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We investigate the quantum dynamics of a spin coupling to a bath of independent spins via the dissipaton equation of motion (DEOM) approach. The bath, characterized by a continuous spectral density function, is composed of spins that are independent level systems described by the s u (2) Lie algebra, representing an environment with a large magnitude of anharmonicity. Based on the previous work by Suarez and Silbey [J. Chem. Phys. 95, 9115 (1991)] and by Makri [J. Chem. Phys. 111, 6164 (1999)] that the spin bath can be mapped to a Gaussian environment under its linear response limit, we use the time-domain Prony fitting decomposition scheme to the bare–bath time correlation function (TCF) given by the bosonic fluctuation–dissipation theorem to generate the exponential decay basis (or pseudo modes) for DEOM construction. The accuracy and efficiency of this strategy have been explored by a variety of numerical results. We envision that this work provides new insights into extending the hierarchical equations of motion and DEOM approach to certain types of anharmonic environments with arbitrary TCF or spectral density. [ABSTRACT FROM AUTHOR]

Exactly solvable Hamiltonians are useful in the study of quantum many-body systems using quantum computers. In the variational quantum eigensolver, a decomposition of the target Hamiltonian into exactly solvable fragments can be used for the evaluation of the energies via repeated quantum measurements. In this work, we apply more general classes of exactly solvable qubit Hamiltonians than previously considered to address the Hamiltonian measurement problem. The most general exactly solvable Hamiltonians we use are defined by the condition that within each simultaneous eigenspace of a set of Pauli symmetries, the Hamiltonian acts effectively as an element of a direct sum of so(N) Lie algebras and can, therefore, be measured using a combination of unitaries in the associated Lie group, Clifford unitaries, and mid-circuit measurements. The application of such Hamiltonians to decomposing molecular electronic Hamiltonians via graph partitioning techniques shows a reduction in the total number of measurements required to estimate the expectation value compared to previously used exactly solvable qubit Hamiltonians. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the class of Lie-admissible algebras (A ,.) such that its product is a biderivation of the Lie algebra (A , [ , ]) , where [ , ] is the commutator of the algebra (A ,.). First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Lie algebras, symmetric Leibniz algebras, Lie-admissible left (or right) Leibniz algebras, Milnor algebras, and LR-algebras. Then, we establish results on the structure of these algebras in the case that the underlying Lie algebras are perfect (in particular, semisimple Lie algebras). In addition, we then study flexible A B D -algebras, showing in particular that these algebras are extensions of Lie algebras in the category of flexible A B D -algebras. Finally, we study left-symmetric A B D -algebras, in particular we are interested in flat pseudo-Euclidean Lie algebras where the associated Levi-Civita products define A B D -algebras on the underlying vector spaces of these Lie algebras. In addition, we obtain an inductive description of all these Lie algebras and their Levi-Civita products (in particular, for all signatures in the case of real Lie algebras). [ABSTRACT FROM AUTHOR]

In this paper, we construct a new class of simple smooth modules over the affine algebra A 1 (1). Specifically, we present a family of sl (2) ˆ -modules W φ for functions φ which are not Whittaker functions, and provide a simplicity criterion as well as an isomorphism theorem for such modules. For each simple sl (2) ˆ -module W φ , a class of simple sl (2) ˜ -module W φ , γ is also constructed for any constant γ. The sl (2) ˆ -modules W φ are weight modules, while the sl (2) ˜ -modules W φ , γ are not. [ABSTRACT FROM AUTHOR]

We present a comprehensive study of nearly associative algebras. Our research shows that these algebras are power associative Lie-admissible for which the related Lie algebra is solvable, and are also Jordan-admissible. Furthermore, we describe the features of finite-dimensional nearly associative algebras that are nilpotent. In addition, we define the concepts of radical and semisimplicity for this type of algebras. We give a characterization of semisimple nearly associative algebras and prove the Wedderburn Principal Theorem for them. Lastly, we deal with quadratic nearly associative algebras. We provide a characterization and then an inductive description of them through the process of double extension, enriching our understanding of the structural intricacies of this class of algebras. [ABSTRACT FROM AUTHOR]

