Algebraic Questions Research Articles

Group gradings on the Lie algebra of upper triangular matrices

The algebras UTn of the n×n upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. We classify the group gradings on the Lie algebra UTn(−). It was proved by Valenti and Zaicev in 2007 that every group grading on the associative algebra UTn is isomorphic to an elementary grading. The elementary gradings on UTn are also well understood, see [6]. It follows from our results that there are nonelementary gradings on UTn(−). Thus the gradings on the Lie algebra UTn(−) are much more intricate than those in the associative case.

Elementary gradings on the Lie algebra [formula omitted

The algebras UTn(K) of the upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. In this paper we consider the Lie algebra UTn(K)(−) of all upper triangular matrices of order n. We study the group gradings on this algebra. It turns out that the gradings on the Lie algebra UTn(K) are much more intricate than those in the associative case. In this paper we describe the elementary gradings on the Lie algebra UTn(K)(−). Finally we study the canonical grading on UTn(K)(−) by the cyclic group Zn of order n. We produce a (finite) basis of the graded polynomial identities satisfied by this grading.

A CHARACTERISATION OF CENTRAL ELEMENTS IN -ALGEBRAS

Wu [‘An order characterization of commutativity for$C^{\ast }$-algebras’,Proc. Amer. Math. Soc.129(2001), 983–987] proved that if the exponential function on the set of all positive elements of a$C^{\ast }$-algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in$C^{\ast }$-algebras in terms of the order.

An analogue of the Aluffi algebra for modules

P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism A↠B by taking a suitable inverse limit of graded algebras, one for each representation of A as a residue ring of a given “ambient” ring R. Since giving a ring surjection A↠B is tantamount to giving an ideal I⊂A, it would seem natural to ask for an analogous notion for A-modules. This is the central purpose of this work. Since a given module may not admit any embedding into a free module, a preparatory toil includes dealing with this technical point at the outset. On the bright side, the intrusion of modules raises a few algebraic questions interesting on their own. It is to expect that this extension to modules may be transcribed in terms of coherent sheaves, thus possibly providing an answer to a question by Aluffi in this regard. Two main bodies of examples are treated in detail to illustrate how the theory works and to show the relation to finer properties of other algebras.

Semantical conditions for the definability of functions and relations

Let \({\mathcal{L}\subseteq \mathcal{L}^\prime}\) be first order languages, let \({R \in \mathcal{L}^\prime- \mathcal{L}}\) be a relation symbol, and let \({\mathcal{K}}\) be a class of \({\mathcal{L}^\prime}\)-structures. In this paper, we present semantical conditions equivalent to the existence of an \({\mathcal{L}}\)-formula \({\varphi(\vec{x})}\) such that \({\mathcal{K}\vDash \varphi(\vec{x}) \leftrightarrow R(\vec{x})}\), where \({\varphi}\) has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential Horn, etc.). For each of these definability results for relations, we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley’s theorem for quasiprimal algebras, and the Baker–Pixley Theorem for finite algebras with a majority term.

The Coadjoint Operator, Conjugate Points, and the Stability of Ideal Fluids

We give a new description of the coadjoint operator \(\mathrm {Ad}^{*}_{\eta ^{-1}(t)}\) along a geodesic \(\eta (t)\) of the \(L^{2}\) metric in the group of volume-preserving diffeomorphisms, important in hydrodynamics. When the underlying manifold is two dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations modulo a compact operator; when the manifold is three dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations plus a bounded operator. We give two applications of this result when the underlying manifold is two dimensional: conjugate points along geodesics of the \(L^{2}\) metric are characterized in terms of the coadjoint operator and thus determining the conjugate locus is a purely algebraic question. We also prove that Eulerian and Lagrangian stability of the 2D Euler equations are equivalent and that instabilities in the 2D Euler equations are contained and small.

The complexity of divisibility

We address two sets of long-standing open questions in linear algebra and probability theory, from a computational complexity perspective: stochastic matrix divisibility, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic matrices is an NP-complete problem, and extend this result to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochastic matrices. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is in P, but decomposability is NP-hard. For the former, we give an explicit polynomial-time algorithm. All results on distributions extend to weak-membership formulations, proving that the complexity of these problems is robust to perturbations.

