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Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions { c n gri (A) } n ≥ 1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T 2 ⁎ -equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums. [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let m ≥ 2 be an integer. In this paper, we prove that if a group grading on U T m (F) admits a graded involution then this grading is a coarsening of a Z ⌊ m 2 ⌋ -grading on U T m (F) and the graded involution is equivalent to the reflection or symplectic involution on U T m (F) , this grading is called the finest grading on U T m (F). Furthermore, if m ≤ 4 the algebra U T m (F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z ⌊ m 2 ⌋ , ⁎) -identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z ⌊ m 2 ⌋ , ⁎) -codimensions is determined. As a consequence, we prove that for any G -grading on U T m (F) and any graded involution the (G , ⁎) -exponent is m. Finally, we study the algebra U T 3 (F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z -grading and the Z 2 -grading induced by (0 , 1 , 0). We determine a basis for the (Z 2 , ⁎) -identities and we compute the codimension sequence for the (Z 2 , ⁎) -graded identities for U T 3 (F). [ABSTRACT FROM AUTHOR]

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]

Let n be an integer greater than 1, and M n (C) be the algebra of all n × n -complex matrices. Let x 0 ∈ C n be a nonzero vector, and Φ be a linear map on M n (C) such that Φ (I) is invertible. For any matrix T ∈ M n (C) , let γ (T , x 0) denote the local reduced minimum modulus of T at x 0. In this paper, we show that Φ satisfies γ (T , x 0) = 0 ⇔ γ (Φ (T) , x 0) = 0 , (T ∈ M n (C)) , if and only if there are two invertible matrices A , B ∈ M n (C) such that A x 0 = A ⁎ x 0 = x 0 and Φ (T) = B T A for all T ∈ M n (C). When n = 2 , we show that the invertibility hypothesis of Φ (I) is redundant. [ABSTRACT FROM AUTHOR]

Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and addition, respectively. We say that the columns of a real matrix A are strongly independent if the max-plus linear system A ⊗ x = b has a unique solution for at least one real vector b. A square matrix A with strongly independent columns is called strongly regular. The investigation of the properties of regularity is important for applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. The present paper studies three versions of the regularity of matrices and interval matrices, namely, strong regularity, von Neumann regularity and Gondran-Minoux regularity. For each concept of regularity we will present equivalent conditions which can be verified in polynomial time. [ABSTRACT FROM AUTHOR]

In geometric tolerancing, functional specification is becoming challenging as the complex mutual situations of geometrical features increase and the requisites for complex assembly accuracy escalate. A computational representation of sets of situation features is proposed in this paper to manage functional specification of complex assemblies. A minimum triplet set of a point, a straight line and a plane called ToLiP, is introduced to represent situation features. Functional specification operations can thus be mathematically established using projective geometric algebra (PGA). The effectiveness of the proposed approach is highlighted through a case study using quadrupole magnets of a particle accelerator. [ABSTRACT FROM AUTHOR]

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra. [ABSTRACT FROM AUTHOR]

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring. [ABSTRACT FROM AUTHOR]

In this paper, first we give the notion of a compatible 3-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible 3-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible 3-Lie algebra. Then we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible 3-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible 3-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible 3-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible 3-Lie algebra using the second cohomology group. [ABSTRACT FROM AUTHOR]

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local "spin variables" can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent "dual" formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra U q (s l ˆ (2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l − l ′) –th modified KP hierarchy. Firstly, the representation of g l ˆ ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ , which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L ∞ [ 1 ] -algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ , the corresponding twisted L ∞ [ 1 ] -algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version. [ABSTRACT FROM AUTHOR]

In this paper, we address the computation of the symmetries of polynomial (and thus also rational) planar vector fields using elements from Computer Algebra. We show that they can be recovered from the symmetries of the roots of an associated univariate complex polynomial which is constructed as a generator of a certain elimination ideal. Computing symmetries of the roots of the auxiliary polynomial is a task considerably simpler than the original problem, which can be done efficiently working with classical Computer Algebra tools. Special cases, in which the group of symmetries of the polynomial roots is infinite, are separately considered and investigated. The presented theory is complemented by illustrative examples. The main steps of the procedure for investigating the symmetries of a given polynomial vector field are summarized in a flow chart for clarity. • Computation of the symmetries of polynomial/rational planar vector fields using Computer Algebra is addressed. • These symmetries can be recovered from the symmetries of the roots of an associated univariate complex polynomial. • Special cases, in which the symmetry group of the polynomial roots is infinite, are separately considered and investigated. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new invariant for a non-degenerate evolution algebra, which consists of an ordered sequence of evolution algebras of lower dimension, belonging all of them to a specific family. We use this invariant to propose a method to classify non-degenerate evolution algebras, and we apply it up to dimension 3. We also use it to describe the derivations of some families of evolution algebras and the degenerations of the variety of evolution algebras with square not greater than 1. [ABSTRACT FROM AUTHOR]

