Algebraic Invariants Research Articles

On the Generators of Quantum Dynamical Semigroups with Invariant Subalgebras

The problem of characterizing GKLS-generators and CP-maps with an invariant von Neumann algebra [Formula: see text] appeared in different guises in the literature. We prove two unifying results, which hold even for weakly closed *-algebras: first, we show how to construct a normal form for [Formula: see text]-invariant GKLS-generators, if a normal form for [Formula: see text]-invariant CP-maps is known — rendering the two problems essentially equivalent. Second, we provide a normal form for [Formula: see text]-invariant CP-maps if [Formula: see text] is atomic (which includes the finite-dimensional case). As an application we reproduce several results from the literature as direct consequences of our characterizations and thereby point out connections between different fields.

TIDAL: Topology-Inferred Drug Addiction Learning.

Drug addiction is a global public health crisis, and the design of antiaddiction drugs remains a major challenge due to intricate mechanisms. Since experimental drug screening and optimization are too time-consuming and expensive, there is urgent need to develop innovative artificial intelligence (AI) methods for addressing the challenge. We tackle this challenge by topology-inferred drug addiction learning (TIDAL) built from integrating multiscale topological Laplacians, deep bidirectional transformer, and ensemble-assisted neural networks (EANNs). Multiscale topological Laplacians are a novel class of algebraic topology tools that embed molecular topological invariants and algebraic invariants into its harmonic spectra and nonharmonic spectra, respectively. These invariants complement sequence information extracted from a bidirectional transformer. We validate the proposed TIDAL framework on 22 drug addiction related, 4 hERG, and 12 DAT data sets, which suggests that the proposed TIDAL is a state-of-the-art framework for the modeling and analysis of drug addiction data. We carry out cross-target analysis of the current drug addiction candidates to alert their side effects and identify their repurposing potentials. Our analysis reveals drug-mediated linear and bilinear target correlations. Finally, TIDAL is applied to shed light on relative efficacy, repurposing potential, and potential side effects of 12 existing antiaddiction medications. Our results suggest that TIDAL provides a new computational strategy for pressingly needed antisubstance addiction drug development.

Invariants of formal pseudodifferential operator algebras and algebraic modular forms

We study from an algebraic point of view the question of extending an action of a group Γ on a commutative domain R to a formal pseudodifferential operator ring B = R((x ; d)) with coefficients in R, as well as to some canonical quadratic extension C = R((x 1/2 ; 1 2 d))2 of B. We give a necessary and sufficient condition of compatibility between the action and the derivation d of R for such an extension to exist, and we determine all possible extensions of the action to B and C. We describe under suitable assumptions the invariant subalgebras B Γ and C Γ as Laurent series rings with coefficients in R Γ. The main results of this general study are applied in a numbertheoretical context to the case where Γ is a subgroup of SL(2, C) acting by homographies on an algebra R of functions in one complex variable. Denoting by Mj the vector space of algebraic modular forms in R of weight j (even or odd), we build for any nonnegative integer k a linear isomorphism between the subspace C Γ k of invariant operators of order ≥ k in C Γ and the product space M k = j≥k Mj, which can be identified with a space of algebraic Jacobi forms of weight k. It results in particular a structure of noncommutative algebra on M0 and an algebra isomorphism Ψ : M0 → C Γ 0 , whose restriction to the particular case of even weights was previously known in the litterature. We study properties of this correspondence combining arithmetical arguments and the use of the algebraic results of the first part of the article.

The complexity of solving Weil restriction systems

The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Gröbner basis methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension k↪K, associates to a system F defined over K a system Weil(F) defined over k, in such a way that the solutions of F over K and those of Weil(F) over k are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system Weil(F) obtained via Weil restriction in terms of algebraic invariants of the system F.

