Infinite Lie Algebras Research Articles

A conjecture of Dekking on the dimensions of the lower central series factors of a certain just infinite Lie algebra

Revisiting just infinite nil graded fractal Lie superalgebras studied by de Morais Costa and Petrogradsky, and more precisely a Lie algebra R in characteristic 2, we give a simple combinatorial construction of the sequence of dimensions of the lower central series factors of R in terms of automatic and morphic sequences. In particular we prove a conjecture of Dekking about this sequence of dimensions.

Modified DKP hierarchy as modified BKP hierarchy

In this paper, we investigate some topics involving the modified D-type Kadomtsev-Petviashvili (DKP) hierarchy. Note that the modified DKP (mDKP) hierarchy is just the modified B-type Kadomtsev-Petviashvili (BKP) hierarchy in the bosonic forms. Based on this fact, we first proved the conjecture in You [Physica D 50, 429–462 (1991)], that is, the product of the two tau functions of the mDKP hierarchy is some tau function of the KP hierarchy. Then, we investigate the Darboux transformations of the DKP and mDKP hierarchy. Finally, the solutions of the constrained BKP hierarchy are given in the context of the fermionic representation of infinite Lie algebra d∞.

Quantum W1+∞ subalgebras of BCD type and symmetric polynomials

The infinite affine Lie algebras of type ABCD, also called gl̂(∞), ô(∞), and sp̂(∞), are equivalent to subalgebras of the quantum W1+∞ algebras. They have well-known representations on the Fock space of a Dirac fermion (Â∞), a Majorana fermion (B̂∞ and D̂∞), or a symplectic boson (Ĉ∞). Explicit formulas for the action of the quantum W1+∞ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the vertical presentation of the quantum toroidal gl(1) algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the horizontal presentation. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum W1+∞ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of B̂∞, a q-difference equation is obtained for Q-Schur polynomials indexed by strict partitions.

Projective modules over classical Lie algebras of infinite rank in the parabolic category

We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a,b,c,d. Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O‾ over infinite rank Lie superalgebra of types a,b,c,d through the super duality is also a Koszul category.

On conservation laws for a viscous fluid

The equation of motion of viscous fluid have the divergence type of the conservation laws and admit the infinite Lie algebra of point symmetries. Canonical operators of the algebra give rise to conservation laws of the first order. This was tested for the basic operators of admitted algebra. Any conservation law may be turned by doing some canonical operator on the nontrivial conservation law. The canonical operators of the zero order of conservation laws were calculated. The problem about the calculation of canonical operators of the first order of conservation laws are supplied.

Lie Symmetries, One-Dimensional Optimal System and Optimal Reduction of (2 + 1)-Coupled nonlinear Schrödinger Equations

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.

О серии подмногообразий многообразия, порожденного простой бесконечномерной алгеброй картановского типа общей серии W2

О серии подмногообразий многообразия, порожденного простой бесконечномерной алгеброй картановского типа общей серии W2

Infinite Lie Algebras Generated by Supersymmetric Hypermatrices

In this paper we study the structure and properties of complex infinite supersymmetric hypermatrices generated by a semisimple basis, exponential sets of hypermatrices, hypermatrix Lie algebra and elements of the group of complex matrices of order two and determinant one. We study the hypermatrix Lie algebra generated by the polygons on analytic torus of genus g. By using new algebraic tools, namely cubic hypermatrices we study the algebraic structures associated with the hypermatrices of certain Lie algebras e.g. $\{sl2; f, \infty\}$; $\{sl2; \infty, \infty\}$ and $\{SL2; f, \infty\}$; $\{SL2; \infty, \infty\}$ and we construct generators of infinite periodic hypermatrix Lie algebraic structures which have classical Lie algebra decomposition; specifically a set of Lie algebras composed of hypermatrices. We study the exponential of a complex analytic Lie algebra, rotations of hypermatrices, and relations between hypermatrix groups, hypermatrix Lie Algebra, Fourier hypermatrices and the Laurent hypermatrix. Finally, as an application we will show that there is an isomorphism of the hypermatrix Lie algebra associated with a set of polygons on the torus of genus g and analytic functions associated with a countable set of solutions of a meromorphic function on the torus. In conclusion we will present a Riemann type isomorphism theorem for hypermatrices on a torus and the convoluted complex plane, generated by holomorphic functions, based on the equivalent relations of the geometry and the algebra of the torus of dimension three and genus g.