The homology of algebras of pseudo-differential symbols and the noncommutative residue

We will perform the calculation by two completely different methods, each of which has some advantages. In the first, we filter the algebra in question as in [2], and use a homogeneity argument to show that the resulting spectral sequence degenerates. From this point of view, the homology is seen to be a “semi-classical” invariant, since it is calculated at first order in Planck’s constant, as the homology of the differential forms on T ∗M with respect to the operator δ : Ω∗(T ∗M) → Ω∗−1(T ∗M) (defined by Koszul [14] and Brylinski [2]; see §1). The other method is a sheaf-theoretic calculation, modeled on Weil’s proof of de Rham’s Theorem, which uses a form of Poincare lemma for Hochschild homology of symbols. In the case of differential operators, this Poincare lemma states that the Hochschild homology of D(R), the algebra of differential operators on R, satisfies HH∗(D(R)) = H2n−∗(Rn,C). For pseudo-differential operators, the Poincare lemma is formulated for the sheaf ER of micro-differential operators on S∗M ×S, introduced in [16]; the copy of S comes from the fact that the sheaf ER lives on the cotangent bundle of a complexification of M . In the course of the paper, we also perform a number of other such calculations of Hochschild homology, for algebras of differential operators, operators with compact support, and formal deformations of C∞(X) for X a conic symplectic manifold.

Topological aperiodicity for product systems over semigroups of Ore type

We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik. They also give interesting conditions for topological higher rank graphs and $P$-graphs, and apply to the new Cuntz C*-algebra $\mathcal{Q}_\mathbb{N}$ arising from the "$ax+b$"-semigroup over $\mathbb{N}$.

An Almost Optimal Algorithm for Computing Nonnegative Rank

Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly exponential in $r$. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this task run in time exponential in the number of variables but polynomial in all of the other parameters (the number of inequalities and the maximum degree). Hence, these algorithms motivate natural algebraic questions whose solution have immediate algorithmic implications: How many variables do we need to represent the decision problem, and does $M$ have nonnegative rank at most $r$? A naive formulation uses $nr + mr$ variables and yields an algorithm that is exponential in $n$ and $m$ even for constant $r$. Arora et al. [Proceedings of STOC, 2012, pp. 145--162] recently reduced the number of variables to $2r^2 2^r$, and here we exponentially reduce the number of variables to $2r^2$ and this yields our main algorithm. In fact, the algorithm that we obtain is nearly optimal (under the exponential time hypothesis) since an algorithm that runs in time $(nm)^{o(r)}$ would yield a subexponential algorithm for $3$-SAT [Proceedings of STOC, 2012, pp. 145--162]. Our main result is based on establishing a normal form for nonnegative matrix factorization---which in turn allows us to exploit algebraic dependence among a large collection of linear transformations with variable entries. Additionally, we also demonstrate that nonnegative rank cannot be certified by even a very large submatrix of $M$, and this property also follows from the intuition gained from viewing nonnegative rank through the lens of systems of polynomial inequalities.

Fruit Flies and Moduli: Interactions between Biology and Mathematics

Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathematics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.

A proof of Schwartz's conjecture about the eigenvalues of Lannes' T-functor

This paper gives an algebraic proof of a conjecture due to Lionel Schwartz which asserts that the linear operator induced by Lannes' T-functor on the Grothendieck group Knred generated by indecomposable summands of the mod p cohomology of a rank n elementary abelian p-group is diagonalizable over Q, with eigenvalues 1,p,…,pn and with multiplicities pn−pn−1,pn−1−pn−2,…,p−1,1, respectively. Using work of Harris and Shank, we first reduce this to an algebraic question involving the Grothendieck ring G0(Mn,p) of modules over the semigroup ring Fp[End(Fp⊕n)], showing that the induced action of T on Knred corresponds to the multiplication by an explicit element. In the second step, we establish the separability of the algebra C⊗G0(Mn,p), from which the diagonalizability and the computation of the eigenvalues and their multiplicities follow easily. The arguments use ingredients from the theory of Brauer characters of finite groups.

