahou-Kervaire resolution over a skew polynomial ring

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings.

Pre-twisted calculus and differential equations

In this paper we introduce the φA-differentiability for functions f:U⊂Rk→Rn, where U is an open set, A is the linear space Rn endowed with a unital associative commutative algebra product, and φ:U⊂Rk→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.

Generalized Cohomological Field Theories in the Higher Order Formalism

In the classical Batalin--Vilkovisky formalism, the BV operator $\Delta$ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by $\Delta$ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin--Vilkovisky algebras where the operator $\Delta$ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever $\Delta$ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s.

Improving Quantum Measurements by Introducing "Ghost" Pauli Products.

Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers individual fragment variances by modifying the fragments without changing the total observable expectation value. Our approach is based on adding Pauli products ("ghosts") that are compatible with members of multiple fragments. The total expectation value does not change because a sum of coefficients for each "ghost" Pauli product introduced to several fragments is zero. Yet, these additions change individual fragment variances because of the non-vanishing contributions of "ghost" Pauli products within each fragment. The proposed algorithm minimizes individual fragment variances using a classically efficient approximation of the quantum wavefunction for variance estimations. Numerical tests on a few molecular electronic Hamiltonian expectation values show several-fold reductions in the number of measurements in the "ghost" Pauli algorithm compared to those in the other recently developed techniques.

On picture fuzzy ideals on commutative rings

In this paper, we focus on combining the theories of picture fuzzy sets on rings and establishing a new framework for picture fuzzy sets on commutative rings. The aim of this manuscript is to apply picture fuzzy set for dealing with several kinds of theories in commutative rings. Moreover, we introduce the notions of picture fuzzy ideals on commutative rings and some properties of them are obtained. Finally, we give suitable definitions of the operations of picture fuzzy ideals over a commutative ring, as composition, product and intersection.

The Trace Field Theory of a Finite Tensor Category

Given a finite tensor category $\mathcal {C}$ , we prove that a modified trace on the tensor ideal of projective objects can be obtained from a suitable trivialization of the Nakayama functor as right $\mathcal {C}$ -module functor. Using a result of Costello, this allows us to associate to any finite tensor category equipped with such a trivialization of the Nakayama functor a chain complex valued topological conformal field theory, the trace field theory. The trace field theory topologically describes the modified trace, the Hattori-Stallings trace, and also the structures induced by them on the Hochschild complex of $\mathcal {C}$ . In this article, we focus on implications in the linear (as opposed to differential graded) setting: We use the trace field theory to define a non-unital homotopy commutative product on the Hochschild chains in degree zero. This product is block diagonal and can be described through the handle elements of the trace field theory. Taking the modified trace of the handle elements recovers the Cartan matrix of $\mathcal {C}$

P-nilpotency criteria for some verbal subgroups

In this paper we address the problem of understanding when a verbal subgroup of a finite group is p-nilpotent, with p a prime, that is, when all its elements of p′-order determine a subgroup. We provide two p-nilpotency criteria, one for the terms of the lower central series of any finite group and one for the terms of the derived series of any finite soluble group, which relies on arithmetic properties related to the order of products of commutators.