Commutative Algebra Research Articles

Fp-Projective periodicity

The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the behavior of the objects of cocycles in acyclic complexes. It is known that any flat Proj-periodic module is projective, any fp-injective Inj-periodic module is injective, and any Cot-periodic module is cotorsion. It is also known that any pure PProj-periodic module is pure-projective and any pure PInj-periodic module is pure-injective. Generalizing a result of Šaroch and Št'ovíček, we show that every FpProj-periodic module is weakly fp-projective. The proof is quite elementary, using only a strong form of the pure-projective periodicity and the Hill lemma. More generally, we prove that, in a locally finitely presentable Grothendieck category, every FpProj-periodic object is weakly fp-projective. In a locally coherent category, all weakly fp-projective objects are fp-projective. We also present counterexamples showing that a non-pure PProj-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.

Conformable fractional derivative in commutative algebras

Conformable fractional derivative in commutative algebras

Joining Iso-Structured Models with Commutative Orthogonal Block Structure

In this work, we focus on a special class of mixed models, named models with commutative orthogonal block structure (COBS), whose covariance matrix is a linear combination of known pairwise orthogonal projection matrices that add to the identity matrix, and for which the orthogonal projection matrix on the space spanned by the mean vector commutes with the covariance matrix. The COBS have least squares estimators giving the best linear unbiased estimators for estimable vectors. Our approach to COBS relies on their algebraic structure, based on commutative Jordan algebras of symmetric matrices, which proves to be advantageous as it leads to important results in the estimation. Specifically, we are interested in iso-structured COBS, applying to them the operation of models joining. We show that joining iso-structured COBS gives COBS and that the estimators for the joint model may be obtained from those for the individual models.

Structure of primitive axial algebras

A “fusion law” is a collection of multiplication rules among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras are closely related to 3-transposition groups and vertex operator algebras.In earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite-dimensional noncommutative algebras, and commutative axial algebras generated by primitive axes of the same type. In particular, all primitive axial algebras are flexible. They also have Frobenius forms. We give a precise description of all the primitive axes of axial algebras generated by two primitive axes.

WIENER TAUBERIAN THEOREMS FOR CERTAIN BANACH ALGEBRAS ON REAL RANK ONE SEMISIMPLE LIE GROUPS

AbstractWe prove Wiener Tauberian theorem type results for various spaces of radial functions, which are Banach algebras on a real-rank-one semisimple Lie group G. These are natural generalizations of the Wiener Tauberian theorem for the commutative Banach algebra of the integrable radial functions on G.

Module braces: relations between the additive and the multiplicative groups

In this paper, we define a class of braces that we call module braces or R-braces, which are braces for which the additive group has also a module structure over a ring R, and for which the values of the gamma functions are automorphisms of R-modules. This class of braces has already been considered in the literature in the case where the ring R is a field; we generalise the definition to any ring R, reinterpreting it in terms of the so-called gamma function associated with the brace, and prove that this class of braces enjoys all the natural properties one can require. We exhibit explicit example of R-braces, and we study the splitting of a module braces in relation to the splitting of the ring R, generalising thereby Byott’s result on the splitting of a brace with nilpotent multiplicative group as a sum of its Sylow subgroups. The core of the paper is in the last two sections, in which, using methods from commutative algebra and number theory, we study the relations between the additive and the multiplicative groups of an R-brace showing that if a certain decomposition of the additive group is small (in some sense which depends on R), then the additive and the multiplicative groups have the same number of elements of each order. In some cases, this result considerably broadens the range of applications of the results already known on this issue.

Invariant metrics on current Lie algebras

In this work we state conditions for a current Lie algebra to admit an invariant metric, where is a quadratic Lie algebra and is an associative and commutative algebra with unit. We show that if is an indecomposable quadratic Lie algebra, then admits an invariant metric if and only if also admits an invariant metric. In addition, we prove a theorem similar to the double extension for , where is an indecomposable, nilpotent and quadratic Lie algebra.

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

Positivstellensätze for semirings

In this paper we develop a number of results and notions concerning Positivstellensätze for semirings (preprimes) of commutative unital real algebras. First we reduce the Archimedean Positivstellensatz for semirings to the corresponding result for quadratic modules. Various applications of the Archimedean Positivstellensatz for semirings are investigated. A general Positivstellensatz with denominators is proved for filtered algebras with semirings. As an application we derive a denominator-free Positivstellensatz for the cylindrical extension of an algebra with Archimedean semiring. A large number of illustrating examples are given.

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.

New Kind of Banach Algebra Via Proximit structure

This article introduces the concept of Banach proximit algebra and examines several results in this new context. We give new definitions for the terms "proximit algebra," "commutative proximit algebra," "identity proximit algebra," and "normed proximit algebra." However, we introduce the concepts of subproximit algebra, proximit homomorphism, and proximit isomorphism of Banach proximit algebras in a natural way. Finally, we offer a few intriguing possibilities to support our arguments.