Commutators, commutativity and dimension in the socle of a Banach algebra: A generalized Wedderburn–Artin and Shoda's theorem

As a follow-up to work done in [7], some new insights to the structure of the socle of a semisimple Banach algebra are obtained. In particular, it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals. Remarkably, the finite-dimensional case here reduces to the classical Wedderburn–Artin Theorem, and this approach does not use any continuous irreducible representations of the algebra in question. Furthermore, the structure of the socles for which the classical Shoda's Theorem for matrices can be extended, is characterized exactly as those socles which are minimal two-sided ideals. It is then shown that the set of commutators in the socle (i.e. {xy−yx:x,y∈SocA}) is a vector subspace. Finally, we characterize those socles which belong to the center of a Banach algebra and obtain results which suggest that the dimension of certain subalgebras of the socle in fact provides a measure, to some extent, of commutativity.

Moduli Spaces of Modules of Schur-Tame Algebras

In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and 𝜃-stable irreducible component C of a module variety of A-modules, the moduli space \(\mathcal {M}(C)^{ss}_{\theta }\) is either a point or a rational projective curve.

Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture

We prove the Mirkovic–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.

Generalized Coefficients for Hopf Cyclic Cohomology

A category of coefficients for Hopf cyclic cohomology is defined. It is shown that this category has two proper subcategories of which the smallest one is the known category of stable anti Yetter-Drinfeld modules. The middle subcategory is comprised of those coefficients which satisfy a generalized SAYD condition depending on both the Hopf algebra and the (co)algebra in question. Some examples are introduced to show that these three categories are different. It is shown that all components of Hopf cyclic cohomology work well with the new coefficients we have defined.

An algebraic study of linkages with helical joints

Methods from algebra and algebraic geometry have been used in various ways to study linkages in kinematics. These methods have failed so far for the study of linkages with helical joints (joints with screw motion), because of the presence of some non-algebraic relations. In this article, we explore a delicate reduction of some analytic equations in kinematics to algebraic questions via a theorem of Ax. As an application, we give a classification of mobile closed 5-linkages with revolute, prismatic, and helical joints.

A Study of the Impact of Inquiry-Based Professional Development Experiences on the Beliefs of Intermediate Science Teachers about '€œBest Practices' for Classroom Teaching

Open-ended interviews were used to study differences in Grade 7-8 teachers' beliefs about exemplary classroom practices produced by exposure to in-service professional development programs based on different theoretical frameworks. Teachers who had participated in an intensive, inquiry-based, in-service professional development program that focused on preferred teaching methodologies from a constructivist, inquiry-based perspective described in the '€œbest practices' literature were compared with teachers who had been through traditional in-service professional development workshops. Analysis of interviews based on the teachers’ responses to both conceptual and algebraic questions that involved the concept of density revealed insight into the teachers’ preferred teaching methodologies, their beliefs about how students learn, their attitudes toward their own learning, and their depth of understanding of the concept of density. This study suggests that, for some teachers, extensive in-service professional development can produce a substantive change in teachers' beliefs about optimum teaching practice. But this change can have unforseen consequences; in this case, the formation of a set of beliefs or values that forgot the importance of building quantitative skills as well a conceptual understanding of the concepts being taught.

Global Weyl Modules for Equivariant Map Algebras

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra A that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of A-modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e. untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that A is finitely generated and the global Weyl module is a finitely generated A-module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in a previous paper. Finally, we show that A is the algebra of coinvariants of the analogous algebra in the untwisted case.

COMPUTATIONAL COMPLEXITY OF VARIOUS MAL'CEV CONDITIONS

This paper examines the computational complexity of determining whether or not an algebra satisfies a certain Mal'Cev condition. First, we define a class of Mal'Cev conditions whose satisfaction can be determined in polynomial time (special cube term satisfying the DCP) when the algebra in question is idempotent and provide an algorithm through which this determination may be made. The aforementioned class notably includes near unanimity terms and edge terms of fixed arity. Second, we define a different class of Mal'Cev conditions whose satisfaction, in general, requires exponential time to determine (Mal'Cev conditions satisfiable by CPB0 operations).