Classical Lie Algebras Research Articles

Classification of dynamical Lie algebras of 2-local spin systems on linear, circular and fully connected topologies

Much is understood about 1-dimensional spin chains in terms of entanglement properties, physical phases, and integrability. However, the Lie algebraic properties of the Hamiltonians describing these systems remain largely unexplored. In this work, we provide a classification of all Lie algebras generated by the terms of 2-local spin chain Hamiltonians, or so-called dynamical Lie algebras, on 1-dimensional linear and circular lattice structures. We find 17 unique dynamical Lie algebras. Our classification includes some well-known models such as the transverse-field Ising model and the Heisenberg chain, and we also find more exotic classes of Hamiltonians that appear new. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches. We discuss the practical implications of our work in the context of variational quantum computing, quantum control and the spin chain literature.

The algebraic classification of low-dimensional nilpotent n-Lie algebras

In this paper, we study the low-dimensional nilpotent [Formula: see text]-Lie algebras and classify nilpotent [Formula: see text]-Lie algebras of dimension [Formula: see text] and nilpotent [Formula: see text]-Lie algebras of class two with dimension [Formula: see text].

Schur-Weyl duality for toroidal algebras of type A

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.

A Class of Multi-Component Non-Isospectral TD Hierarchies and Their Bi-Hamiltonian Structures

By using the classical Lie algebra, the stationary zero curvature equation, and the Lenard recursion equations, we obtain the non-isospectral TD hierarchy. Two kinds of expanding higher-dimensional Lie algebras are presented by extending the classical Lie algebra. By solving the expanded non-isospectral zero curvature equations, the multi-component non-isospectral TD hierarchies are derived. The Hamiltonian structure for one of them is obtained by using the trace identity.

Sandwich elements and the Richardson property

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.

Gelfand–Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Gelfand–Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Shapovalov elements of classical and quantum groups

Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a simple Lie algebra g are parameterized by a positive root β and a positive integer m. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a matrix element of the inverse Shapovalov form via a generalized Nagel-Moshinsky algorithm. This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl basis in g. In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum L-operator) in fundamental representations.

Hermitians in matrix algebras with operator norm and associated Lie algebras

This is a continuation of Crabb et al. (2021) [3] and Duncan and McGregor (2022) [4] in which we began an investigation of the real space H of Hermitian matrices in Mn(C) with respect to norms on Cn. Here, we introduce a family of Lie algebras derived from the classical real Lie algebras, with the aim of determining whether or not they are Lie isomorphic to ideals of some H.

Homotopy type of the nilpotent orbits in classical Lie algebras

Homotopy type of the nilpotent orbits in classical Lie algebras

Multicomponent KP type hierarchies and their reductions, associated to conjugacy classes of Weyl groups of classical Lie algebras

This, to a large extent, expository paper describes the theory of multicomponent hierarchies of evolution equations of XKP type, where X = A, B, C, or D, and AKP = KP and their reductions, associated with the conjugacy classes of the Weyl groups of classical Lie algebras of type X. As usual, the main tool is the multicomponent boson–fermion correspondence, which leads to the corresponding tau-functions, wave functions, dressing operators, and Lax operators.

One dimensional representations of finite W-algebras, Dirac reduction and the orbit method

In this paper we study the variety of one dimensional representations of a finite W-algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a conjecture of Losev describing the image of his orbit method map. In order to do so we first establish new Yangian-type presentations of semiclassical limits of the W-algebras attached to distinguished nilpotent elements in classical Lie algebras, using Dirac reduction.

WKB analysis of the linear problem for modified affine Toda field equations

We study the WKB analysis of the solutions to the linear problem for a modified affine Toda field equation, which is equivalent to the higher-order ordinary differential equation (ODE) studied in the ODE/IM correspondence. After gauge transformation, we diagonalize the flat connection of the linear problem to reduce the latter to a set of independent first-order linear differential equations. We explicitly perform this procedure for classical affine Lie algebras with lower ranks. In particular, we study the WKB solutions of the {D}_r^{(1)} - and {D}_{r+1}^{(2)} -type linear problems, which correspond to the higher-order ODEs with the pseudo-differential operator. The diagonalized connection is obtained from the Riccati equations of the adjoint linear problem and related to the conserved currents of the integrable hierarchy constructed by Drinfeld and Sokolov up to total derivatives.

Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras

Abstract Let W ⁢ ( n ) {W(n)} be the Jacobson–Witt algebra over algebraic closed field 𝕂 {\mathbb{K}} with characteristic p > 2 {p>2} . In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W ⁢ ( n ) {W(n)} , which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety ℬ {\mathcal{B}} consisting of all B-subalgebras of W ⁢ ( n ) {W(n)} in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that ℬ {\mathcal{B}} is related to the variety of all full flags in 𝕂 n + 1 {\mathbb{K}^{n+1}} . Additionally, we provide a detailed description of the varieties for W ⁢ ( 1 ) {W(1)} as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson–Witt algebras.