An algebraic model for the multiple Meixner polynomials of the first kind

An interpretation of the multiple Meixner polynomials of the first kind is provided through an infinite Lie algebra realized in terms of the creation and annihilation operators of a set of independent oscillators. The model is used to derive properties of these orthogonal polynomials

Solvable infinite filiform Lie algebras

An infinite filiform Lie algebra L is residually nilpotent and its graded associated with respect to the lower central series has smallest possible dimension in each degree but is still infinite. This means that gr(L) is of dimension two in degree one and of dimension one in all higher degrees. We prove that if L is solvable, then already [L,L] is abelian. The isomorphism classes in this case are given in [1], but the proof is incomplete. We make the necessary additional computations and restate the result in [1] when the ground field is the complex numbers.

Gause Symmetry and Howe Duality in 4D Conformal Field Theory Models

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.The paper reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren and the author.

Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space‐time dimensions the associated dual pairs are constructed and classified.

Infinite Lie algebras and dual pairs in 4D CFT models

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

INFINITE DIMENSIONAL LIE ALGEBRAS IN 4D CONFORMAL FIELD THEORY

It is known that there are no scalar Lie fields in more than two space-time dimensions [4]. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. Recent work, [2, 3], is reviewed, in which we classify such algebras and their unitary positive energy representations in a theory of a system of scalar fields of dimension two. The results are linked to the Doplicher–Haag–Roberts theory of superselection sectors governed by a (global) compact gauge group.

Certain Classes of Automorphisms of Infinite Rank Affine Lie Algebras#

ABSTRACT An infinite rank affine Lie algebra 𝔤 is a Kac-Moody algebra associated with an infinite affine matrix. For each nonnegative integer l, 𝔤 contains a subalgebra 𝔤l which is a classical finite dimensional simple Lie algebra, 𝔤 0 ⊂ 𝔤 1 ⊂ ± b ··· and 𝔤 is the inductive limit of the set {𝔤i, i = 0, 1,…} of these subalgebras. In the present article, we will determine all automorphisms of 𝔤 leaving 𝔤ni invariant for each {ni} in a set {ni}, where the set {ni, i = 1, 2,…} is any given nonnegative integer sequence with n1 < n2 < ···. These automorphisms are generalizations of automorphisms of classical finite dimensional Lie algebras.

CKP and BKP Equations Related to the Generalized Quartic Hénon–Heiles Hamiltonian

In their classification of soliton equations from a group theoretical standpoint according to the representation of infinite Lie algebras, Jimbo and Miwa listed bilinear equations of low degree for the KP and the modified KP hierarchies. In this list, we consider the (1+1)-dimensional reductions of three particular equations of special interest for establishing some new links with the generalized Henon–Heiles Hamiltonian, possibly useful for integrating the latter with functions having the Painleve property. Two of those partial differential equations have N-soliton solutions that, as for the Kaup–Kupershmidt equation, can be written as the logarithmic derivative of a Grammian. Moreover, they can describe head-on collisions of solitary waves of different type and shape.

Insertion and Elimination: the Doubly Infinite Lie Algebra of Feynman Graphs

The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine.

Geometry of the gauge algebra in noncommutative Yang-Mills theory

A detailed description of the infinite-dimensional Lie algebra of star-gauge transformations in noncommutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying deformed algebra of functions on spacetime, of deformed symplectic diffeomorphisms, of the infinite unitary Lie algebra, and of the algebra of compact operators on a quantum mechanical Hilbert space. The spacetime and string interpretations are also elucidated.

Braid groups, infinite Lie algebras of Cartan type and rings of invariants

In this paper we show that each element α of the pure braid group P n or the pure symmetric automorphism group H( n) of the free group F n of rank n can be represented as α= exp(D)= id+D+(D 2/2!)+(D 3/3!)+⋯ , where D= D( α) is an element of an infinite-dimensional Lie algebra h(n) . Each such D is a derivation of the power series ring C[[a 1,…,a r]] , r= n 2− n, which fixes the volume form a 1∧⋯∧ a r and so h(n) is a subalgebra of S n 2− n , the special Lie algebra of Cartan type. There is a corresponding action of these groups on C[[a 1,…,a r]] and C[a 1,…,a r] . We use the representation α=exp( D) to prove results about the ring of invariants for this action of the pure braid group. The Lie algebra h(n) is a subalgebra of a graded Lie algebra h(n) ; we also calculate the Poincaré series of the Lie algebra l(n) and of certain of its subalgebras, and show that these Poincaré series are rational.

Algebraic nature of shape-invariant and self-similar potentials

Self-similar potentials generalize the concept of shape-invariance which was originally introduced to explore exactly-solvable potentials in quantum mechanics. In this article it is shown that previously introduced algebraic approach to the latter can be generalized to the former. The infinite Lie algebras introduced in this context are shown to be closely related to the q-algebras. The associated coherent states are investigated.