The tetrahedron algebra ⊠ is an infinite-dimensional Lie algebra defined by generators { x i j | i , j ∈ { 0 , 1 , 2 , 3 } , i ≠ j } and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in ⊠ that "looks like" a standard generator. For mutually distinct h , i , j , k ∈ { 0 , 1 , 2 , 3 } , consider the standard generator x i j of ⊠. An element ξ ∈ ⊠ is called x i j -like whenever both (i) ξ commutes with x i j ; (ii) ξ and x h k satisfy a Dolan-Grady relation. Pick mutually distinct i , j , k ∈ { 0 , 1 , 2 , 3 }. In our main result, we find an attractive basis for ⊠ with the property that every basis element is either x i j -like or x j k -like or x k i -like. We discuss this basis from multiple points of view. [ABSTRACT FROM AUTHOR]

Graded contractions of the Z 2 3 -grading on the complex exceptional Lie algebra g 2 are classified up to equivalence and up to strong equivalence. The non-toral fine Z 2 3 -grading is highly symmetric, with all the homogeneous components Cartan subalgebras. This makes possible a combinatorial treatment based on certain nice subsets of the set of 21 edges of the Fano plane. There are 24 such nice sets up to collineation. Each of these is the support of an admissible graded contraction, one of which is present in every equivalence class of graded contractions. Each nice set gives rise to a single Lie algebra, except for three of the cases in which families depending on one or two parameters are found. In particular, a large family of 14-dimensional Lie algebras arise, most of which are solvable. The properties of each of these Lie algebras are studied. [ABSTRACT FROM AUTHOR]

For a finite dimensional representation V of a finite reflection group W , we consider the rational Cherednik algebra H t , c (V , W) associated with (V , W) at the parameters t ≠ 0 and c. The Dunkl total angular momentum algebra O t , c (V , W) arises as the centraliser algebra of the Lie superalgebra osp (1 | 2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of H t , c (V , W) and the Clifford algebra generated by V. We show that when dim ⁡ V ≥ 3 and for every value of the parameter c , the centre of O t , c (V , W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (− 1) V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into O t , c (V , W) , we establish results analogous to "Vogan's conjecture" for a family of operators depending on suitable elements of the double cover W ˜. [ABSTRACT FROM AUTHOR]

In this paper, we classify 1-cocycles of the Witt algebra with coefficients in the tensor product of two arbitrary tensor density modules. In a special case, we recover a theorem originally established by Ng and Taft in [24]. Furthermore, by these 1-cocycles, we determine Lie bialgebra structures over certain infinite-dimensional Lie algebras containing the Witt algebra. [ABSTRACT FROM AUTHOR]

Let \mathfrak {g} be a complex simple Lie algebra. A simple \mathfrak {g}-module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for \mathfrak {g}. This classification extends the work of Joseph [Ann. Sci. École Norm. Sup. (4) 31 (1998), 17–45], which focused on categorizing minimal highest weight modules annihilated by completely prime ideals. Furthermore, we have determined the associated varieties of these modules. In other words, we have identified all possible weak quantizations of minimal orbital varieties. [ABSTRACT FROM AUTHOR]

Field theories with kinematic Lie algebras, such as field theories featuring color–kinematics duality, possess an underlying algebraic structure known as BV■‐algebra. If, additionally, matter fields are present, this structure is supplemented by a module for the BV■‐algebra. The authors explain this perspective, expanding on our previous work and providing many additional mathematical details. The authors also show how the tensor product of two metric BV■‐algebras yields the action of a new syngamy field theory, a construction which comprises the familiar double copy construction. As examples, the authors discuss various scalar field theories, Chern–Simons theory, self‐dual Yang–Mills theory, and the pure spinor formulations of both M2‐brane models and supersymmetric Yang–Mills theory. The latter leads to a new cubic pure spinor action for 10‐dimensional supergravity. A homotopy‐algebraic perspective on colour–flavour‐stripping is also given, obtain a new restricted tensor product over a wide class of bialgebras, and it is also show that any field theory (even one without colour–kinematics duality) comes with a kinematic L∞$L_\infty$‐algebra. [ABSTRACT FROM AUTHOR]

