Polynomial Lie Algebras Research Articles

Lax representation and Bäcklund–Darboux transformation for coupled BKP hierarchy

By using vertex operator representation of polynomial Lie algebra b∞(n), a theoretical interpretation of coupled BKP hierarchy is given, where corresponding Hirota bilinear equations are derived. Then based upon this, a wave function matrix is introduced, so that we can construct the corresponding dressing operator matrix and obtain the coupled BKP constraints and Sato equation, which allows us to define the Lax operator matrix and obtain the Lax equation. It is found that the coupled BKP hierarchy is a special case of matrix KP theory. Lastly, we present the Bäcklund–Darboux transformations for the coupled BKP equations from both fermionic and bosonic pictures.

Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov bases

Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's Program on Algebraic Operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Gröbner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by nonassociative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program is formulated in terms convergent rewriting systems and equivalently in terms of Gröbner-Shirshov bases. The relation of this Lie algebra analog of Rota's program with Rota's program for associative algebras is established. Applications are given to modified Rota-Baxter operators, differential type operators and Rota-Baxter type operators.

Sigma Functions and Lie Algebras of Schrödinger Operators

In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $$g > 0$$ , a system of $$2g$$ multidimensional Schrodinger equations in magnetic fields with quadratic potentials was defined. Such systems are equivalent to systems of heat equations in a nonholonomic frame. It was proved that such a system determines the sigma function of the universal hyperelliptic curve of genus $$g$$ . A polynomial Lie algebra with $$2g$$ Schrodinger operators $$Q_0, Q_2, \dots, Q_{4g-2}$$ as generators was introduced. In this work, for each $$g > 0,$$ we obtain explicit expressions for $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ and recurrent formulas for $$Q_{2k}$$ with $$k>2$$ expressing these operators as elements of a polynomial Lie algebra in terms of the Lie brackets of the operators $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ . As an application, we obtain explicit expressions for the operators $$Q_0, Q_2, \dots, Q_{4g-2}$$ for $$g = 1,2,3,4$$ .

Polynomial Lie Algebras and Growth of Their Finitely Generated Lie Subalgebras

The concept of polynomial Lie algebra of finite rank was introduced by V. M. Buchstaber in his studies of new relationships between hyperelliptic functions and the theory of integrable systems. In this paper we prove the following theorem: the Lie subalgebra generated by the frame of a polynomial Lie algebra of finite rank has at most polynomial growth. In addition, important examples of polynomial Lie algebras of countable rank are considered in the paper. Such Lie algebras arise in the study of certain hyperbolic partial differential equations, as well as in the construction of self-similar infinite-dimensional Lie algebras (such as the Fibonacci algebra).

Polynomial Lie algebras and the Zelmanov–Shalev theorem

Polynomial Lie algebras and the Zelmanov–Shalev theorem

Differentiation of genus 3 hyperelliptic functions

In this work we give an explicit solution to the problem of differentiation of hyperelliptic functions in genus $3$ case. It is a genus $3$ analogue of the result of F. G. Frobenius and L. Stickelberger. Our method is based on the series of works by V. M. Buchstaber, D. V. Leikin and V. Z. Enolskii. We describe a polynomial map $p\colon \mathbb{C}^{3g} \to \mathbb{C}^{2g}$. For $g = 1,2,3$ we describe $3g$ polynomial vector fields in $\mathbb{C}^{3g}$ projectable for $p$ and their polynomial Lie algebras. We obtain the corresponding derivations of the field of hyperelliptic functions.

Localization and the Weyl algebras

Let W n (ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in W n (ℝ) for p + q = n with the Lie algebra multiplication being given by the bracket [a, b] = ab − ba for a, b quadratic polynomials in W n (ℝ). What does not seem to be so well known is that the converse statement is, in a certain sense, also true, namely, that, by using extension and localization, it is possible, at least in some cases, to construct homomorphisms of W n (ℝ) onto its image in a localization of U(so(p + 2, q)), the universal enveloping algebra of so(p + 2, q), and m = p + q. Since Weyl algebras are simple, these homomorphisms must either be trivial or isomorphisms onto their images. We illustrate this remark for the so(2, q) case and construct a mappping from W q (ℝ) onto its image in a localization of U(so(2, q)). We prove that this mapping is a homomorphism when q = 1 or q = 2. Some specific results about representations for the lowest dimensional case of W 1(ℝ) and U(so(2, 1)) are given.

Polynomial Lie algebras and the Zelmanov-Shalev theorem

Polynomial Lie algebras and the Zelmanov-Shalev theorem

On realizations of polynomial algebras with three generators via deformed oscillator algebras

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analogue of this construction for the most general polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.

Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

Let W n ( $$ {\mathbb K} $$ ) be the Lie algebra of derivations of the polynomial algebra $$ {\mathbb K} $$ [X] := $$ {\mathbb K} $$ [x 1,…,x n ]over an algebraically closed field $$ {\mathbb K} $$ of characteristic zero. A subalgebra $$ L \subseteq {W_n}(\mathbb{K}) $$ is called polynomial if it is a submodule of the $$ {\mathbb K} $$ [X]-module W n ( $$ {\mathbb K} $$ ). We prove that the centralizer of every nonzero element in L is abelian, provided that L is of rank one. This fact allows one to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.

25 Years of Quantum Groups: from Definition to Classification

In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians.

The 4th structure

In this paper we describe all Lie bialgebra structures on the polynomial Lie algebra $\mathbf{g}[u]$, where $\mathbf{g}$ is a simple, finite dimensional, complex Lie algebra. The results are based on an unpublished paper Montaner and Zelmanov. Further, we introduce quasi-rational solutions of the CYBE and describe all quasi-rational $r$-matrices for $\mathbf{sl}(2)$.

Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra [formula omitted

In a recent paper [P. Casati, G. Ortenzi, New integrable hierarchies from vertex operator representations of polynomial Lie algebras, J. Geom. Phys. 56 (3) (2006) 418–449] Casati and Ortenzi gave a representation-theoretic interpretation of recently discovered coupled soliton equations, which were described by e.g. R. Hirota, X. Hu, X. Tang [A vector potential KdV equation and vector Ito equation: Soliton solutions, bilinear Bäcklund transformations and Lax pairs, J. Math. Anal. Appl. 288 (1) (2003) 326–348. [3]], S. Kakei [Dressing method and the coupled KP hierarchy, Phys. Lett. A 264 (6) (2000) 449–458. [6]] and S.Yu. Sakovich [A note in the Painlevé property of coupled KdV equation, arXiv:nlin.SI/0402004. [7]]. Casati and Ortenzi use vertex operators for these Lie algebras and a boson–fermion type of correspondence to get a hierarchy of coupled Hirota bilinear equations. In this paper we reformulate the Hirota bilinear description for the Lie algebra g l ∞ ( n ) and obtain a bilinear identity for matrix wave functions. From that it is straightforward to deduce the Sato–Wilson, Lax and Zakharov–Shabat equations. Using these wave functions and standard calculus with vertex operators we obtain elementary Bäcklund–Darboux transformations.

Classical quasi-trigonometric r−matrices of Cremmer-Gervais type and their quantization

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.

Classical quasi-trigonometric r-matrices of Cremmer-Gervais type and their quantization

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.

New integrable hierarchies from vertex operator representations of polynomial Lie algebras

We give a representation–theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld–Sokolov of A, B, C, D hierarchies and to construct their soliton solutions.

Polynomial Lie Algebras \(\hat E_{R_1 }^\mathcal{P} (u(n);(m))\) in Nonlinear Models of Quantum Optics: Basic Ideas and Cluster Dynamics in the Heisenberg Picture

We consider applications of polynomial Lie algebras \(\hat E_{R_1 }^\mathcal{P} (u(n);(m))\), which are extensions of the unitary Lie algebras u(n) by their symmetric tensors vm of mth rank, to examine a class of nonlinear models of quantum optics and laser physics. Particular attention is paid to examining integrability of evolution equations arising from the Heisenberg equations for collective (cluster) dynamic variables related to the \(\hat E_{R_1 }^\mathcal{P} (u(2);(2))\) generators. This allows one to examine some peculiarities of cluster dynamics of models incorporating second-harmonic generation together with interference phenomena. Full integrability of such equations at the quantum and quasiclassical levels is shown to be possible for particular values of the model parameters. In general, these equations are integrable only in the quasiclassical (cluster mean-field) approximation. They do not admit a full separation of variables and consequently correspond to very irregular dynamic regimes.

Polynomial Lie algebra methods in solving the second-harmonic generation model: some exact and approximate calculations

We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of a polynomial Lie algebra su pd (2) and related spectral representations of the model evolution operator realized in algorithmic forms. It enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.

Polynomial Lie Algebras

We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x1,...,xn]/(f1,...,fn) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A,B,C, D, and E6.

Polynomial Lie algebras in solving a class of integrable models of quantum optics: exact methods and quasiclassics

A wide class of integrable quantum-optical models with Gi-invariant Hamiltonians H is described in the form when H are linear functions in generators of the polynomial Lie algebras supd(2) and Hilbert spaces L(H) of quantum states are decomposed in direct sums of supd(2)-irreducible subspaces. This yields exact and approximate methods of solving physical problems as well as new (supd(2)-cluster) quasiclassics in original models.