Algebra Sl Research Articles

An alternative approach to systems of second-order ordinary differential equations with maximal symmetry. Realizations of [formula omitted] by special functions

Using the general solution of the differential equation x¨(t)+g1(t)x˙+g2(t)x=0, a generic basis of the point-symmetry algebra sl(3,R) is constructed. Deriving the equation from a time-dependent Lagrangian, the basis elements corresponding to Noether symmetries are deduced. The generalized Lewis invariant is constructed explicitly using a linear combination of Noether symmetries. The procedure is generalized to the case of systems of second-order ordinary differential equations with maximal sl(n+2,R)-symmetry, and its possible adaptation to the inhomogeneous non-linear case illustrated by an example.

Upon Generating (2+1)-dimensional Dynamical Systems

Under the framework of the Adler-Gel’fand-Dikii(AGD) scheme, we first propose two Hamiltonian operator pairs over a noncommutative ring so that we construct a new dynamical system in 2+1 dimensions, then we get a generalized special Novikov-Veselov (NV) equation via the Manakov triple. Then with the aid of a special symmetric Lie algebra of a reductive homogeneous group G, we adopt the Tu-Andrushkiw-Huang (TAH) scheme to generate a new integrable (2+1)-dimensional dynamical system and its Hamiltonian structure, which can reduce to the well-known (2+1)-dimensional Davey-Stewartson (DS) hierarchy. Finally, we extend the binormial residue representation (briefly BRR) scheme to the super higher dimensional integrable hierarchies with the help of a super subalgebra of the super Lie algebra sl(2/1), which is also a kind of symmetric Lie algebra of the reductive homogeneous group G. As applications, we obtain a super 2+1 dimensional MKdV hierarchy which can be reduced to a super 2+1 dimensional generalized AKNS equation. Finally, we compare the advantages and the shortcomings for the three schemes to generate integrable dynamical systems.

Unified Treatment of a Class of Spherically Symmetric Potentials: Quasi-Exact Solution

We investigate the Schrödinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra ofsl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory ofsl(2)Lie algebra.

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l1(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ℓ1(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ℓ1(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ℓ1(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

We consider the problem of determining the multiplicity function \(m_\xi ^{{ \otimes ^p}\omega }\) in the tensor power decomposition of a module of a semisimple algebra g into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element Ψ((Lgω )⊗p) of the module tensor power into singular elements of irreducible submodules and formulate the problem of determining the function \(M_\xi ^{{ \otimes ^p}\omega }\). This function satisfies a system of recurrence relations that corresponds to the procedure for multiplying modules. To solve this problem, we introduce a special combinatorial object, a generalized (g,ω) pyramid, i.e., a set of numbers (p, {mi})g,ω satisfying the same system of recurrence relations. We prove that \(M_\xi ^{{ \otimes ^p}\omega }\) can be represented as a linear combination of the corresponding (p, {mi})g,ω. We illustrate the obtained solution with several examples of modules of the algebras sl(3) and so(5).

Lowering–raising triples and [formula omitted

We introduce the notion of a lowering–raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We describe the LR triples using various maps, such as the reflectors, the inverters, the unipotent maps, and the rotators. We relate the LR triples to the equitable presentation of the quantum algebra Uq(sl2) and Lie algebra sl2.

On special subalgebras of derivations of Lie algebras

Let L be a Lie algebra, and Der (L) and IDer (L) be the set of all derivations and inner derivations of L, respectively. Also, let Der c(L) denote the set of all derivations α ∈ Der (L) for which α(x) ∈ Imad x for all x ∈ L. We give necessary and sufficient conditions under which Der c(L) = Der z(L), where Der z(L) is the set of all derivations of L whose images lie in the center of L. Moreover, it is shown that any two isoclinic Lie algebras L1 and L2 satisfy Der c(L1) ≅ Der c(L2).

Gauge equivalent integrable equations on sl(3) with reductions

We consider an auxiliary linear problem on the algebra sl(3) with reductions of ℤ2 × ℤ2 type introduced recently by Gerdjikov, Mikhailov and Valchev (GMV), and discuss its relation with a Generalized Zakharov-Shabat system having the same type of reduction symmetry. We describe the gauge-equivalent hierarchies and find explicitly the analog of the first nontrivial soliton equation in the hierarchy of equations related to GMV linear problem.

E6, the group: The structure of SL(3, 𝕆)

We present the subalgebra structure of 𝔰𝔩(3, 𝕆), a particular real form of 𝔢6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group 𝔰𝔩(3, 𝕆) to construct the multiplication table of the corresponding Lie algebra 𝔰𝔩(3, 𝕆). Both the multiplication table and the group are then utilized to find various nested chains of subalgebras of 𝔰𝔩(3, 𝕆), in which the corresponding Cartan subalgebras are also nested where possible. Because our construction involves the Lie group, we simultaneously obtain an explicit representation of the corresponding nested chains of subgroups of SL(3, 𝕆).