Given a finite dimensional algebra A over a field k , and a finite acyclic quiver Q , let Λ = A ⊗ k k Q / I , where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q , I) over a subcategory X of A -mod, denoted by smon (Q , I , X). We show that (X , Y) is a (complete) hereditary cotorsion pair in A -mod if and only if (smon (Q , I , X) , rep (Q , I , Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that k Q / I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein. [ABSTRACT FROM AUTHOR]

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes. [ABSTRACT FROM AUTHOR]

Let P k be the graded polynomial algebra F 2 [ x 1 , x 2 , ... , x k ] over the prime field of two elements, F 2 , with the degree of each x i being 1. We study the hit problem , set up by Frank Peterson, of finding a minimal set of generators for P k as a module over the mod-2 Steenrod algebra, A. In this paper, we explicitly determine a minimal set of A -generators for P 5 in the case of the generic degrees n = 2 d + 1 − 1 and n = 2 d + 1 − 2 for all d ⩾ 6. [ABSTRACT FROM AUTHOR]

Consider an input–output system where the output is the tracking error given some desired reference signal. It is natural to consider under what conditions the problem has an exact solution, that is, the tracking error is exactly the zero function. If the system has a well defined relative degree and the zero function is in the range of the input–output map, then it is well known that the system is locally left invertible, and thus, the problem has a unique exact solution. A system will fail to have relative degree when more than one exact solution exists. The general goal of this paper is to describe a decomposition of an input–output system having a Chen-Fliess series representation into a parallel product of subsystems in order to identify possible solutions to the problem of zeroing the output. For computational purposes, the focus is on systems whose generating series are polynomials. It is shown that the shuffle algebra on the set of generating polynomials is a unique factorization domain so that any polynomial can be uniquely factored modulo a permutation into its irreducible elements for the purpose of identifying the subsystems in a parallel product decomposition. This is achieved using the fact that this shuffle algebra is isomorphic to the symmetric algebra over the vector space spanned by Lyndon words. A specific algorithm for factoring generating polynomials into its irreducible factors is presented based on the Chen-Fox-Lyndon factorization of words. [ABSTRACT FROM AUTHOR]

This paper presents a co-rotational beam element based on quaternion algebra as a means of parameterizing large rotations. This co-rotational framework is based on a decomposition of beam kinematics into a rigid element frame, which follows the element and its pure deformation. Once the decomposition between rigid body motion and deformation is obtained, the principle of virtual work allows calculating the element response projected onto large displacements and rotations. This 3D co-rotational element lies within the framework of incremental formulations. The special feature of this formulation pertains to the decomposition of kinematics to extract the rigid body motion and pure deformations through use of quaternion algebra by solving an internal nonlinear kinematic equation. Thus, from the 2 quaternions defining global rotations and the 2 displacement vectors at the ends of the beam, this method is able to extract: 1 quaternion and 1 displacement vector parameterizing the rigid body motion, plus 2 other quaternions defining the internal rotations and beam elongation. The proposed formulation based on quaternion algebra therefore constitutes an alternative to the literature's treatment of large-rotation kinematics using a quaternion algebra. The operators are also linearized in order to obtain the algorithmic stiffness operator. Eleven distinct static numerical applications are presented, along with comparisons from the literature in the aim of demonstrating the efficiency of this proposed element. • Quaternion parametrization of large rotations in a co-rotational framework. • Decomposition of kinematics by solving an internal nonlinear system with quaternions • Linearization of the operators in order to obtain the algorithmic stiffness operator. • Eleven distinct study cases are presented. [ABSTRACT FROM AUTHOR]

Let KQ be a path algebra, where Q is a finite quiver and K is a field. We study K Q / C where C is the two-sided ideal in KQ generated by all differences of parallel paths in Q. We show that K Q / C is always finite dimensional and its global dimension is finite. Furthermore, we prove that K Q / C is Morita equivalent to an incidence algebra. The paper starts with a more general setting, where KQ is replaced by K Q / I with I a two-sided ideal in KQ. [ABSTRACT FROM AUTHOR]

