Algebraic Invariants Research Articles

Finite gentle repetitions of gentle algebras and their Avella-Alaminos–Geiss invariants

Among finite dimensional algebras over a field K, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras A, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of A in terms of some values of its Avella-Alaminos–Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebras indexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos–Geiss invariants of A to the dimensions of Hochschild cohomologies of On the way we will see that the Avella-Alaminos–Geiss invariants of are completely determined by those of A. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the last part of this article, we will see how the generalized Auslander–Platzeck–Reiten reflection can be related to the finite gentle repetition.

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

Invariant submodel of rank 1 and two families of exact solutions of gas dynamics equations with an equation of state of the special form

In this article, the gas dynamics equations with an equation of state of the special form are considered.The equation of state is the pressure which is equal to the sum of two functions, with one being a function of a density, and the other one being a function of an entropy. The system of equations is invariant under the action of 12-parameter transformations group. For three-dimensional subalgebra 3.32 of the 12-dimensional Lie algebra invariants are calculated, an invariant submodel of rank 1 is constructed, and two families of exact solutions are obtained. The obtained solutions specify the motion of particles in space with a linear velocity field with inhomogeneous deformation. The first family of solutions has two moments of time of particles collapse. The second family of solutions has one moment of time of particles collapse on the plane. In the simplest case of second family of solutions, a surface consisting of particle trajectories is constructed.

S-Sequences and Monomial Modules

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noetherian ring with unit, has a specific initial module with respect to an admissible order, provided M is generated by an s-sequence. Significant examples complement the results.

Reduction type of smooth plane quartics

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\bar{K}$, in terms of the valuations of certain algebraic invariants of $C$ when the characteristic of the residue field is not $2,\,3,\,5$ or $7$. On the way, we gather several results of general interest on geometric invariant theory over an arbitrary ring $R$ in the spirit of (Seshadri 1977). For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$. We exhibit explicit ones for ternary quartic forms under the action of $\textrm{SL}_{3,R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give simple criteria for the type of reduction.

Algebraic invariants of weighted oriented graphs

Let $$\mathcal {D}$$ be a weighted oriented graph and let $$I(\mathcal {D})$$ be its edge ideal in a polynomial ring R. We give the formula of Castelnuovo–Mumford regularity of $$R/I(\mathcal {D})$$ when $$\mathcal {D}$$ is a weighted oriented path or cycle such that edges of $$\mathcal {D}$$ are oriented in one direction. Additionally, we compute the projective dimension for this class of graphs.

Algebraic invariants of orbit configuration spaces in genus zero associated to finite groups

We consider orbit configuration spaces associated to finite groups acting freely by orientation preserving homeomorphisms on the 2-sphere minus a finite number of points (eventually none). We compute the cohomology ring and the Poincaré series of these spaces. This generalizes the work of V. Arnold for “classical” configuration spaces of points of the plane. The results imply that the spaces we consider are formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincaré series of such spaces to the ranks of quotients of successive terms of the lower central series of their fundamental group. Such formula is known for the spaces studied by V. Arnold, where fundamental groups are Artin pure braid groups.

Differential invariants of Kundt spacetimes

We find generators for the algebra of rational differential invariants for general and degenerate Kundt spacetimes and relate this to other approaches to the equivalence problem for Lorentzian metrics. Special attention is given to dimensions three and four.

Certain invariant spaces of bounded measurable functions on a sphere

In their 1976 paper, Nagel and Rudin characterize the closed unitarily and Möbius invariant spaces of continuous and \(L^p\)-functions on a sphere, for \(1\le p<\infty \). In this paper we provide an analogous characterization for the weak*-closed unitarily and Möbius invariant spaces of \(L^\infty \)-functions on a sphere. We also investigate the weak*-closed unitarily and Möbius invariant algebras of \(L^\infty \)-functions on a sphere.

Group Classification Invariant Solutions of Burgers' Equation

The aims of the present paper is to solve the problem of the group classification of the general Burgers’ equation u_t=f(x,u) u_x^2+g(x,u)u_xx, where f and g are arbitrary smooth functions of the variables x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification: We followed the analysis mathematical method using the method of preliminary group classification. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. The result of the work is a wide class of equations summarized in table form.

On Metric Invariants of Spherical Harmonics

The algebraic and differential SO3-invariants of spherical harmonics are studied in this work. The fields of rational algebraic and rational differential invariants and their applications for the description of regular SO3-orbits of spherical harmonics are given.

