Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

Novel extended Zn-Heisenberg ferromagnet models

In this paper, in terms of taking values in a commutative subalgebra [Formula: see text] of Lie algebra [Formula: see text], one establishes two novel extended [Formula: see text]-Heisenberg ferromagnet models in both [Formula: see text] and [Formula: see text]-dimensions and derives their corresponding Lax representations. Moreover, we present their geometrical equivalent equations which are [Formula: see text]-nonlinear Schrödinger equations.

Self-adjointness and conservation laws of Burgers-type equations

This paper focuses on the self-adjointness of some Burgers-type equations based on the existing definitions. It follows that the corresponding Frobenius Burgers-type equations are constructed by taking values in a commutative subalgebra [Formula: see text]. In order to investigate the self-adjointness of these Frobenius-type equations, we introduce a few additional notations and definitions. Additionally, the conservation laws are obtained of several equations studied by using symmetries.

Self-Adjointness and Conservation Laws of Frobenius Type Equations

The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra Z2ε in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws.

The Casimir elements of the Racah algebra

Let [Formula: see text] denote a field with [Formula: see text]. The Racah algebra [Formula: see text] is the unital associative [Formula: see text]-algebra defined by generators and relations in the following way. The generators are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The relations assert that [Formula: see text] and each of the elements [Formula: see text] is central in [Formula: see text]. Additionally, the element [Formula: see text] is central in [Formula: see text]. We call each element in [Formula: see text] a Casimir element of [Formula: see text], where [Formula: see text] is the commutative subalgebra of [Formula: see text] generated by [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The main results of this paper are as follows. Each of the following distinct elements is a Casimir element of [Formula: see text]: [Formula: see text] [Formula: see text] [Formula: see text] The set [Formula: see text] is invariant under a faithful [Formula: see text]-action on [Formula: see text]. Moreover, we show that any Casimir element [Formula: see text] is algebraically independent over [Formula: see text]; if [Formula: see text], then the center of [Formula: see text] is [Formula: see text].

Poisson-Commutative Subalgebras of 𝒮(𝔤) associated with Involutions

The symmetric algebra ${\mathcal{S}}({{\mathfrak{g}}})$ of a reductive Lie algebra ${{\mathfrak{g}}}$ is equipped with the standard Poisson structure, that is, the Lie–Poisson bracket. Poisson-commutative subalgebras of ${\mathcal{S}}({{\mathfrak{g}}})$ attract a great deal of attention because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset{\mathcal{S}}({{\mathfrak{g}}})$ is bounded by the “magic number” ${\boldsymbol{b}}({{\mathfrak{g}}})$ of ${{\mathfrak{g}}}$. There are two classical constructions of $\mathcal C$ with ${\textrm{tr.deg}}\,{\mathcal C}={\boldsymbol{b}}({{\mathfrak{g}}})$. The 1st one is applicable to $\mathfrak{gl}_n$ and $\mathfrak{so}_n$ and uses the Gelfand–Tsetlin chains of subalgebras. The 2nd one is known as the “argument shift method” of Mishchenko–Fomenko. We generalise the Gelfand–Tsetlin approach to chains of almost arbitrary symmetric subalgebras. Our method works for all types. Starting from a symmetric decompositions ${{\mathfrak{g}}}={{\mathfrak{g}}}_0\oplus{{\mathfrak{g}}}_1$, Poisson-commutative subalgebras ${{\mathcal{Z}}},\tilde{{\mathcal{Z}}}\subset{\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ of the maximal possible transcendence degree are constructed. If the ${{\mathbb{Z}}}_2$-contraction ${{\mathfrak{g}}}_0\ltimes{{\mathfrak{g}}}_1^{\textsf{ab}}$ has a polynomial ring of symmetric invariants, then $\tilde{{\mathcal{Z}}}$ is a polynomial maximal Poisson-commutative subalgebra of ${\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ and its free generators are explicitly described.

Residual Symmetries and Bäcklund Transformations of (2 + 1)-Dimensional Strongly Coupled Burgers System

In this article, we mainly apply the nonlocal residual symmetry analysis to a (2 + 1)-dimensional strongly coupled Burgers system, which is defined by us through taking values in a commutative subalgebra. On the basis of the general theory of Painlevé analysis, we get a residual symmetry of the strongly coupled Burgers system. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is derived by Lie’s first theorem. Further, the linear superposition of the multiple residual symmetries is localized to a Lie point symmetry, and an N-th Bäcklund transformation is also obtained.

The extended Zn-Heisenberg ferromagnet model

By virtue of taking values in a commutative subalgebra [Formula: see text] of Lie algebra [Formula: see text], we construct the [Formula: see text]-Heisenberg ferromagnet model which contains many Heisenberg ferromagnet-type equations. Moreover, we investigate the integrable properties of the [Formula: see text]-Heisenberg ferromagnet model. In terms of the gauge transformation, the gauge equivalent counterpart of the [Formula: see text]-Heisenberg ferromagnet model has been presented. Based on the differential geometry of curves and surfaces, the corresponding geometrical equivalence between the [Formula: see text]-Heisenberg ferromagnet model and [Formula: see text]-nonlinear Schrödinger equation has also been established. Furthermore, we also discuss the [Formula: see text]-generalized inhomogeneous Heisenberg ferromagnet model.

Residual Symmetries and Bäcklund Transformations of Strongly Coupled Boussinesq–Burgers System

In this article, we construct a new strongly coupled Boussinesq–Burgers system taking values in a commutative subalgebra Z 2 . A residual symmetry of the strongly coupled Boussinesq–Burgers system is achieved by a given truncated Painlevé expansion. The residue symmetry with respect to the singularity manifold is a nonlocal symmetry. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is obtained with the help of Lie’s first theorem. Further, the linear superposition of multiple residual symmetries is localized to a Lie point symmetry, and a N-th Bäcklund transformation is also obtained.

Computing embedding numbers and branches of orders via extensions of the Bruhat-Tits tree

Let [Formula: see text] be a local field and let [Formula: see text] be the two-by-two matrix algebra over [Formula: see text]. In our previous work, we developed a theory that allows the computation of the set of maximal orders in [Formula: see text] containing a given suborder. This set is given as a subgraph of the Bruhat–Tits (BT)-tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the BT-tree in terms of balls in [Formula: see text]. This is easier for orders spanning a split commutative subalgebra, i.e. an algebra isomorphic to [Formula: see text]. In fact, we have successfully used this idea in the past to compute embedding numbers for the split algebra. In the present work, we develop a theory of branches over field extensions that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions, generalizing previous results due to the first author and Saavedra. We assume throughout that [Formula: see text].