Algebraic Construction Research Articles

Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations

Abstract The KdV hierarchy is a paradigmatic example of the rich mathematical structure underlying integrable systems and has far-reaching connections in several areas of theoretical physics. While the positive part of the KdV hierarchy is well known, in this paper we consider an affine Lie algebraic construction for its negative part. We show that the original Miura transformation can be extended to a gauge transformation that implies several new types of relations among the negative flows of the KdV and mKdV hierarchies. Contrary to the positive flows, such a “gauge-Miura” correspondence becomes degenerate whereby more than one negative mKdV model is mapped into a single negative KdV model. For instance, the sine-Gordon and another negative mKdV flow are mapped into a single negative KdV flow which inherits solutions of both former models. The gauge-Miura correspondence implies a rich degeneracy regarding solutions of these hierarchies. We obtain similar results for the generalized KdV and mKdV hierachies constructed with the affine Lie algebra $$ \hat{s\ell}\left(r+1\right) $$ s ℓ ̂ r + 1 . In this case the first negative mKdV flow corresponds to an affine Toda field theory and the gauge-Miura correspondence yields its KdV counterpart. In particular, we show explicitly a KdV analog of the Tzitzéica-Bullough-Dodd model. In short, we uncover a rich mathematical structure for the negative flows of integrable hierarchies obtaining novel relations and integrable systems.

Values of the length function for nonassociative algebras

We study realizable values of the length function for unital possibly nonassociative algebras of a given dimension. To do this we apply the method of characteristic sequences and establish sufficient conditions of realizability for a given value of length. The proposed conditions are based on binary decompositions of the value and algebraic constructions that allow to modify length function of an algebra. Additionally we provide a description of unital algebras of maximal possible length in terms of their bases.

A note on RG domain wall between successive {A}_2^{(p)} minimal models

We investigate the RG domain wall between neighboring {A}_2^{(p)} minimal CFT models and establish the map between UV and IR fields (matrix of mixing coefficients). A particular RG invariant set of six primary and four descendant fields is analyzed in full details. Using the algebraic construction of the RG domain wall we compute the UV/IR mixing matrix. To test our results we show that it diagonalizes the matrix of anomalous dimensions previously known from perturbative analysis. It is important to note that the diagonalizing matrix can not be found from perturbative analysis solely due to degeneracy of anomalous dimensions. The same mixing coefficients are used to explore anomalous W-weights as well.

A new look at Lie algebras

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.

The Lie algebra of the lowest transitively differential group of degree three *

We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous polynomial coefficients of degree up to and including three. (2) The generator of the algebra must include the translation operators for all the variables . (3) The Lie algebra is the smallest indecomposable Lie algebra satisfying (1) and (2). It turns out to be a 39-dimensional Lie algebra in six variables (k = 6) and the construction of this algebra is also the simplest possible case in the general construction of the Lie algebras of the transitively differential groups introduced by Guillemin and Sternberg in 1964 involving the coefficients of degree 3. Those algebras and various subalgebras have similarities with algebras related to different applications in physics such as the Schrödinger, Conformal and Galilei algebras with and without central extension. The paper is devoted to the presentation of the structure and different decompositions of the Lie algebra under investigation. It is also devoted to the presentation of relevant Lie subalgebras and the construction of their Casimir invariants using different methods. We rely, in particular, on differential operator realizations, symbolic computation packages, the Berezin bracket and virtual copies of the Lie algebras.

Construction of Rank-One Solvable Rigid Lie Algebras with Nilradicals of a Decreasing Nilpotence Index

It is shown that for any integers k≥2, q≥2k and N≥k+q+2, there exists a real solvable Lie algebra of the first rank with a maximal torus of derivations t possessing the eigenvalue spectrum spec(t)=1,2,⋯,k,q,q+1⋯,N, a nilradical of the nilpotence index N−k and a characteristic sequence (N−k,1k).

A survey of metaheuristic algorithms for the design of cryptographic Boolean functions

Boolean functions are mathematical objects used in diverse domains and have been actively researched for several decades already. One domain where Boolean functions play an important role is cryptography. There, the plethora of settings one should consider and cryptographic properties that need to be fulfilled makes the search for new Boolean functions still a very active domain. There are several options to construct appropriate Boolean functions: algebraic constructions, random search, and metaheuristics. In this work, we concentrate on metaheuristic approaches and examine the related works appearing in the last 25 years. To the best of our knowledge, this is the first survey work on this topic. Additionally, we provide a new taxonomy of related works and discuss the results obtained. Finally, we finish this survey with potential future research directions.

Hall algebras and quantum symmetric pairs of Kac-Moody type

We extend our ıHall algebra construction from acyclic to arbitrary ıquivers, where the ıquiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal ıquantum group of Kac-Moody type arising from quantum symmetric pairs to the ıHall algebra associated to a virtually acyclic ıquiver.