Conformable fractional derivative in commutative algebras

Conformable fractional derivative in commutative algebras

Numerical methods for detecting symmetries and commutant algebras

Numerical methods for detecting symmetries and commutant algebras

Graded Lie-Rinehart algebras

The present work introduces the class of graded Lie-Rinehart algebras as a natural generalization of graded Lie algebras. It is demonstrated that a tight G-graded Lie-Rinehart algebra L over a commutative and associative G-graded algebra A, where G is an abelian group, can be decomposed into the orthogonal direct sums L=⨁i∈IIi and A=⨁j∈JAj, where each Ii and Aj is a non-zero ideal of L and A, respectively. Additionally, both decompositions satisfy that for any i∈I, there exists a unique j∈J such that AjIi≠0 and that any Ii is a graded Lie-Rinehart algebra over Aj. In the case of maximal length, the aforementioned decompositions of L and A are through indecomposable (graded) ideals, and the (graded) simplicity of any Ii and any Aj are also characterized.

Dirac Geometry I: Commutative Algebra

AbstractThe purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of differential graded algebras and, more generally, as the homotopy of algebras in spectra. The commutative algebras in question are those in the symmetric monoidal category of graded abelian groups, and, being commutative, they form the affine building blocks of a geometry, as commutative rings form the affine building blocks of algebraic geometry. We name this geometry Dirac geometry, because the grading exhibits the hallmarks of spin in that it is a remnant of the internal structure encoded by anima, it distinguishes symmetric and anti-symmetric behavior, and the coherent cohomology of Dirac schemes and Dirac stacks, which we develop in the sequel, admits half-integer Serre twists. Thus, informally, Dirac geometry constitutes a “square root” of $$\mathbb {G}_m$$ G m -equivariant algebraic geometry.

From symmetries to commutant algebras in standard Hamiltonians

In this work, we revisit several families of standard Hamiltonians that appear in the literature and discuss their symmetries and conserved quantities in the language of commutant algebras. In particular, we start with families of Hamiltonians defined by parts that are local, and study the algebra of operators that separately commute with each part. The families of models we discuss include the spin-1/2 Heisenberg model and its deformations, several types of spinless and spinful free-fermion models, and the Hubbard model. This language enables a decomposition of the Hilbert space into dynamically disconnected sectors that reduce to the conventional quantum number sectors for regular symmetries. In addition, we find examples of non-standard conserved quantities even in some simple cases, which demonstrates the need to enlarge the usual definitions of symmetries and conserved quantities. In the case of free-fermion models, this decomposition is related to the decompositions of Hilbert space via irreducible representations of certain Lie groups proposed in earlier works, while the algebra perspective applies more broadly, in particular also to arbitrary interacting models. Further, the von Neumann Double Commutant Theorem (DCT) enables a systematic construction of local operators with a given symmetry or commutant algebra, potentially eliminating the need for “brute-force” numerical searches carried out in the literature, and we show examples of such applications of the DCT. This paper paves the way for both systematic construction of families of models with exact scars and characterization of such families in terms of non-standard symmetries, pursued in a parallel paper Moudgalya and Motrunich (2022) (https://arxiv.org/abs/2209.03377).

Isomorphism between the Algebra of Measurable Functions and its Subalgebra of Approximately Differentiable Functions

The present paper is devoted to study of certain classes of homogeneous regular subalgebras of the algebra of all complex-valued measurable functions on the unit interval. It is known that the transcendence degree of a commutative unital regular algebra is one of the important invariants of such algebras together with Boolean algebra of its idempotents. It is also known that if $(\Omega, \Sigma, \mu)$ is a Maharam homogeneous measure space, then two homogeneous unital regular subalgebras of $S(\Omega)$ are isomorphic if and only if their Boolean algebras of idempotents are isomorphic and transcendence degrees of these algebras coincide. Let $S(0,1)$ be the algebra of all (classes of equivalence) measurable complex-valued functions and let $AD^{(n)}(0,1)$ ($n\in \mathbb{N}\cup\{\infty\}$) be the algebra of all (classes of equivalence of) almost everywhere $n$-times approximately differentiable functions on $[0,1].$ We prove that $AD^{(n)}(0,1)$ is a regular, integrally closed, $\rho$-closed, $c$-homogeneous subalgebra in $S(0,1)$ for all $n\in \mathbb{N}\cup\{\infty\},$ where $c$ is the continuum. Further we show that the algebras $S(0,1)$ and $AD^{(n)}(0,1)$ are isomorphic for all $n\in \mathbb{N}\cup\{\infty\}.$ As an application of these results we obtain that the dimension of the linear space of all derivations on $S(0,1)$ and the order of the group of all band preserving automorphisms of $S(0,1)$ coincide and are equal to $2^c.$ Finally, we show that the Lie algebra $\operatorname{Der} S(0, 1)$ of all derivations on $S(0,1)$ contains a subalgebra isomorphic to the infinite dimensional Witt algebra.

Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras

We introduce a dual notion of the Poisson algebra, called the transposed Poisson algebra, by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. The transposed Poisson algebra shares common properties of the Poisson algebra and arises naturally from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic construction of a Poisson algebra from a commutative algebra with two commuting derivations similarly applies to a transposed Poisson algebra. More broadly, the transposed Poisson algebra captures the algebraic structures when the commutator is taken in pre-Lie Poisson algebras and two other Poisson type algebras. Furthermore, the transposed Poisson algebra improves two processes that produce 3-Lie algebras from Poisson algebras with a strongness condition.

A lower bound for the dimension of tetrahedral splines in large degree

Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.

On integral extensions of Banach algebras

We consider countable extensions of commutative and unital Banach algebras. We study these Banach algebra structures with or without assuming the continuity of the canonical injection. We also prove that a countable extension endowed with a Banach algebra norm with continuous injection is actually a finite extension.