Wigner–Eckart theorem for the non-compact algebra 𝔰𝔩(2, ℝ)

The Wigner–Eckart theorem is a well known result for tensor operators of 𝔰𝔲(2) and, more generally, any compact Lie algebra. In this paper, the theorem will be generalized to the particular non-compact case of 𝔰𝔩(2, ℝ). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely, between finite-dimensional and infinite-dimensional representations. As an application, the Wigner–Eckart theorem will be used to construct an analogue of the Jordan–Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.

On nilpotent commuting varieties of r-tuples in the Witt algebra

Let g be the Witt algebra over an algebraically closed field k of characteristic p>3, and G=Aut(g) be the automorphism group of g with Lie(G)=g0. A result [12, Theorem 5.2] of A. Suslin, E. Friedlander and C. Bendel implies that the spectrum of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r-tuples of nilpotent elements in g0. As an analogue of Ngo's result [5, Theorem 1.2.1] in the case of the classical Lie algebra sl2 and a generalization of our previous work [15], in this paper we show that the varieties of r-tuples of nilpotent elements in g, as well as certain subalgebras, are reducible. Irreducible components and their dimensions are precisely presented. Moreover, these nilpotent commuting varieties of r-tuples are neither normal nor Cohen–Macaulay. These results are different from those in the case of the classical Lie algebra sl2.

Unique characterization of the Fourier transform in the framework of representation theory

In this paper we elaborate upon the investigation initiated in [3] of typical and distinctive properties of the Fourier transform (FT), in particular the crucial role played by the Howe dual pair (O(m),sl2). We prove in detail a result on the unique characterization of the FT making extensive use of a representation of the Lie algebra sl2. As an example, we consider the case m = 1. We refer to [3] for a detailed study involving the derivation of a class of operators portraying FT symmetry properties.

Hom-structures on semi-simple Lie algebras

Abstract A Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

Some Weighted Group Algebras are Operator Algebras

AbstractLet G be a finitely generated group with polynomial growth, and let ω be a weight, i.e. a sub-multiplicative function on G with positive values. We study when the weighted group algebra ℓ1 (G, ω) is isomorphic to an operator algebra. We show that ℓ1 (G, ω) is isomorphic to an operator algebra if ω is a polynomial weight with large enough degree or an exponential weight of order 0 < α < 1. We demonstrate that the order of growth of G plays an important role in this problem. Moreover, the algebraic centre of ℓ1 (G, ω) is isomorphic to a Q-algebra, and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when G consists of the d-dimensional integers ℤd or the three-dimensional discrete Heisenberg group ℍ3(ℤ). The case of the free group with two generators is considered as a counter-example of groups with exponential growth.

Module amenability of restricted semigroup algebras under module actions

In this article, weshow that module amenability with the canonical action of restricted semigroup algebra l1r (S) and semigroup algebra l1(Sr) are equivalent, where Sr is the restricted semigroup of associated to the inverse semigroup S. We use this to give a characterization of module amenability of restricted semigroup algebra l1r (S) with the canonical action, where S is a Clifford semigroup.

Bi-integrable couplings of a Kaup–Newell type soliton hierarchy and their bi-Hamiltonian structures

A new Kaup–Newell type soliton hierarchy is generated from an asymmetric matrix spectral problem associated with the three-dimensional special linear Lie algebra sl(2,R). Then based on semi-direct sums of matrix Lie algebras consisting of 3×3 block matrix Lie algebras, corresponding bi-integrable couplings of this hierarchy are constructed. Each equation in the resulting system has a bi-Hamiltonian structure furnished by the variational identity, which lead to Liouville integrability.

Multiplet classification for SU(n,n)

In the present paper we review our project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n,n) for n = 2, 3,4. We give explicitly the main multiplets of indecomposable elementary representations and some reduced multiplets. We give explicitly the minimal representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras sl(2n, R) and when n = 2k for the algebras su* (4k) with suitably chosen maximal parabolic subalgebras.

An integrable Hierarchy on the Perfect Schrödinger Lie Algebra

We construct an integrable hierarchy of coupled integrable solitons equations on the non semisimple perfect Schrodinger Lie algebra sl(2) ⋉ V(1).

G-identities for the Lie algebra [formula omitted

In this paper we study the G-identities for the Lie algebra sl2(C) over the complex field C. If sl2(C) is acted on faithfully by a finite group G then G is isomorphic to one of the following groups: Cn, Dn, A4, S4, A5. Here Cn is the cyclic group of order n, Dn is the dihedral group (of order 2n), An and Sn stand for the alternating and for the symmetric group permuting n letters, respectively.In each one of the above cases we describe a basis of the G-identities for sl2(C). In order to do that we use the explicit form of the graded identities for sl2(C) as well as properties of the group algebras of the corresponding groups. The same problem for the associative algebra M2(C) of the 2×2 matrices was settled by A. Berele in 2004.

Billiard Arrays and finite-dimensional irreducible [formula omitted]-modules

We introduce the notion of a Billiard Array. This is an equilateral triangular array of one-dimensional subspaces of a vector space V, subject to several conditions that specify which sums are direct. We show that the Billiard Arrays on V are in bijection with the 3-tuples of totally opposite flags on V. We classify the Billiard Arrays up to isomorphism. We use Billiard Arrays to describe the finite-dimensional irreducible modules for the quantum algebra Uq(sl2) and the Lie algebra sl2.