Algebraic Diagrammatic Construction Schemes Employing the Intermediate State Formalism: Theory, Capabilities, and Interpretation.

Algebraic diagrammatic construction (ADC) schemes represent a family of ab initio methods for the calculation of excited electronic states and electron-detached and -attached states. All ADC methods have been demonstrated to possess great potential for molecular applications, e.g., for the calculation of absorption or photoelectron spectra or electron attachment processes. ADC originates from Green's function or propagator theory; however, most recent ADC developments heavily rely on the intermediate state representation or effective Liouvillian formalisms, which comprise new ADC methods and computational schemes for high-order properties. The different approaches for the calculation of excitation energies, ionization potentials, and electron affinities are intimately related, and they provide a coherent description of these quantities at equivalent levels of theory and with comparable errors. Most quantum chemical program packages contain ADC methods; however, the most complete ADC suite of methods can be found in the recent release of Q-Chem.

A Hammett's analysis of the substituent effect in functionalized diketopyrrolopyrrole (DPP) systems: Optoelectronic properties and intramolecular charge transfer effects.

Diketopyrrolopyrrole (DPP) systems have promising applications in different organic electronic devices. In this work, we investigated the effect of 20 different substituent groups on the optoelectronic properties of DPP-based derivatives as the donor ( )-material in an organic photovoltaic (OPV) device. For this purpose, we employed Hammett's theory (HT), which quantifies the electron-donating or -withdrawing properties of a given substituent group. Machine learning (ML)-based , , , , , , , and Hammett's constants previously determined were used. Mono- (DPP-X1 ) and di-functionalized (DPP-X2 ) DPPs, where X is a substituent group, were investigated using density functional theory (DFT), time-dependent DFT (TDDFT), and ab initio methods. Several properties were computed using CAM-B3LYP and the second-order algebraic diagrammatic construction, ADC(2), an ab initio wave function method, including the adiabatic ionization potential ( ), the electron affinity ( ), the HOMO-LUMO gaps ( ), and the maximum absorption wavelengths ( ), the first excited state transition 1 S0 → 1 S1 energies ( ) (the optical gap), and exciton binding energies. From the optoelectronic properties and employing typical acceptor systems, the power conversion efficiency ( ), open-circuit voltage ( ), and fill factor ( ) were predicted for a DPP-based OPV device. These photovoltaic properties were also correlated with the machine learning (ML)-based Hammett's constants. Overall, good correlations between all properties and the different types of constants were obtained, except for the constants, which are related to inductive effects. This scenario suggests that resonance is the main factor controlling electron donation and withdrawal effects. We found that substituent groups with large values can produce higher photovoltaic efficiencies. It was also found that electron-withdrawing groups (EWGs) reduced and considerably compared to the unsubstituted DPP-H. Moreover, for every decrease (increase) in the values of a given optoelectronic property of DPP-X1 systems, a more significant decrease (increase) in the same values was observed for the DPP-X2 , thus showing that the addition of the second substituent results in a more extensive influence on all electronic properties. For the exciton binding energies, an unsupervised machine learning algorithm identified groups of substituents characterized by average values (centroids) of Hammett's constants that can drive the search for new DDP-derived materials. Our work presents a promising approach by applying HT on molecular engineering DPP-based molecules and other conjugated molecules for applications on organic optoelectronic devices.

Bootstrapping string dynamics in the 6d \U0001d4a9 = (2, 0) theories

We present two complementary approaches to calculating the 2-point function of stress tensors in the presence of a 1/2 BPS surface defect of the 6d \U0001d4a9 = (2, 0) theories. First, we use analytical bootstrap techniques at large N to obtain the first nontrivial correction to this correlator, from which we extract the defect CFT (dCFT) data characterising the 2d dCFT of the 1/2 BPS plane. Along the way we derive a supersymmetric inversion formula, obtain the relevant superconformal blocks and check that crossing symmetry is satisfied. Notably our result features a holomorphic function whose appearance is related to the chiral algebra construction of Beem, Rastelli and van Rees. Second, we use that chiral algebra description to obtain exact results for the BPS sector of the dCFT, valid at any N and for any choice of surface operator. These results provide a window into the dynamics of strings of the mysterious 6d theories.

Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

Graph test of controllability in qubit arrays: a systematic way to determine the minimum number of external controls

The ability to implement any desired quantum logic gate on a quantum processing unit is equivalent to evolution-operator controllability of the qubits. Conversely, controllability analysis can be used to minimize the resources, i.e. the number of external controls and qubit–qubit couplings, required for universal quantum computing. Standard controllability analysis, consisting in the construction of the dynamical Lie algebra, is, however, impractical already for a comparatively small number of qubits. Here, we show how to leverage an alternative approach, based on a graph representation of the Hamiltonian, to determine controllability of arrays of coupled qubits. We provide a complete computational framework and exemplify it for arrays of five qubits, inspired by the ibmq_quito architecture. We find that the number of controls can be reduced from five to one for complex qubit–qubit couplings and to two for standard qubit–qubit couplings.

