Lie algebras of classical and stochastic electrodynamics

The Lie algebras associated with infinitesimal symmetry transformations of third-order differential equations of interest to classical electrodynamics and stochastic electrodynamics have been obtained. The structure constants for a general case are presented and the Lie algebra for each particular application is easily achieved. By the method used here it is not necessary to know the explicit expressions of the infinitesimal generators in order to determine the structure constants of the Lie algebra.     

Vertex representations forN-toroidal Lie algebras and a generalization of the Virasoro algebra

Vertex representations are obtained for toroidal Lie algebras for any number of variables. These representations afford representations of certainn-variable generalizations of the Virasoro algebra that are abelian extensions of the Lie algebra of vector fields on a torus.Work supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Integrable quantum systems and classical Lie algebras

We have obtained six new infinite series of trigonometric solutions to triangle equations (quantumR-matrices) associated with the nonexceptional simple Lie algebras:sl(N),sp(N),o(N). TheR-matrices are given in two equivalent representations: in an additive one (as a sum of poles with matrix coefficients) and in a multiplicative one (as a ratio of entire matrix functions). TheseR-matrices provide an exact integrability of anisotropic generalizations ofsl(N),sp(N),o(N) invariant one-dimensional lattice magnetics and two-dimensional periodic Toda lattices associated with the above algebras.     

Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

Using deformation theory, Braverman and Joseph constructed certain primitive ideals in the enveloping algebras of the simple Lie algebras. Except in the case sl(2,C)$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$, there is a special value of the deformation parameter giving an ideal of infinite codimension. For the classical Lie algebras, the uniqueness of the special value is equivalent to the existence of tensors with very particular properties. The existence of these tensors was concluded abstractly by Braverman and Joseph but here we present explicit formulae. This allows a rather direct computation of the special value of the deformation parameter.     

Noncommutative symmetric functions and laplace operators for classical Lie algebras

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).     

Pseudo-differential equations,and the Bethe ansatz for the classical Lie algebras

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.     

Schubert polynomials and the Littlewood-Richardson rule

The decomposition of a product of two irreducible representations of a linear group Gl(N, ) is explicitly given by the Littlewood-Richardson rule, which amounts to finding how many Young tableaux satisfy certain conditions. We obtain more general multiplicities by generating vexillary permutations and by using partially symmetrical polynomials (Schubert polynomials).A la mémoire de S. Ulam, exemple et ami.     

Solvable lattice models related to the vector representation of classical simple Lie algebras

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraX _n ⁽¹⁾ =A _n ⁽¹⁾ ,B _n ⁽¹⁾ ,C _n ⁽¹⁾ ,D _n ⁽¹⁾ . The local states taken on by the fluctuation variables are the dominant integral weights ofX _n ⁽¹⁾ of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofX _n . The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.     

Bicovariant quantum algebras and quantum Lie algebras

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun toU _q g, given by elements of the pure braid group. These operators—the reflection matrixYL ⁺ SL ^– being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO _q (N).This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139     

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA ₂,B ₂, andG ₂. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.     

Standard complex for quantum Lie algebras

For a quantum Lie algebra Γ, let Γ^ be its exterior extension (the algebra Γ^ is canonically defined). We introduce a differential on the exterior extension algebra Γ^ which provides the structure of a complex on Γ^. In the situation when Γ is a usual Lie algebra, this complex coincides with the “standard complex.” The differential is realized as a commutator with a (BRST) operator Q in a larger algebra Γ^[Ω], with extra generators canonically conjugated to the exterior generators of Γ^. A recurrent relation which uniquely defines the operator Q is given.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras continuum Lie algebras. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.     

Lie algebras and quasifield operators

It is shown that realisations of any Lie algebra by means of bilinear polynomials of quasifield operators exist. These realisations are used to find some class of representations of the algebra.     

Target-space duality between simple compact Lie groups and Lie algebras under the Hamiltonian formalism: I. Remnants of duality at the classical level

It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie groupG with a bi-invariant metric and a generating function suggested in the physics literature, we follow the above line of thought and work out the canonical transformation generated by together with an Ad-invariant metric and a B-field on the associated Lie algebra ofG so thatG and form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation including a careful analysis of its domain and image. The geometry of the T-dual structure on is lightly touched. We leave the task of tracing back the Hamiltonian formalism at the quantum level to the sequel of this paper.     

黑克、布劳、及伯曼-温采尔代数的诱导及分导系数和量子经典李代数的耦合与重新耦合系数

本文总结了计算黑克、布劳、及伯曼 温采尔代数在各种工数链下诱导及分导系数的线性方程方法(LEM)。特别强调了关于A,B,C,D型李代数及其量子情形与其中心代数之间的舒尔 魏尔 布劳双关性关系。这一关系使我们能够利用相应中心代数的诱导及分导系数计算出经典李代数及其量子情形的耦合与重新耦合系数。讨论了从该方法得到B,C,D型李代数不可约表示克罗内克积分解的应用。基于LEM还得到了处理对应于置换群CG系列问题的黑克代数张量积的方法。     

Rational forms for twistings of enveloping algebras of simple Lie algebras

We give rational forms for twistings of classical enveloping algebras. We also remark a link with the generalized formalism of Gurevich, Manin, and Cartier.     

Step algebras of semi-simple subalgebras of Lie algebras

For each pair (G,K) where G is a complex finite-dimensional Lie algebra and K a semi-simple subalgebra of G, we construct an associative algebra (step algebra) $\text{Y}$ (G,K) and a homomorphism i_*: $\text{Y}$ (G,K)→E(G) is the enveloping algebra of G. $\text{Y}$ (G,K) has the following properties: (1) If V is any G-module and x ? V a K-maximal vector, then sx = i_* (s)x is K-maximal for any s ? $\text{Y}$ (G,K); (2) If V is irreducible and a certain simple criteria is fulfilled, then any K-maximal vector can be written in the form sx_m, s ? $\text{Y}$ (G,K), where x_m is some fixed K-maximal vector. Because of these properties $\text{Y}$ (G,K) has great practical value when constructing irreducible representations of Lie algebras in a form which makes the reduction with respect to a semi-simple subalgebra explicit.     

Nilpotent locally convex Lie algebras and Lie field structures

The purpose of this work is to join Lie field structures with certain infinite-dimensional Lie algebras with locally convex topology. These topological Lie algebras allow topological groups which are a generalization of the connected nilpotent Lie groups. We showed the existence of the continuous unitary representations of the gained groups and then we proved the analogue of Gårding theorem. Using this theorem we established the existence of representations of Lie field structures into Lie algebras of skew-symmetric operators on Hilbert spaces.Work supported by National Science Foundation.On leave of absence from the Institute Rudjer Bokovi, Zagreb.