Hilbert–Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. In this paper, we study unitary representations of these groups from various perspectives. First, we address norm-continuous, also called bounded, representations. These are well known for simple groups, but the general picture is more complicated. Our first main result is a characterization of the discrete decomposability of all bounded representations in terms of boundedness of the set of coroots. We also show that bounded representations of type II and III exist if the set of coroots is unbounded. Second, we use covariance with respect to a one-parameter group of automorphisms to implement some regularity. Here we develop some perturbation theory based on half-Lie groups that reduces matters to the case where a “maximal torus” is fixed, so that compatible weight decompositions can be studied. Third, we extend the context to projective representations which are covariant for a one-parameter group of automorphisms. Here important families of representations arise from “bounded extremal weights”, and for these, the corresponding central extensions can be determined explicitly, together with all one-parameter groups for which a covariant extension exists. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a Kaup–Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct its associated Liouville integrable Hamiltonian hierarchy, through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian formulation. An illustrative example of combined derivative nonlinear Schrödinger equations with two arbitrary constants is explicitly presented. [ABSTRACT FROM AUTHOR]

We generalize an algorithm of Leclerc [ 6 ] describing explicitly the bijection of Lalonde–Ram [ 5 ] from finite to affine Lie algebras. In type |$A_{n}^{(1)}$|⁠ , we compute all affine standard Lyndon words for any order of the simple roots and establish some properties of the induced orders on the positive affine roots. [ABSTRACT FROM AUTHOR]

A new vector-product Lie algebra is constructed for a reductive homogeneous space, which can lead to the presentation of two corresponding loop algebras. As a result, two integrable hierarchies of evolution equations are derived from a new form of zero-curvature equation. These hierarchies can be reduced to the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation. Notably, one of them exhibits a pseudo-Hamiltonian structure, which is derived from a new vector-product identity proposed in this paper. [ABSTRACT FROM AUTHOR]

Let be a finite-dimensional simple Lie algebra over ℂ. A Y () -module is said to be weight if it is a weight -module. We give a complete classification of simple weight modules for Y ( 2) which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical 2 -representation theory, we show that there exists a class of Y ( 2) irreducible modules which have uniformly two-dimensional weight spaces. [ABSTRACT FROM AUTHOR]

Fix a simply-laced semisimple Lie algebra. We study the crystal B (n λ) , were λ is a dominant minuscule weight and n is a natural number. On one hand, B (n λ) can be realized combinatorially by height n reverse plane partitions on a heap associated to λ . On the other hand, we use this heap to define a module over the preprojective algebra of the underlying Dynkin quiver. Using the work of Saito and Savage–Tingley, we realize B (n λ) via irreducible components of the quiver Grassmannian of n copies of this module. In this paper, we describe an explicit bijection between these two models for B (n λ) and prove that our bijection yields an isomorphism of crystals. Our main geometric tool is Nakajima's tensor product quiver varieties. [ABSTRACT FROM AUTHOR]

We study coloured invariants of torus knots T (p , p ′) (where p , p ′ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when p , p ′ ≥ h ∨ , we use the representation theory of the corresponding principal affine algebras to understand the trailing monomials of the coloured invariants. In these cases, we show that the appropriate limits of the renormalized invariants are equal to the characters of certain algebra modules (up to some factors); this result on limits rests on a purely Lie-algebraic conjecture on asymptotic weight multiplicities which we verify in some examples. [ABSTRACT FROM AUTHOR]

