Search

In this paper, we give an alternative construction of a certain class of deformed double current algebras. These algebras are deformations of |$ U(\textrm {End}(\Bbbk ^r)[x,y]) $| and they were initially defined and studied by N. Guay in his papers. Here, we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank. [ABSTRACT FROM AUTHOR]

Let U q − be the negative part of the quantum enveloping algebra, and σ the algebra automorphism on U q − induced from a diagram automorphism. Let U _ q − be the quantum algebra obtained from σ , and B ˜ (resp. B _ ˜) the canonical signed basis of U q − (resp. U _ q −). Assume that U q − is simply-laced of finite or affine type. In our previous papers [10] , [11] , we have proved by an elementary method, that there exists a natural bijection B ˜ σ ≃ B _ ˜ in the case where σ is admissible. In this paper, we show that such a bijection exists even if σ is not admissible, possibly except some small rank cases. [ABSTRACT FROM AUTHOR]

The estimation algebra plays an important role in classification of finite dimensional filters. When finite dimensional estimation algebra has maximal rank, Yau et al. [Yau (2003). Complete classification of finite-dimensional estimation algebras of maximal rank. International Journal of Control, 76(7), 657–677; Yau & Hu (2005). Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state-space dimension and Mitter conjecture. International Journal of Control, 78(10), 689–705.] have proved that η must be a degree 2 polynomial. In this paper, we study the structure of finite dimensional exact estimation algebra with state dimension 3 and rank 2. We establish a sufficient and necessary condition for estimation algebra with nonmaximal rank to be finite dimensional. Importantly, in the new filtering system, η needs not to be a degree 2 polynomial and can be of any degree 4 n 1 + 2 , n 1 ∈ Z + . It is the first time to systematically analyse nonmaximal rank exact estimation algebra in which η is a polynomial of any degree 4 n 1 + 2 , n 1 ∈ Z + . For Riccati-type equation, estimates have been done from the viewpoints of both classical solution and weak solution respectively. Finally, finite dimensional filters of Benés type are constructed successfully. [ABSTRACT FROM AUTHOR]

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras. [ABSTRACT FROM AUTHOR]

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component" of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component" to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical presymplectic 2-form on the loop space. This construction is a crucial ingredient for the definition of the supersymmetric path integral on the loop space. [ABSTRACT FROM AUTHOR]

Several structural properties of an algebraic structure can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In this paper, we exhibit finite representations of this sequence for finite algebras from congruence modular varieties. [ABSTRACT FROM AUTHOR]

Every finite dimensional normed algebra is isomorphic to the finite direct product of or, it is also proved these algebras are ultrasemiprime algebras. In this paper, the ultrasemiprime proof of the finite direct product of and is generalized to the finite direct product of any ultrasemiprime algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study some classes of terms, called trees, which play a key role in both universal algebra and theoretical computer science. We introduce the concept of terms defined by singular transformations and provide some concrete examples. We also prove that the set of such terms together with an operation of type (n + 1) for a fixed natural number n defined on that set forms an algebra of type (n + 1) satisfying the axiom of superassociativity. [ABSTRACT FROM AUTHOR]

In this paper, we show that the finite subalgebra \mathcal {A}^\mathbb {R}(1), generated by \mathrm {Sq}^1 and \mathrm {Sq}^2, of the \mathbb {R}-motivic Steenrod algebra \mathcal {A}^\mathbb {R} can be given 128 different \mathcal {A}^\mathbb {R}-module structures. We also show that all of these \mathcal {A}-modules can be realized as the cohomology of a 2-local finite \mathbb {R}-motivic spectrum. The realization results are obtained using an \mathbb {R}-motivic analogue of the Toda realization theorem. We notice that each realization of \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an \mathbb {R}-motivic v_1-self-map. The {\mathrm {C}_2}-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the \mathrm {RO}({\mathrm {C}_2})-graded Steenrod operations on a {\mathrm {C}_2}-equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the {\mathrm {C}_2}-equivariant realizations of \mathcal {A}^{\mathrm {C}_2}(1). We find another application of the \mathbb {R}-motivic Toda realization theorem: we produce an \mathbb {R}-motivic, and consequently a {\mathrm {C}_2}-equivariant, analogue of the Bhattacharya-Egger spectrum \mathcal {Z}, which could be of independent interest. [ABSTRACT FROM AUTHOR]