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [1,2]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3]. These models are based on the A 2 , G 2 and B 2 Lie algebras, and their local vertex configurations are intertwiners of the corresponding q -deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of B 2 also on the square lattice. For specific root-of-unity choices of q , we show the equivalence to a number of three- and four-state spin models on the dual lattice. The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For q on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix. In the A 2 case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the G 2 case we identify four regimes numerically. The B 2 case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra. [ABSTRACT FROM AUTHOR]

A Boolean algebra A equipped with a (finitely-additive) positive probability measure m can be turned into a metric space (A , d m) , where d m (a , b) = m ((a ∧ ¬ b) ∨ (¬ a ∧ b)) , for any a , b ∈ A , sometimes referred to as metric Boolean algebra. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in R N (for a certain N) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra A such that the metric space (At (A) , d m) embeds isometrically in R N (with the Euclidean metric). [ABSTRACT FROM AUTHOR]

In this paper, we present a method to describe (functionally) free algebras in some subvarieties of BL-algebras. Particularly, we give a description of free algebras in subvarieties of the subvariety MG , where MG is the subvariety of BL generated by the algebra [ 0 , 1 ] MV ⊕ [ 0 , 1 ] G. [ABSTRACT FROM AUTHOR]

The rigidity degree of a generator-cogenerator determines the dominant dimension of its endomorphism algebra, and is closely related to a recently introduced homological dimension — rigidity dimension. In this paper, we give explicit formulae for the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras by developing combinatorial methods from the Euclidean algorithm. As an application, the rigidity dimensions of some algebras of types A and E are given. [ABSTRACT FROM AUTHOR]

The statements of Theorem 6.11 and Theorem 7.8 in the published paper are not correct: each have a missing hypothesis which is needed for their proof. We restate both theorems (with the required extra hypothesis), and give corrected proofs of each. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the Nijenhuis operators on Hom-Lie conformal algebras. We construct a graded Lie algebra whose Maurer-Cartan elements are given by Nijenhuis operators with the aid of Frölicher-Nijenhuis bracket. After constructing graded Lie algebra, we define the cohomology associated with a Nijenhuis operator. We further introduce Hom- NS Lie conformal algebra as an algebraic structure behind Nijenhuis operators on Hom-Lie conformal algebra. We provide various examples of Hom- NS Lie conformal algebras. Finally, as an application to cohomology, we study formal deformations of Nijenhuis operators. [ABSTRACT FROM AUTHOR]

In the present paper we construct separation of variables (SoV) for all Lax-integrable systems, two by two Lax matrix of which enjoys classical reflection equation algebra with the elliptic r − s matrices. We show that, similar to the cases of SoV for the classical XYZ and XXZ models [1,2] , the constructed SoV admits two types of momenta, which are important in the quantum case [1,2]. We consider two examples of such the integrable hamiltonian systems governed by the simplest Poisson brackets connected with classical reflection equation algebra with the elliptic r − s matrices. They are the classical Sklyanin algebra and its four-dimensional extension which provide a quadratic structure for the classical Steklov top. We explicitly construct the variables of separation, Abel equations and reconstruction formulae for the corresponding integrable models. [ABSTRACT FROM AUTHOR]

In this paper, we study non-weight representations of another two classes of Lie superalgebras T (p , s) of Block type, where p ∈ C and s = 0 , 1. We completely classify the free U (h) -modules of rank (1 + 1) over T (p , s). As byproducts, we also obtain the non-weight representation results for four classes of non-semisimple finitely graded super-extensions of the Virasoro algebra. [ABSTRACT FROM AUTHOR]

It was demonstrated recently that the W 1 + ∞ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized W ˜ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the W ˜ algebras studied earlier. We suggest a definition of the generalized W ˜ algebra as differential operators in variables p k basing on the matrix realization of the W 1 + ∞ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the W 1 + ∞ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with W ˜ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric τ -functions corresponding to the completed cycles. [ABSTRACT FROM AUTHOR]

In 1992, Terwilliger introduced the notion of the Terwilliger algebra in order to study association schemes. The Terwilliger algebra of an association scheme A is the subalgebra of the complex matrix algebra, generated by the Bose-Mesner algebra of A and its dual idempotents with respect to a point x. In Bannai and Munemasa (1995) [3] determined the dimension of the Terwilliger algebra of abelian groups and dihedral groups, by showing that they are triply transitive (i.e., triply regular and dually triply regular). In this paper, we give a generalization of their results to the group association scheme of semidirect products of the form C n ⋊ C 2 , where C m is a cyclic group of order m ≥ 2. Moreover, we will give the complete characterization of the Wedderburn components of the Terwilliger algebra of these groups. [ABSTRACT FROM AUTHOR]