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold L , there exists a global trivialization of the tangent bundle, which defines a map ρ p : l ⟶ T p L for each point p ∈ L , where l is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of l. Furthermore, flows of these vector fields give rise to a product between elements of l and L , which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on l. We will describe the properties and examples of these constructions. [ABSTRACT FROM AUTHOR]

A pseudo-Euclidean left-symmetric algebra (A ,. , 〈 , 〉) is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form 〈 , 〉 such that left multiplications by any element of A are skew-symmetric with respect to 〈 , 〉. We recall that a pseudo-Euclidean Lie algebra (g , [ , ] , 〈 , 〉) is flat if and only if (g ,. , 〈 , 〉) , its underlying vector space endowed with the Levi-Civita product associated with 〈 , 〉 , is a pseudo-Euclidean left-symmetric algebra. In this paper, we study pseudo-Euclidean left-symmetric algebras (A ,. , 〈 , 〉) such that commutators of all elements of A are contained in the left annihilator of (A ,.) , these algebras will be called pseudo-Euclidean left-symmetric L -algebras. Next, we introduce and study pseudo-Euclidean modules of pseudo-Euclidean left-symmetric L -algebras. We develop double extension processes that allow us to have inductive descriptions of all pseudo-Euclidean left-symmetric L -algebras and of all its pseudo-Euclidean modules of any signature. This, in particular, improves some of the results obtained in [11]. [ABSTRACT FROM AUTHOR]

For any positive integer n , let W n = Der (C [ t 1 , ... , t n ]). The subspaces h n = Span { t 1 ∂ ∂ t 1 , ... , t n ∂ ∂ t n } and Δ n = Span { ∂ ∂ t 1 , ... , ∂ ∂ t n } are two abelian subalgebras of W n. We show that a full subcategory Ω 1 of the category of W n -modules M which are locally finite over Δ n is equivalent to some full subcategory of weight W n -modules M which are cuspidal modules when restricted to the subalgebra sl n + 1 of W n. [ABSTRACT FROM AUTHOR]

We consider algebras embeddable into free bicommutative algebras with respect to commutator and anti-commutator products over a field of characteristic zero. We show that every metabelian Lie algebra can be embedded into a bicommutative algebra with respect to the commutator product. Furthermore, we prove that the class of commutative algebras embeddable into bicommutative algebras with respect to the anti-commutator product forms a variety. As a consequence, we obtain that every metabelian Lie algebra can be embedded into a Novikov algebra with respect to the commutator product. [ABSTRACT FROM AUTHOR]

We state and prove an analog of the Schur-Weyl duality for a quotient of the classical 2-toroidal Lie algebra of type A. We then provide a method to extend this duality to the m-toroidal case, m > 2. [ABSTRACT FROM AUTHOR]

Let L be a Lie algebra and let A be an associative commutative algebra with unity, both over the same field F. We consider the following question. Is every generalized derivation (resp. quasiderivation) of L ⊗ A the sum of a derivation and a map from the centroid of L ⊗ A , if the same holds true for L? [ABSTRACT FROM AUTHOR]

If A is an associative algebra, then we can define the adjoint Lie algebra A(−) and Jordan algebra A(+). It is easy to see that any associative Rota–Baxter (RB) operator on A induces a Lie and Jordan RB operator on A(−) and A(+), respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra H4, which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra H4(−). [ABSTRACT FROM AUTHOR]

AbstractWe study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of derivations by considering a suitable notion of equivalence when we restrict attention to sucssh algebras. Afterwards, we investigate a link between the well-known (α,β,γ) -derivations and some new extended derivations obtained in our classification. Finally, we present applications of extended derivations to study degenerations of algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study strong Kähler with torsion (SKT) and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of J$J$‐invariant nilradical and non‐J$J$‐invariant nilradical. A classification of such SKT Lie algebras in dimension 6 is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds. [ABSTRACT FROM AUTHOR]