We introduce the following combinatorial problem. Let G be a triangle-free regular graph with edge density ρ. (In this paper all densities are normalized by n , n 2 2 etc. rather than by n − 1 , ( n 2 ) , ...) What is the minimum value a (ρ) for which there always exist two non-adjacent vertices such that the density of their common neighbourhood is ≤ a (ρ) ? We prove a variety of upper bounds on the function a (ρ) that are tight for the values ρ = 2 / 5 , 5 / 16 , 3 / 10 , 11 / 50 , with C 5 , Clebsch, Petersen and Higman-Sims being respective extremal configurations. Our proofs are entirely combinatorial and are largely based on counting densities in the style of flag algebras. For small values of ρ , our bound attaches a combinatorial meaning to so-called Krein conditions that might be interesting in its own right. We also prove that for any ϵ > 0 there are only finitely many values of ρ with a (ρ) ≥ ϵ but this finiteness result is somewhat purely existential (the bound is double exponential in 1 / ϵ). [ABSTRACT FROM AUTHOR]

The concept of the direct product of finite family of B-algebras is introduced by Lingcong and Endam[J. A. V. Lingcong and J. C. Endam, Direct product of B-algebras, Int. J. Algebra, 10(1):33-40, 2016.]. In this paper, we introduce the concept of the direct product of infinite family of B-algebras, we call the external direct product, which is a generalization of the direct product in the sense of Lingcong and Endam. Also, we introduce the concept of the weak direct product of B-algebras. Finally, we provide several fundamental theorems of (anti-)B-homomorphisms in view of the external direct product B-algebras. [ABSTRACT FROM AUTHOR]

In 1988, Beck (J Algebra 116:208–226, 1988) conjectured that the chromatic number and clique number is the same for the graphs associated to any commutative ring with unity. In 1993, Anderson and Naseer (J Algebra 159:500–514, 1993) disproved the conjecture with a counterexample. In this paper, we find the clique number and bounds for the chromatic number of the graph associated with the finite product of commutative rings with unity. Also we show that some of the results, proved in (Anderson and Naseer in J Algebra 159:500–514, 1993, Beck in J Algebra 116:208–226, 1988) are consequences of our results. In addition, we have constructed a new family of counterexamples of the above conjecture. [ABSTRACT FROM AUTHOR]

In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra. [ABSTRACT FROM AUTHOR]

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent F p -braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite F p -braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent F p -braces generated by one element of cardinality p 4 is obtained. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left nilpotent ideal, therefore every finite brace contains the largest left nilpotent ideal. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the link between evolution algebras and (pseudo)digraphs. We study some theoretical properties about this association and determine the properties of the (pseudo)digraphs associated with each type of evolution algebras. We also analyze the isomorphism classes for each configuration associated with these algebras providing a new method to classify them, and we compare our results with the current classifications of two‐ and three‐dimensional evolution algebras. In order to complement the theoretical study, we have designed and performed the implementation of an algorithm, which constructs and draws the (pseudo)digraph associated with a given evolution algebra and another procedure to study the solvability of a given evolution algebra. [ABSTRACT FROM AUTHOR]

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]

In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated. [ABSTRACT FROM AUTHOR]

We consider a nontrivial action of C2 on the type 1 spectrum Y := M2(1) Λ C(η), which is well-known for admitting a 1-periodic v1-self-map. The resultant finite C2-equivariant spectrum Y... can also be viewed as the complex points of a finite ℝ-motivic spectrum Yℝ. In this paper, we show that one of the 1-periodic v1-self-maps of Y can be lifted to a self-map of Y... as well as Yℝ. Further, the cofiber of the self-map of Yℝ is a realization of the subalgebra Aℝ (1) of the ℝ-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of Y.... [ABSTRACT FROM AUTHOR]

In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A , we show that the finite generation condition on A -modules can be replaced by a condition on any affine commutative A -module algebra R under the assumption that R is integral over its invariant subring R A. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A ′ over a finite field, where A can be viewed as a deformation of A ′ , and prove that if the finite generation condition holds for A ′ , then the same condition holds for A. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which is known as Huneke's criterion. [ABSTRACT FROM AUTHOR]

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples. [ABSTRACT FROM AUTHOR]

In this paper we generalize classical results regarding minimal realizations of non‐commutative (nc) rational functions using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. We prove the existence and uniqueness of a minimal realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity. Moreover, we show that using this realization we can evaluate the function on all of its domain (of matrices of all sizes) and also with respect to any stably finite algebra. As a corollary we obtain a new proof of the theorem by Cohn and Amitsur, that equivalence of two rational expressions over matrices implies that the expressions are equivalent over all stably finite algebras. Applications to the matrix valued and the symmetric cases are presented as well. [ABSTRACT FROM AUTHOR]