Algebraic invariants of projections of varieties and partial elimination ideals

In this paper, we are interested in the properties of inner and outer projections with a view toward the Eisenbud-Goto regularity conjecture or the characterization of varieties satisfying certain extremal conditions. For example, if X is a quadratic scheme, the depth and regularity X and those of its inner projection from a smooth point are equal. In general, the above equalities do not hold for non-quadratic schemes. Therefore it is natural to investigate the algebraic invariants (e.g., depth and regularity) of X and its projected image in general.We develop a framework which provides partial answers and explains their relations using the partial elimination ideal theory. Our main theorems recover several preceding results in the literature. We also give some interesting examples and applications to illustrate our results.

On the solvability of graded Novikov algebras

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a [Formula: see text]-graded Novikov algebra [Formula: see text] over a field [Formula: see text] with solvable [Formula: see text]-component [Formula: see text] is solvable, where [Formula: see text] is a finite additive abelean group and the characteristic of [Formula: see text] does not divide the order of the group [Formula: see text]. We also show that any Novikov algebra [Formula: see text] with a finite solvable group of automorphisms [Formula: see text] is solvable if the algebra of invariants [Formula: see text] is solvable.

Algebraic Invariants in Abstract Cellular Complex

Algebraic Invariants in Abstract Cellular Complex

Cohomological invariants for central simple algebras of degree 8 and exponent 2

Cohomological invariants for central simple algebras of degree 8 and exponent 2

JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS

For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

Currents, charges and algebras in exceptional generalised geometry

A classical Ed(d)-invariant Hamiltonian formulation of world-volume theories of half-BPS p-branes in type IIb and eleven-dimensional supergravity is proposed, extending known results to d ≤ 6. It consists of a Hamiltonian, characterised by a generalised metric, and a current algebra constructed s.t. it reproduces the Ed(d) generalised Lie derivative. Ed(d)-covariance necessitates the introduction of so-called charges, specifying the type of p-brane and the choice of section. For p > 2, currents of p-branes are generically non- geometric due to the imposition of U-duality, e.g. the M5-currents contain coordinates associated to the M2-momentum.A derivation of the Ed(d)-invariant current algebra from a canonical Poisson structure is in general not possible. At most, one can derive a current algebra associated to para-Hermitian exceptional geometry.The membrane in the SL(5)-theory is studied in detail. It is shown that in a generalised frame the current algebra is twisted by the generalised fluxes. As a consistency check, the double dimensional reduction from membranes in M-theory to strings in type IIa string theory is performed. Many features generalise to p-branes in SL(p + 3) generalised geometries that form building blocks for the Ed(d)-invariant currents.

On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis

Modern weather and climate prediction models are based on a system of nonlinear partial differential equations called the primitive equations. Lie symmetries of the primitive equations with zero external heating rate are computed and the structure of its maximal Lie invariance algebra, which is infinite-dimensional, is studied. The maximal Lie invariance algebra for the case of a nonzero constant Coriolis parameter is mapped to the case of vanishing Coriolis force. The same mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame. This mapping is used to obtain exact solutions for the rotating case from exact solutions for the nonrotating equations. Another important result of the paper is the computation of the complete point symmetry group of the primitive equations using the algebraic method.

CONSTRUCTING TRI-TOPOLOGICAL NETWORK SPACE MODEL USING CONNECTED COMPONENT GRAPH THEORY (T3-C2G) BASED ON HOMOTOPY ALGEBRAIC INVARIANCE MODEL

The complex network contains non-deterministic topological spaces under an invariance structural approach to create failures on a continual link during communication. The non-lineardynamic topological structure leads to problematic threading links on network nodes due to a non-identical path to route the data. To resolve this problem, we propose atri-logical algebraic mathematical construction model called homotopy based tri-topological network spa- ce using connected component graph $(T^3-C^2G)$ under network nonlinear structure,The Algebraic Invariance Linear Queuing Theory (HA/I/LQT) is used to resolve the link failure route propagation to make improved communication performance. This homotopy reduction to reduce the complex nature to make continual link based on Quillen topological structure space under the covariance tri-topological structure. Further, this makes tri-logical structure resembles the sequence of triangle structured route space to make the nearest point of node adjustment from the nearest path. This balances the M/M/G-$T^3$-Max queuing theory on triangular weightage in routing schemes to specify the dynamic homotopy topological structure to make continuous routing links to reduce the complex nature of network routing.

Conditional probabilities via line arrangements and point configurations

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterization of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables; however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.