Core-Excited States and X-ray Absorption Spectra from Multireference Algebraic Diagrammatic Construction Theory.

We report the development and benchmark of multireference algebraic diagrammatic construction theory (MR-ADC) for the simulations of core-excited states and X-ray absorption spectra (XAS). Our work features an implementation that incorporates core-valence separation into the strict and extended second-order MR-ADC approximations (MR-ADC(2) and MR-ADC(2)-X), providing efficient access to high-energy excited states without including inner-shell orbitals in the active space. Benchmark results on a set of small molecules indicate that at equilibrium geometries, the accuracy of MR-ADC is similar to that of single-reference ADC theory when static correlation effects are not important. In this case, MR-ADC(2)-X performs similarly to single- and multireference coupled cluster methods in reproducing the experimental XAS peak spacings. We demonstrate the potential of MR-ADC for chemical systems with multiconfigurational electronic structure by calculating the K-edge XAS spectrum of the ozone molecule with a multireference character in its ground electronic state and the dissociation curve of core-excited molecular nitrogen. For ozone, the MR-ADC results agree well with the data from experimental and previous multireference studies of ozone XAS, in contrast to the results of single-reference methods, which underestimate relative peak energies and intensities. The MR-ADC methods also predict the correct shape of the core-excited nitrogen potential energy curve, and are in good agreement with accurate calculations using driven similarity renormalization group approaches. These findings suggest that MR-ADC(2) and MR-ADC(2)-X are promising methods for the XAS simulations of multireference systems and pave the way for their efficient computer implementation and applications.

Various instances of Harish-Chandra pairs

In this paper we address several algebraic constructions in the context of groupoids, algebroids and Z-graded manifolds. We generalize the results of integration of N-graded Lie algebras to the honest Z-graded case and provide some examples of application of the technique based on Harish–Chandra pairs. We extend the construction to the action Lie algebroid setting.

A double construction of quadratic anticenter-symmetric Jacobi-Jordan algebras

This work addresses some relevant characteristics and properties of anticenter-symmetric Jacobi-Jordan algebras such as bimodules, matched pairs. Besides, the Jacobi-Jordan admissible algebra is defined; a special emphasis is given to a double construction of quadratic anticenter-symmetric algebras. We then follow this theory with the main properties and related algebraic structures of an anticenter-symmetric JJ algebra, namely the anti-Zinbiel algebras. Finally, we discuss the double construction of some classes of the two dimensional anticenter-symmetric JJ algebras.

Convergence acceleration of preconditioned conjugate gradient solver based on error vector sampling for a sequence of linear systems

AbstractIn this article, we focus on solving a sequence of linear systems that have identical (or similar) coefficient matrices. For this type of problem, we investigate subspace correction (SC) and deflation methods, which use an auxiliary matrix (subspace) to accelerate the convergence of the iterative method. In practical simulations, these acceleration methods typically work well when the range of the auxiliary matrix contains eigenspaces corresponding to small eigenvalues of the coefficient matrix. We develop a new algebraic auxiliary matrix construction method based on error vector sampling in which eigenvectors with small eigenvalues are efficiently identified in the solution process. We use the generated auxiliary matrix for convergence acceleration in the following solution step. Numerical tests confirm that both SC and deflation methods with the auxiliary matrix can accelerate the solution process of the iterative solver. Furthermore, we examine the applicability of our technique to the estimation of the condition number of the coefficient matrix. We also present the algorithm of the preconditioned conjugate gradient method with condition number estimation.

A General Construction of Integer Codes Correcting Specific Errors in Binary Communication Channels

Integer codes have been successfully applied to various areas of communication and computer technology. They demonstrate good performance in correcting specific kinds of errors. In many cases, the used integer codes are constructed by computer search. This paper presents an algebraic construction of integer codes over the ring of integers modulo A=2n+1 capable of correcting at least up to two bit errors in a single b-byte. Moreover, the codes can correct some configurations of three or more erroneous bits, but not all possible ones. The construction is based on the use of cyclotomic cosets of 2 modulo A.

An algebraic construction technique for codes over Hurwitz integers

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

Equivariant K-theory of the semi-infinite flag manifold as a nil-DAHA module

The equivariant K-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a q-Heisenberg algebra. The action of the latter generates a free submodule of rank |W|, where W is the (finite) Weyl group. We show that this submodule is stable under the nil-DAHA, which enables one to express the nil-DAHA action in terms of |W|times |W| matrices over the q-Heisenberg algebra. Our main result gives an explicit algebraic construction of these matrices as a limit from the (non-nil) DAHA in simply-laced type. This construction reveals that multiplication by equivariant scalars, when expressed in terms of the Heisenberg algebra, is given by the nonsymmetric q-Toda system introduced by Cherednik and the author.