In this paper, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey–Sauter, Nguyen–Reiten–Todorov–Zhu and Pressland–Sauter. Moreover, we give characterizations of almost n -Auslander–Gorenstein algebras and almost n -Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost 1-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules. [ABSTRACT FROM AUTHOR]

In this paper, we give categorical properties such as pullback, finite product, finite limit, coproduct, colimit and pushout in XHom -- Lie/A of the category of Hom-Lie crossed modules. [ABSTRACT FROM AUTHOR]

Let (X , ≤) be a finite preordered set, R a 2-torsion free commutative ring with unity and I (X , R) the incidence algebra of X over R. In this paper we prove that every nonlinear Lie derivation of I (X , R) is of the standard form. More explicitly, each nonlinear Lie derivation of I (X , R) is a sum of an inner derivation, a transitive induced derivation, an additive induced derivation and a central-valued map. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of a self-similar action of a group |$G$| on a |$k$| -graph |$\Lambda $| and associate it a universal C |$^\ast $| -algebra |${{\mathcal{O}}}_{G,\Lambda }$|⁠. We prove that |${{\mathcal{O}}}_{G,\Lambda }$| can be realized as the Cuntz–Pimsner algebra of a product system. If |$G$| is amenable and the action is pseudo free, then |${{\mathcal{O}}}_{G,\Lambda }$| is shown to be isomorphic to a "path-like" groupoid C |$^\ast $| -algebra. This facilitates studying the properties of |${{\mathcal{O}}}_{G,\Lambda }$|⁠. We show that |${{\mathcal{O}}}_{G,\Lambda }$| is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of |${{\mathcal{O}}}_{G,\Lambda }$| in terms of the underlying action, and we prove that, whenever |${{\mathcal{O}}}_{G,\Lambda }$| is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether |$\Lambda $| has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a finite Lie conformal superalgebra called the Heisenberg-Virasoro Lie conformal superalgebra s by using a class of Heisenberg-Virasoro Lie conformal modules. The super Heisenberg-Virasoro algebra of Ramond type S is defined by the formal distribution Lie superalgebra of s. Then we construct a class of simple S -modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras. These modules are isomorphic to simple restricted S -modules, and include the highest weight modules, Whittaker modules and high order Whittaker modules. As a byproduct, we present a subalgebra of S , which is isomorphic to the super Heisenberg-Virasoro algebra of Neveu-Schwarz type. [ABSTRACT FROM AUTHOR]

Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup 〈 x , x α 〉 is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: On a class of normal endomorphisms of groups, J. Algebra and its Applications 13, (2014), 6pp] the authors proved that every cyclic automorphism is central, namely, that every cyclic automorphism acts trivially on the factor group G / Z (G) . In this paper, the class F W of groups in which every element induces by conjugation a cyclic automorphism on a (normal) subgroup of finite index will be investigated. [ABSTRACT FROM AUTHOR]

For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injection. We call such a set a monobrick. In this paper, we provide a uniform method to study torsion-free classes and wide subcategories via monobricks. We show that monobricks are in bijection with left Schur subcategories, which contains all subcategories closed under extensions, kernels and images, thus unifies torsion-free classes and wide subcategories. Then we show that torsion-free classes bijectively correspond to cofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free) classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories, without using τ -tilting theory. For Nakayama algebras, left Schur subcategories are the same as subcategories closed under extensions, kernels and images, and we show that its number is related to the large Schröder number. [ABSTRACT FROM AUTHOR]

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K -group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17] , the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17] , our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a class of infinite Lie conformal superalgebras S (p) , which are closely related to Lie conformal algebras of extended Block type defined in [6]. Then all finite non-trivial irreducible conformal modules over S (p) for p ∈ C ⁎ are completely classified. As an application, we also present the classifications of finite non-trivial irreducible conformal modules over finite quotient algebras s (n) for n ≥ 1 and sh which is isomorphic to a subalgebra of Lie conformal algebra of N = 2 superconformal algebra. Moreover, as a generalized version of S (p) , the infinite Lie conformal superalgebras GS (p) are constructed, which have a subalgebra isomorphic to the finite Lie conformal algebra of N = 2 superconformal algebra. [ABSTRACT FROM AUTHOR]

The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element. [ABSTRACT FROM AUTHOR]