We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems A 3 , C 2 , B 3 and D 4 . We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type A 3 , which already appeared in our previous work, we show that there is a commuting map of Quispel–Roberts–Thompson (QRT) type which is built from a composition of mutations and a permutation applied to the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed A 3 map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the q -Painlevé III quiver found by Okubo. The deformed integrable maps of types C 2 , B 3 and D 4 are also related to elliptic surfaces. From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we discuss Lie nilpotency and Lie solvability of non-abelian tensor product of multiplicative Lie algebras. As our main results, we provide upper bounds on the Lie nilpotency class and Lie solvability length of quotient of tensor product of multiplicative Lie algebras. [ABSTRACT FROM AUTHOR]

This article is devoted to investigation into the notion of multiplicative generalized (σ, τ)-reverse derivations associated with (σ, τ)-reverse derivations of semiprime ∗--rings is characterized. The action of these derivations on ∗--Lie ideals of semiprime ∗--rings is also consideration. Moreover, the commutativity of semiprime ∗--rings admitting multiplicative generalized (σ, τ)-reverse derivations associated with (σ, τ)-reverse derivations satisfying certain algebraic identities on ∗--Lie ideals is explored. [ABSTRACT FROM AUTHOR]

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such that the codimension of nilradical is equal to the number of generators of the nilradical. [ABSTRACT FROM AUTHOR]

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional sl (2 , R) and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is a contact 1-form, conditions under which the Ricci curvature tensor is v-parallel are given. Ricci solitons for Lorentzian–Sasakian Hom-Lie algebras are also studied. It is shown that a Ricci soliton vector field ζ is conformal whenever the Lorentzian–Sasakian Hom-Lie algebra is Ricci semisymmetric. To illustrate the use of the theory, a two-parameter family of three-dimensional Lorentzian–Sasakian Hom-Lie algebras which are not Lie algebras is given and their Ricci solitons are computed. [ABSTRACT FROM AUTHOR]

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group G A (1 , 5) ⊂ S 5 . We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed. [ABSTRACT FROM AUTHOR]

On the basis of a specific matrix Lie algebra, we propose a Kaup–Newell-type matrix eigenvalue problem with four potentials and compute an associated soliton hierarchy within the zero-curvature formulation. A hereditary recursion operator and a bi-Hamiltonian structure are presented to show the Liouville integrability of the resulting soliton hierarchy. An illustrative example is a novel model consisting of combined derivative nonlinear Schrödinger equations with two arbitrary constants. [ABSTRACT FROM AUTHOR]

In recent years, the simplified computation of position and velocity changes in nonlinear systems using Lie groups and Lie algebra has been widely used in the study of robot localization systems. The unscented Kalman filter (UKF) can effectively deal with nonlinear systems through the unscented transformation, and in order to more accurately describe the robot localization system, the UKF method based on Lie groups has been studied successively. The computational complexity of the UKF on Lie groups is high, and in order to simplify its computation, the Lie groups are applied to the manifold, which efficiently handles the state and uncertainty and ensures that the system maintains the geometric constraints and computational simplicity during the updating process. In this paper, a multi-sensor fusion localization method based on an unscented Kalman filter on manifolds (UKF-M) is investigated. Firstly, a system model and a multi-sensor model are established based on an Autonomous Underwater Vehicle (AUV), and a corresponding UKF-M is designed for the system. Secondly, the multi-sensor fusion method is designed, and the fusion method is applied to the UKF-M. Finally, the proposed method is validated using an underwater cave dataset. The experiments demonstrate that the proposed method is suitable for underwater environments and can significantly correct the cumulative error in the trajectory estimation to achieve accurate underwater localization. [ABSTRACT FROM AUTHOR]