3D geometric moment invariants from the point of view of the classical invariant theory

The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory.Using the remarkable fact that the complex groups $SO(3,\mathbb{C})$ and $SL(2,\mathbb{C})$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory.
 We give a precise statement of the 3D geometric invariant moments computation, intro\-ducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2,\mathbb{C})$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3,\mathbb{C})$ to equivalent action of the complex Lie algebra $\mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in thefields of image analysis and pattern recognition.

Publisher Correction: On the algebraic invariants of certain affine semigroup rings

Publisher Correction: On the algebraic invariants of certain affine semigroup rings

О базисных инвариантах некоторых конечных подгрупп в SL3(C)

The work is devoted to the study of algebras of invariants of finite unitary groups G' = G∩SL3(C), where G is a finite unitary irreducible primitive group generated by reflections in the unitary space U3. It is known that the system of invariants of the group G' that form an algebra is obtained from the system of invariants of the group G that form the algebra by adding all semi-invariants of the group G of a special form. In the paper, generators of the algebras of invariants of all the indicated groups G' are constructed.

Some algebraic invariants of the edge ideals of perfect $ [h, d] $-ary trees and some unicyclic graphs

<abstract><p>This article is mainly concerned with computations of some algebraic invariants of quotient rings of edge ideals of perfect $ [h, d] $-ary trees and unicyclic graphs. We compute exact values of depth and Stanley depth and consequently projective dimension for above mentioned quotient rings, except for the one special case of unicyclic graph for which best possible bounds of Stanley depth are given.</p></abstract>

Remark on complements on surfaces

Abstract We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is $\frac {1}{42}$ -lc. As corollaries, we show that any $\mathbb R$ -complementary surface X has an n-complement for some integer $n\leq 192\cdot 84^{128\cdot 42^5}\approx 10^{10^{10.5}}$ , and Tian’s alpha invariant for any surface is $\leq 3\sqrt {2}\cdot 84^{64\cdot 42^5}\approx 10^{10^{10.2}}$ . Although the latter two values are expected to be far from being optimal, they are the first explicit upper bounds of these two algebraic invariants for surfaces.

Generic Gelfand-Tsetlin modules of quantized and classical orthogonal algebras

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard q-deformed enveloping algebra Uq′(son) defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra U(son). In this paper we only consider the case when q is not a root of unity, and q→1 for the classical case. Extending work by Mazorchuk on son, we provide rational matrix coefficients for these infinite-dimensional modules of both Uq′(son) and U(son). We use these modules with rationalized formulas to embed the respective algebras into skew group algebras of shift operators. Casimir elements of Uq′(son) were given by Gavrilik and Iorgov, and we consider the commutative subalgebra Γ⊂Uq′(son) generated by these elements and the corresponding subalgebra Γ1⊂U(son). The images of Γ and Γ1 under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show that Γ is a Harish-Chandra subalgebra of Uq′(son) and Γ1 is a Harish-Chandra subalgebra of U(son).

Secondary higher invariants and cyclic cohomology for groups of polynomial growth

We prove that if \Gamma is a group of polynomial growth, then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocycle, we thus define a higher analogue of Lott’s delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if \Gamma is of polynomial growth, then there is a well-defined pairing between delocalized cyclic cocycles and K -theory classes of C^* -algebraic secondary higher invariants. When this K -theory class is that of a higher rho invariant of an invertible differential operator, we show this pairing is precisely the aforementioned higher analogue of Lott’s delocalized eta invariant. As an application of this equivalence, we provide a delocalized higher Atiyah–Patodi–Singer index theorem, given that M is a compact spin manifold with boundary, equipped with a positive scalar metric g and having fundamental group \Gamma=\pi_1(M) which is finitely generated and of polynomial growth.

Some algebraic Invariants of the residue class rings of the edge ideals of perfect semiregular trees

Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I.