Hedge algebra is gaining increasing attention due to its high efficiency and fast calculation speed compared to fuzzy logic. This paper proposes the first algebra-based formation controller with a leader–follower approach to maintain the desired formation of multiple robots. In each time step, the controller computes the desired positions of the followers and strives to minimize the distance between them and their desired positions. We evaluate the performance of this controller on our robots using ROS in a virtual environment with Gazebo. Our work proves hedge algebra's significant potential in the mobile robots' domain, specifically in formation control. HAFC allows not only the precise formation of a robot swarm into any desired formation, with minimal deviation but also adaptive transition to the different formations. Our results show that HAFC can effectively maintain the desired formation of multiple robots, even in certain extraordinary situations. Furthermore, HAFC exhibits higher efficiency than the fuzzy logic-based controller, as evidenced by its faster adaptation rate and shorter computation time. The proposed approach has the potential to be applied in various real-world scenarios, such as surveillance, transportation, and exploration, where the robot has to move in formation and change formation accordingly with the terrain. • This work is the first study in formation control of mobile robots and adaptive formation translation based on the Hedge-algebras theory. • HAFC exhibits higher efficiency compared to the fuzzy logic-based controller, as evidenced by its faster adaptation rate and shorter computation time. • The simulation is implemented on ROS and obtain a high accuracy and miniature errors. A structure is proposed so as to conduct real formation experiments. • The proposed approach has the potential to be applied in various real-world scenarios, such as surveillance, transportation, and exploration, where the robots have to move in formation and change formation accordingly with the terrain. [ABSTRACT FROM AUTHOR]

In this paper, we explore the existence of nontrivial monadic operators on linearly ordered monadic nilpotent minimum algebras (NM-algebras, for short) and achieve the necessary and sufficient conditions for a linearly ordered monadic NM-algebra having monadic operators. We verify that each finite linearly ordered monadic NM-algebra with more than 4 elements has nontrivial monadic operators. We study the simple and subdirectly irreducible monadic NM-algebras and derive the necessary and sufficient conditions for a monadic NM-algebra to be simple and subdirectly irreducible, respectively. In addition, we discuss finite monadic involutive monoidal t-norm based algebras (IMTL-algebras, for short) and give some examples to show that there exist finite linearly ordered monadic IMTL-algebras having only trivial monadic operators. We establish the conditions for finite linearly ordered perfect monadic IMTL-algebras having nontrivial monadic operators. [ABSTRACT FROM AUTHOR]

In [16] , we studied deformations A α , β of the second Weyl algebra and computed their derivations. In the present paper, we identify the semiclassical limits A α , β of these deformations and compute their Poisson derivations. Our results show that the first Hochschild cohomology group HH 1 (A α , β) is isomorphic to the first Poisson cohomology group HP 1 (A α , β). [ABSTRACT FROM AUTHOR]

Let d be a positive integer. The Yangian Y d = Y (gl (d , C)) of the general linear Lie algebra gl (d , C) has countably many generators and quadratic-linear defining relations, which can be packed into a single matrix relation using the Yang matrix — the famous RTT presentation. Alternatively, Y d can be built from certain centralizer subalgebras of the universal enveloping algebras U (gl (N , C)) , with the use of a limit transition as N → ∞. This approach is called the centralizer construction. The paper shows that a generalization of the centralizer construction leads to a new family { Y d , L : L = 1 , 2 , 3 , ... } of Yangian-type algebras (the Yangian Y d being the first term of this family). For the new algebras, the RTT presentation seems to be missing. Nevertheless, the algebras Y d , L share a number of properties of the Yangian Y d , including the existence of defining quadratic-linear commutation relations. [ABSTRACT FROM AUTHOR]

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As applications of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras. [ABSTRACT FROM AUTHOR]

In this paper, we characterize suitable partial (co)actions of Taft and Nichols Hopf algebras on algebras, and moreover we get that such partial (co)actions are symmetric. For certain algebras, these partial (co)actions obtained are, indeed, all of them. This work generalizes the results obtained by the authors in [16]. [ABSTRACT FROM AUTHOR]

The main goals for this paper are i) to study of the algebraic structure of N -graded vertex algebras V B associated with vertex A -algebroids B when B are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras V B and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex A -algebroids B associated with given cyclic non-Lie left Leibniz algebras B. Next, we use the constructed vertex A -algebroids B to create a family of indecomposable non-simple vertex algebras V B. Finally, we use the algebraic structure of the unital commutative associative algebras A that we found to study relations between certain types of the vertex algebras V B and the vertex operator algebra associated to the rank one Heisenberg algebra. [ABSTRACT FROM AUTHOR]