Commutative Algebra Research Articles

Leibniz rule for the high q$$ q $$‐derivatives of a quotient of two functions

In this paper, we introduce some explicit and recursive formulas for the th ‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of ‐differential operator. One of the formulas involves the determinant of the functions and their ‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting ‐identities.

Relative perinormality and relative global perinormality in a new pullback construction and idealization

If (S, J) is a zero-dimensional local ring, K = S / J its residue class field and D an integral domain with quotient field K, we show that twin notions of relative perinormality (N. Epstein, J. Shapiro, Perinormality in pullbacks Journal of Commutative Algebra 11 (3):341–362, 2019) and relative global perinormality (H. Klawa, Globally perinormal domains and graded perinormality, Doctoral Dissertation, George Mason University, 2023) are preserved by a pullback construction R = D × K S . As a byproduct, we recover the preservation of these notions in a new pullback construction of Chang et al. and idealization.

Universal early-time growth in quantum circuit complexity

We show that quantum circuit complexity for the unitary time evolution operator of any time-independent Hamiltonian is bounded by linear growth at early times, independent of any choices of the fundamental gates or cost metric. Deviations from linear early-time growth arise from the commutation algebra of the gates and are manifestly negative for any circuit, decreasing the linear growth rate and leading to a bound on the growth rate of complexity of a circuit at early times. We illustrate this general result by applying it to qubit and harmonic oscillator systems, including the coupled and anharmonic oscillator. By discretizing free and interacting scalar field theories on a lattice, we are also able to extract the early-time behavior and dependence on the lattice spacing of complexity of these field theories in the continuum limit, demonstrating how this approach applies to systems that have been previously difficult to study using existing techniques for quantum circuit complexity.

On $3$-generated axial algebras of Jordan type $\frac{1}{2}$

Axial algebras of Jordan type $\eta$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras. A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field.

Cyclotomic Generating Functions

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5)⊂S5. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed.

Transferring algebra structures on complexes

AbstractWith the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher $$A_\infty $$ A ∞ products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of $$A_\infty $$ A ∞ algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions.

Biderivations, commuting linear maps, post-Lie algebra structure on solvable Lie algebras of maximal rank

In this paper, all biderivations of solvable Lie algebras of maximal rank g = Q ⊕ N are characterized. Namely, we consider a solvable Lie algebra of the form g = Q ⊕ N , where Q is the maximal torus subalgebra of g , N is the nilradical of g and dim Q = dim N / N 2 .In the case dim Q = N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given.

About Supports of Local Cohomology Modules

The article provides the relation between the theory of local cohomology modules, and vanishing results, and also about the theory of support of such modules. Here, we put results about the theory, and also we provide a relation of local cohomology in the theory of commutative algebra and homological algebra.

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

Bipartite Sachdev-Ye models with Read-Saleur symmetries

We introduce an SU(M)-symmetric disordered bipartite spin model with unusual characteristics. Although superficially similar to the Sachdev-Ye (SY) model, it has several markedly different properties for M≥3. In particular, it has a large nontrivial nullspace whose dimension grows exponentially with system size. The states in this nullspace are frustration-free and are ground states when the interactions are ferromagnetic. The exponential growth of the nullspace leads to Hilbert-space fragmentation and a violation of the eigenstate thermalization hypothesis. We demonstrate that the commutant algebra responsible for this fragmentation is a nontrivial subalgebra of the Read-Saleur commutant algebra of certain nearest-neighbor models such as the spin-1 biquadratic spin chain. We also discuss the low-energy behavior of correlations for the disordered version of this model in the limit of a large number of spins and large M, using techniques similar to those applied to the SY model. We conclude by generalizing the Shiraishi-Mori embedding formalism to nonlocal models, and apply it to turn some of our nullspace states into quantum many-body scars. Published by the American Physical Society 2024

Equivariant prime ideals for infinite dimensional supergroups

Let A A be a commutative algebra equipped with an action of a group G G . The so-called G G -primes of A A are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When G G is an infinite dimensional group, these ideals can be very subtle: for instance, distinct G G -primes can have the same radical. In previous work, the second author showed that if G = G L ∞ G=\mathbf {GL}_{\infty } and A A is a polynomial representation, then these pathologies disappear when G G is replaced with the supergroup G L ∞ | ∞ \mathbf {GL}_{\infty |\infty } and A A with a corresponding algebra; this leads to a geometric description of G G -primes of A A . In the present paper, we construct an abstract framework around this result, and apply the framework to prove analogous results for other (super)groups. We give some applications to the isomeric determinantal ideals (commonly known as “queer determinantal ideals”).

Commutative avatars of representations of semisimple Lie groups

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials.

Inner isotopes associated with automorphisms of commutative associative algebras

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.

Diagonal tensor algebra and naïve liftings

The notion of naïve lifting of differential graded (DG) modules along certain DG algebra extensions was introduced by the authors for the purpose of studying problems in homological commutative algebra that involve self-vanishing of Ext. Our goal in this paper is to deeply study naïve lifting property using the diagonal tensor algebra. Our main result provides several characterizations of naïve liftability of DG modules under certain Ext vanishing conditions. As an application, we affirmatively answer a question posed by the authors regarding a specific characterization of naïve liftability.

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

Abstract The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl -regular Nijenhuis operators. We prove the splitting theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl -regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl -regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl -regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e. its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.

Automorphism groups of axial algebras

Axial algebras are a class of commutative non-associative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151.

Generalized Hom–Lie–Rinehart algebras

The first-order Hom-differential operators on a commutative Hom-associative algebra with unit are introduced. We describe Hom–Lie algebra associated with a module of first-order Hom-differential operators. We give the notion of a generalized Hom–Lie–Rinehart algebra and we show the characterization of generalized Hom–Lie–Rinehart algebras in terms of Hom–Gerstenhaber–Jacobi algebras. We prove the necessary and sufficient condition to obtain a generalized Hom–Lie–Rinehart algebra structure on a Hom–Lie algebroid.

On the pre-commutative envelopes of commutative algebras

On the pre-commutative envelopes of commutative algebras

Representation-reduced stated skein modules and algebras

For any marked three manifold (M,N) and any quantum parameter q12 (a nonzero complex number), we use Sq1/2(M,N) to denote the stated skein module of (M,N). When q12 is a root of unity of odd order, the commutative algebra S1(M,N) acts on Sq1/2(M,N). For any maximal ideal ρ of S1(M,N), define Sq1/2(M,N)ρ=Sq1/2(M,N)⊗S1(M,N)(S1(M,N)/ρ).We prove the splitting map for Sq1/2(M,N) respects the S1(M,N)-module structure, so it reduces to the splitting map for Sq1/2(M,N)ρ. We prove the splitting map for Sq1/2(M,N)ρ is injective if there exists at least one component of N such that this component and the boundary of the splitting disk belong to the same component of ∂M. We also prove the representation-reduced stated skein module of the marked handlebody is an irreducible Azumaya representation of the stated skein algebra of its boundary.Let M be an oriented connected closed three manifold. For any positive integer k, we use Mk to denote the marked three manifold obtained from M by removing k open three dimensional balls and adding one marking to each newly created sphere boundary component. We prove dimCSq1/2(Mk)ρ=1 for any maximal ideal ρ of S1(Mk).