An Algorithm for Berenstein–Kazhdan Decoration Functions and Trails for Classical Lie Algebras

Abstract For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^-:=B^-\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure $\theta ^-_{\textbf{i}}:(\mathbb{C}^{\times })^{l(w_0)}\rightarrow B_{w_0}^-$ for each reduced word $\textbf{i}$ of the longest element $w_0$ of Weyl group. A rational function $\Phi ^h_{BK}=\sum _{i\in I}\Delta _{w_0\Lambda _i,s_i\Lambda _i}$ on $B_{w_0}^-$ is called a half-potential, where $\Delta _{w_0\Lambda _i,s_i\Lambda _i}$ is a generalized minor. Computing $\Phi ^h_{BK}\circ \theta ^-_{\textbf{i}}$ explicitly, we get an explicit form of string cone or polyhedral realization of $B(\infty )$ for the finite dimensional simple Lie algebra $\mathfrak{g}=\textrm{Lie}(G)$. In this paper, for an arbitrary reduced word $\textbf{i}$, we give an algorithm to compute the summand $\Delta _{w_0\Lambda _i,s_i\Lambda _i}\circ \theta ^-_{\textbf{i}}$ of $\Phi ^h_{BK}\circ \theta ^-_{\textbf{i}}$ in the case $i\in I$ satisfies that for any weight $\mu $ of $V(-w_0\Lambda _i)$ and $t\in I$, it holds $\langle h_t,\mu \rangle \in \{2,1,0,-1,-2\}$. In particular, if $\mathfrak{g}$ is of type $\textrm{A}_n$, $\textrm{B}_n$, $\textrm{C}_n$ or $\textrm{D}_n$ then all $i\in I$ satisfy this condition so that one can completely calculate $\Phi ^h_{BK}\circ \theta ^-_{\textbf{i}}$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type $\textrm{G}_2$.

On classical Z2×Z2-graded Lie algebras

We construct classes of Z2×Z2-graded Lie algebras corresponding to the classical Lie algebras in terms of their defining matrices. For the Z2×Z2-graded Lie algebra of type A, the construction coincides with the previously known class. For the Z2×Z2-graded Lie algebra of types B, C, and D, our construction is new and gives rise to interesting defining matrices closely related to the classical ones but undoubtedly different. We also give some examples and possible applications of parastatistics.

Lie Structures and Chain Ideal Lattices

The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to show that the number of Lie algebras of this class is large and they support other types of Lie structures. Beginning with general examples and algebraic decompositions, we focus on computational algorithms to build Lie algebras in which the lattice of ideals is a chain. The chain condition forces gradings on the nilradicals of this class of algebras. Our algorithms yield to several positive naturally graded parametric families of Lie algebras. Further generalizations and other kind of structures will also be discussed.

Controllability and Accessibility on Graphs for Bilinear Systems Over Lie Groups

This article presents graph theoretic conditions for the controllability and accessibility of bilinear systems over the special orthogonal group, the special linear group and the general linear group, respectively, in the presence of drift terms. The controlled terms are assumed to take place between pairwise states. Such bilinear systems naturally induce two interaction graphs: one graph from the drift, and another from the controlled dynamics. As a result, the system controllability or accessibility becomes a property of the two graphs in view of the classical Lie algebra rank condition. We establish a systemic way of transforming the Lie bracket operations in the underlying Lie algebra, into specific operations of removing or creating links over the drift and controlled interaction graphs. As a result, we establish a series of graphical conditions for the controllability and accessibility of such bilinear systems, which rely only on the connectivity of the union of the drift and controlled interaction graphs. We present examples to illustrate the validity of the established results, and show that the proposed conditions are in fact considerably tight.

Deformations and rigidity in varieties of Lie algebras

We present a novel construction of linear deformations for Lie algebras and use it to prove the non-rigidity of several classes of Lie algebras in different varieties. In particular, we address the problem of k -rigidity for k -step nilpotent Lie algebras and k -solvable Lie algebras. We show that Lie algebras with an abelian factor are not rigid, even for the case of a 1-dimensional abelian factor. This holds in the more restricted case of k -rigidity. We also prove that the k -step free nilpotent Lie algebras are not ( k + 1 ) -rigid, but however they are k -rigid.

Index of seaweed subalgebras of classical Lie algebras

We generalize the results in [3] giving a reduction algorithm allowing to compute the index of seaweed subalgebras of classical simple Lie algebras. We thus are able to obtain the index of some interesting families of seaweed subalgebras and to give new examples of large classes of Frobenius Lie algebras among them.

Non-weight representations of Lie superalgebras of Block type, I

In this paper, we study non-weight representations of two classes of Lie algebras and two classes of Lie superalgebras of Block type. More precisely, we completely classify the free U(h)-modules of rank one or two over these Lie superalgebras. Then we determine the simplicity and isomorphism classes of these modules. As a byproduct, we also obtain the non-weight representation results for a class of non-semisimple finitely graded extensions of the Virasoro algebra.