Through the generalized fractional derivative, it is studied how the decay term and the fractional parameter affect the quantum system, specifically the interaction between the SU(1,1) algebraic system and a three‐level atom. By transforming the differential equations into fractional differential equations, general fractional solutions are obtained. The influence of decay and fractional parameter on phenomena such as revival and collapse, entropy squeezing, purity, and concurrence are investigated. The results demonstrate how both decay and fractal parameter affect periods of collapse and revival. It is worth noting that the decay parameter shortens the collapse periods, while an increase in the fractional parameter leads to longer collapse periods. The decay parameter also reduces the degree of entanglement between the different components of the quantum system, while increasing the fractional parameter enhances the entanglement within the quantum system. Hence, it can be concluded that the fractional parameter plays a crucial role in the observed effects on the studied properties. [ABSTRACT FROM AUTHOR]

A wide class of two‐component evolution systems is constructed admitting an infinite‐dimensional Lie algebra. Some examples of such systems that are relevant to reaction–diffusion systems with cross‐diffusion are highlighted. It is shown that a nonlinear evolution system related to the Ricci flow on warped product manifold, which has been extensively studied by several authors, follows from the above‐mentioned class as a very particular case. The Lie symmetry properties of this system and its natural generalization are identified and a wide range of exact solutions is constructed using the Lie symmetry obtained. Moreover, a special case is identified when the system in question is reducible to the fast diffusion equation in one space dimension. Finally, another class of two‐component evolution systems with an infinite‐dimensional Lie symmetry that possess essentially different structures is presented. [ABSTRACT FROM AUTHOR]

This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end. [ABSTRACT FROM AUTHOR]

We study continuous selections of the set-valued map that takes every skew-symmetric bilinear form on a vector space to its corresponding set of maximal isotropic subspaces. Applications are made to establishing continuity properties of the Vergne polarizing subalgebras of completely solvable Lie algebras in terms of Schubert cells of suitable Grassmann manifolds. [ABSTRACT FROM AUTHOR]

A restricted Lie algebra L over an algebraically closed field k of characteristic p > 0 is said to possess the Richardson property if there exists a finite dimensional faithful restricted L -module V with associated representation ρ : L → gl (V) such that gl (V) = ρ (L) ⊕ R where R is a subspace of gl (V) such that [ ρ (L) , R ] ⊆ R. In [31] , the author conjectured that if L has the Richardson property with respect to an irreducible restricted L -module V then there exists a reductive algebraic group G over k such that L ≅ Lie (G) as restricted Lie algebras. In this note we confirm this conjecture under the assumption that p > 3. This assumption is needed since our proof relies in a crucial way on the classification of Lie algebras without strong degeneration obtained in [25] for p > 5 and in [24] for p = 5. [ABSTRACT FROM AUTHOR]

We study a family of completely integrable systems attached to Takiff algebras g N , extending Toda systems for split simple Lie algebras g. With respect to Darboux coordinates on coadjoint orbits O , the potentials of the hamiltonians are products of polynomial and exponential functions. Explicit general solutions for their equations of motion are obtained using differential operators called jet transformations. The classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, quantizing the Poisson algebra of functions on O. These results are illustrated with concrete examples, including a 3-body problem based on sl (2) and an extension of soliton solutions for A ∞ to associated Takiff algebras. [ABSTRACT FROM AUTHOR]

We provide a criterion for certain algebraic objects over Jacobson schemes to be forms of each other based on their behaviour at closed fibres. This criterion permits to answer a question related to current and affine Kac-Moody Lie algebras that I. Burban had asked the authors. [ABSTRACT FROM AUTHOR]

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra A. The algebra A is associative, noncommutative, and infinite-dimensional. It is defined by two generators A , B and two relations called the down-up relations. In the present paper, we introduce the Z 3 -symmetric down-up algebra A. We define A by generators and relations. There are three generators A , B , C and any two of these satisfy the down-up relations. We describe how A is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras sl 2 and sl 3 , the sl 3 loop algebra, the Kac-Moody Lie algebra A 2 (1) , the q -Weyl algebra, the quantized enveloping algebra U q (sl 2) , and the quantized enveloping algebra U q (A 2 (1)). We give some open problems and conjectures. [ABSTRACT FROM AUTHOR]