On the algebraic invariants of certain affine semigroup rings

On the algebraic invariants of certain affine semigroup rings

Weierstrass sections for parabolic adjoint action in type A

The notion of “Weierstrass Section”, comes from Weierstrass canonical form for elliptic curves. In celebrated work Kostant (1963) [26] constructed such a section for the action of a semisimple Lie algebra on its dual using a principal s-triple. Actually it is enough to have an “adapted pair” and indeed the construction in Joseph and Shafrir (2010) [25] works rather well for the co-adjoint action of an algebraic, but not necessarily reductive Lie algebra.In the present work a Weierstrass section is constructed for the adjoint action of a parabolic subgroup P on the nilradical m of its Lie algebra in type A. The starting point is Richardson's theorem which asserts that m admits a dense P orbit (in all types).For type A an explicit generator of this dense orbit was given in Brüstle et al. (1999) [4]. It involves the joining boxes in the Young tableau associated to P.On the other hand Richardson's theorem implies the polynomiality of the (semi)-invariant subalgebra, where the generators themselves are given through ideals of definition of hypersurface orbital varieties Joseph and Melnikov (2003) [24]. (Again this relationship holds in all types.)In this adapted pairs seldom exist.Thus an entirely new construction is developed and this is mainly combinatorial.The initial step is the construction of Ringel et al. ([24]). In this the joining of boxes is significantly modified to take account of the explicit form of the generators of the invariant algebra suggested in Benlolo and Sanderson (2001) [3].Indications are given as to how one might extend this construction of a Weierstrass section to other types.

On the classification of 3-dimensional complex hom-Lie algebras

Let hom-Ln(C) be the algebraic set of complex n-dimensional hom-Lie algebras. The group GL(n,C) acts on it via change of basis. We classify hom-Lie structures with nilpotent twisting map on 3-dimensional complex Lie algebras, up to isomorphism, and we give the classification of orbit closures in such family. For this purpose, we introduce some invariants of hom-Lie algebras to study both problems. The ideas and techniques presented here can be easily extrapolated to study similar problems on other linear algebraic structures.

Benney–Roskes/Zakharov–Rubenchik system: Lie symmetries and exact solutions

We investigate Lie symmetry algebra of the Benney–Roskes/Zakharov–Rubenchik systems. The invariance algebra turns out to be infinite-dimensional. We also find several exact solutions of periodic, line soliton and stationary types.

GEOMETRY OF THE FOCAL SURFACES OF THE INVERSE SURFACE OF A REGULAR SURFACE IN E3

The purpose of this paper, first, is to give a definition of the focal surfaces of the inverse of a given regular surface in E3. Second, some new characteristic properties of the focal surfaces are to express depending on the algebraic invariants of the inverse surface of a given regular surface. In the last part of the study, we gave examples supporting our claims and plotted their graphics with the help of Maple software.

Projective invariants of images

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$ , it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras . In this approach, we deal with explicit polynomials in the enveloping algebra of . We present explicit examples of low-dimensional Lie algebras (up to dimension six) to show how they can display different behaviours and can lead to abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the Levi factor of a Levi decomposable Lie algebra can be used as a tool to construct ‘copies’ of polynomial algebras. Different schemes to obtain polynomial algebras associated to algebraic Hamiltonians have been proposed in the literature, among them the use of commutants of various type. The present approach is different and relies on the construction of intermediate Casimir invariants in the enveloping algebra .

Centralizer matrix algebras and symmetric polynomials of partitions

For a field R of characteristic p≥0 and a matrix c in the full n×n matrix algebra Mn(R) over R, let Sn(c,R) be the centralizer algebra of c in Mn(R). We show that Sn(c,R) is a Frobenius-finite, 1-Auslander-Gorenstein, and gendo-symmetric algebra, and that the extension Sn(c,R)⊆Mn(R) is separable and Frobenius. Further, we study the isomorphism problem of invariant matrix algebras. Let σ be a permutation in the symmetric group Σn and cσ the corresponding permutation matrix in Mn(R). We give sufficient and necessary conditions for the invariant algebra Sn(cσ,R) to be semisimple. If R is an algebraically closed field, we establish a combinatoric characterization of when two semisimple invariant R-algebras are isomorphic in terms of the cycle types of permutations.