Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs†

Let Q be a m × m real matrix and f _j  : ? → ?, j = 1, …, m, be some given functions. If x and f(x) are column vectors whose j-coordinates are x _j and f _j (x _j ), respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system Qx = f(x) for Q symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by Neuberger. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph 𝒢 is just any linear operator D:C ⁰(𝒢) → C ⁰(𝒢), where C ⁰(𝒢) is the real vector space of real maps defined on the vertex set V of the graph. Given a derivation D and a function F:V × ? → ?, one has associated a partial difference equation Dμ = F(v,μ), and one searches for solutions μ ∈ C ⁰(𝒢). Sufficient conditions in order to have non-trivial solutions of partial difference equations on any finite, connected simple graph for D symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair (𝒢, d), where 𝒢 is a connected finite degree simple graph and d is a positive function on the set of edges of the graph. The metric d permits to consider some classical derivations, such as the Laplacian operator ?₂. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91–107) was considered the nonlinear elliptic partial difference equations ?₂ u = F(u), for the metric d = 1.     

Preprojective Cluster Variables of Acyclic Cluster Algebras

It is proved that any cluster-tilted algebra defined in the cluster category 𝒞(H) has the same representation type as the initial hereditary algebra H. For any valued quiver (Γ, Ω), an injection from the subset 𝒫?(Ω) of the cluster category 𝒞(Ω) consisting of indecomposable preprojective objects, preinjective objects, and the first shifts of indecomposable projective modules to the set of cluster variables of the corresponding cluster algebra 𝒜_Ω is given. The images are called “preprojective cluster variables”. It is proved that all preprojective cluster variables other than u_i have denominators u ^dim M in their irreducible fractions of integral polynomials, where M is the corresponding preprojective module or preinjective module. In case the valued quiver (Γ, Ω) is of finite type, the denominator theorem holds with respect to any cluster. Namely, let x = (x₁,…, x_n) be a cluster of the cluster algebra 𝒜_Ω, and V the cluster tilting object in 𝒞(Ω) corresponding to x, whose endomorphism algebra is denoted by Λ. Then the denominator of any cluster variable y other than x_i is x ^dim M, where M is the indecomposable Λ-module corresponding to y. This result is a generalization of the corresponding result of Caldero–Chapoton–Schiffler to the non-simply-laced case.     

Exceptional Pairs in Hereditary Categories

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ?. If r = dim _k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H _r ) over the r-Kronecker algebra H _r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ? is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ?¹ or to mod(H _r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.     

Smash Products of H-Simple Module Algebras

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End _A′(V ? H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.     

Generalized skew derivations on triangular algebras determined by action on zero products

For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜→𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈𝒜, where D:𝒜→𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.     

On Galois Comodules

Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring 𝒞 for which P _A is finitely generated and projective and the evaluation map μ𝒞:Hom ^𝒞 (P, 𝒞) ?  _S P → 𝒞 is an isomorphism (of corings) where S = End ^𝒞 (P). It has been observed that for such comodules the functors ? ?  _A 𝒞 and Hom _A (P, ?) ?  _S P from the category of right A-modules to the category of right 𝒞-comodules are isomorphic. In this note we use this isomorphism related to a comodule P to define Galois comodules without requiring P _A to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. Galois comodules are close to being generators and have common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.     

On the spectral norms of Cauchy–Toeplitz and Cauchy–Hankel matrices

In this study, we have found upper and lower bounds for the spectral norm of Cauchy–Toeplitz and Cauchy–Hankel matrices in the forms T_n=[1/(a+(i−j)b)]ⁿ_i,j=1, H_n=[1/(a+(i+j)b)]ⁿ_i,j=1.     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.     

Non-abelian Tensor Product of Leibniz Algebras and an Exact Sequence in Leibniz Homology

Abstract

Let 𝔪 and 𝔫 be two-sided ideals of a Leibniz algebra 𝔤 such that 𝔤 = 𝔪 + 𝔫. The goal of the paper is to achieve the exact sequence Ker(𝔪  𝔫 + 𝔫  𝔪 → 𝔤) → HL ₂(𝔤) → HL ₂(𝔤/𝔪) ⊕ HL ₂(𝔤/𝔫) → 𝔪 ∩ 𝔫/ [𝔪,𝔫] → HL ₁(𝔤) → HL ₁(𝔤/𝔪) ⊕ HL ₁(𝔤/𝔫) → 0, where HL denotes the Leibniz homology with trivial coefficients of a Leibniz algebra and denotes a non-abelian tensor product of Leibniz algebras.     

Some Maximal Function Fields and Additive Polynomials

We derive explicit equations for the maximal function fields F over 𝔽 _{q ²ⁿ} given by F = 𝔽 _{q ²ⁿ} (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 _{q ²ⁿ} , and where A(Y) is q-additive and deg(f) = q ⁿ  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 _{q ²ⁿ} (i.e., the extension H/F is Galois).     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Irreducible Quasi-Finite Representations of a Block Type Lie Algebra

Let B be the Block type Lie algebra over ? with basis {L _{α, i} , C ₁, C ₂ | (α, i) ∈ ? × ? \ {(0, ? 2)}} and Lie bracket [L _{α, i} , L _{β, j} ] = (β(i + 1) ? α(j + 1))L _{α+β, i+j}  + αδ_{α+β, 0}δ _{i+j, ?2} C ₁ + (i + 1)δ_{α+β, 0}δ _{i+j, ?2} C ₂, where C ₁, C ₂ are central elements. In this paper, it is proved that a quasi-finite irreducible B-module is either a highest or a lowest weight module. We also give a classification of all highest/lowest weight B-modules.     

Solvability of universal bol 2-loops

ABSTRACT

Let (A, ^?) be a structurable algebra. Then the opposite algebra (A ^op , ^?) is structurable, and we show that the triple system B ^op _A(x, y, z):=V^opx,y(z)=x(y¯z)+z(y¯x)?y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸₁?𝔸₂ denotes tensor products of composition algebras, (^?) is the standard conjugation, and (^∧) denotes a certain pseudoconjugation on A, we show that the triple systems B ^op _𝔸1?𝔸2 ( x , y¯^∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸₁, dim𝔸₂) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras.     

Hopf-Type Algebras

In this article, we provide an alternative approach to the definition of a weak Hopf algebra (WHA). For an associative unital algebra A with a coassociative comultiplication Δ ∈Alg ^u (A, A ? A), the set of homomorphisms from A to A ? A, which do not preserve the units. If the linear maps Ξ₁, Ξ₂ ∈ End(A ? A), defined by Ξ₁(a ? b) = Δ(a)(1 ? b), Ξ₂(a ? b) = (a ? 1)Δ(b), are von Neumann regular elements in the ring End(A ? A) of endomorphisms of A ? A satisfying some appropriate assumptions, we call the A a Hopf-type algebra. We show the existence of a target, a source, a counit, and an antipode of A as in the usual WHA.     

The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.