We prove that every simple finite dimensional binary Lie superalgebra over the complex numbers field C with non-zero odd part is either a Lie superalgebra or has a solvable even part. [ABSTRACT FROM AUTHOR]

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and -operators are studied. [ABSTRACT FROM AUTHOR]

Let (Z , ω) be a Kähler manifold and let U be a compact connected Lie group with Lie algebra acting on Z and preserving ω. We assume that the U -action extends holomorphically to an action of the complexified group U ℂ and the U -action on Z is Hamiltonian. Then there exists a U -equivariant momentum map μ : Z → . If G ⊂ U ℂ is a closed subgroup such that the Cartan decomposition U ℂ = U exp (i) induces a Cartan decomposition G = K exp () , where K = U ∩ G , = ∩ i and = ⊕ is the Lie algebra of G , there is a corresponding gradient map μ : Z → . If X is a G -invariant compact and connected real submanifold of Z , we may consider μ as a mapping μ : X →. Given an Ad (K) -invariant scalar product on , we obtain a Morse like function f = 1 2 ∥ μ ∥ 2 on X. We point out that, without the assumption that X is a real analytic manifold, the Lojasiewicz gradient inequality holds for f. Therefore, the limit of the negative gradient flow of f exists and it is unique. Moreover, we prove that any G -orbit collapses to a single K -orbit and two critical points of f which are in the same G -orbit belong to the same K -orbit. We also investigate convexity properties of the gradient map μ in the Abelian case. In particular, we study two-orbit variety X and we investigate topological and cohomological properties of X. [ABSTRACT FROM AUTHOR]

Let D be a Noetherian infinite integral domain, denote by M 2 (D) and by s l 2 (D) the 2 × 2 matrix algebra and the Lie algebra of the traceless matrices in M 2 (D) , respectively. In this paper we study the weak polynomial identities for the natural grading by the cyclic group Z 2 of order 2 on M 2 (D) and on s l 2 (D). We describe a finite basis of the graded polynomial identities for the pair (M 2 (D) , s l 2 (D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M 2 (D) , s l 2 (D)) , is finitely generated. The polynomial identities for M 2 (D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra s l 2 (D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M 2 (D) , g l 2 (D)) were also described, by Krasilnikov and the second author. In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets. [ABSTRACT FROM AUTHOR]

For an arbitrary q ≥ 2 , we find an upper bound for the dimension of a subalgebra of the full matrix algebra M n (K) over an arbitrary field K satisfying the identity [ [ x 1 , y 1 ] , z 1 ] ⋅ [ [ x 2 , y 2 ] , z 2 ] ⋅ ⋯ ⋅ [ [ x q , y q ] , z q ] = 0 , and we show that this upper bound is sharp by presenting an example in block triangular form of a subalgebra of M n (K) with dimension equal to the obtained upper bound. We apply this result to Lie solvable algebras of index 2, i.e., algebras satisfying the identity [ [ x 1 , y 1 ] , [ x 2 , y 2 ] ] = 0. To be precise, for n ≤ 4 , we find the sharp upper bound for the dimension of a Lie solvable subalgebra of M n (K) of index 2, and for n > 4 , we obtain the relatively tight (at least for small values of n > 4) interval [ 2 + ⌊ 3 n 2 8 ⌋ , 2 + ⌊ 5 n 2 12 ⌋ ] for the maximum dimension of a Lie solvable subalgebra of M n (K) of index 2, the exact value of which is not known. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and A , B be unital algebras over R . Let M be a unital (A , B) -bimodule, which is faithful as a left A -module and also as a right B -module. Let T = Tri (A , M , B) be an (n − 1) -torsion free triangular algebra. In this note, we investigate the structure of local Lie n-derivations under certain mild conditions and 2-local Lie n-derivations under weaker conditions than that of local Lie n-derivations with n ≥ 3 on T . Under their corresponding conditions, we conclude that they are both standard on T . [ABSTRACT FROM AUTHOR]

Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞ . Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q . [ABSTRACT FROM AUTHOR]