A Class of New Lie Algebra,the Corresponding g-mKdV Hierarchy and Its Hamiltonian Structure

A class of new Lie algebra B ₃ is constructed, which is far different from the known Lie algebra A _n−1. Based on the corresponding loop algebra [(B₃)\tilde]\tilde{B_{3}}, the generalized mKdV hierarchy is established. In order to look for the Hamiltonian structure of such integrable system, a generalized trace functional of matrices is introduced, whose special case is just the well-known trace identity. Finally, its expanding integrable model is worked out by use of an enlarged Lie algebra.     

Hamiltonian and Super-Hamiltonian Extensions Related to Broer-Kaup-Kupershmidt System

Based on the Lie algebra A ₁, the integrable Broer-Kaup-Kupershmidt (BKK) system is revisited. The bi-Hamiltonian structure is constructed by the trace identity. Two extensions of the Lie algebra A ₁ are considered, i.e., the non-semi-simple Lie algebra of 4×4 matrix and the super-Lie algebra of 3×3 matrix, from which two hierarchies of soliton equations related to BKK system are given. With the aid of the generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures of the resulting systems are constructed.     

Kontsevich Deformation Quantization and Flat Connections

In Torossian (J Lie Theory 12(2):597–616, 2002), the second author used the Kontsevich deformation quantization technique to define a natural connection ω _n on the compactified configuration spaces [`(C)]_n,0{\overline{C}_{n,0}} of n points on the upper half-plane. Connections ω _n take values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ω _n is flat.     

Dynamical Lie algebra method for highly excited vibrational state of asymmetric linear tetratomic molecules

The highly excited vibrational states of asymmetric linear tetratomic molecules are studied in the framework of Lie algebra. By using symmetric groupU ₁(4)U ₂(4)⊗U ₃(4), we construct the Hamiltonian that includes not only Casimir operators but also Majorana operators M₁₂, M₁₃ and M₂₃, which are useful for getting potential energy surface and force constants in Lie algebra method. By Lie algebra treatment, we obtain the eigenvalues of the Hamiltonian, and make the concrete calculation for molecule C₂HF.     

A new integrable case of the motion of the 4-dimensional rigid body

A Lax pair for a new family of integrable systems on SO(4) is presented. The construction makes use of a twisted loop algebra of theG ₂ Lie algebra. We also describe a general scheme producing integrable cases of the generalized rigid body motion in an external field which have a Lax representation with spectral parameter. Several other examples of multi-dimensional tops are discussed.     

Solvable lattice models whose states are dominant integral weights of A it−1 (1)

A new hierarchy of solvable IRF models is presented. It is generated from Belavin's Z _n ×Z _n symmetric model. The site variables take values in the set of level l dominant integral weights of A _–1 ⁽¹⁾ . It is conjectured that the local state probabilities are given through the irreducible decomposition of characters for the affine Lie algebra pair (A _n–1 ⁽¹⁾ A _n–1 ⁽¹⁾ ,A _n–1 ⁽¹⁾ ).     

Construction of the elliptic Gaudin system based on Lie algebra

Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics. The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra. Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, A _n , B _n , C _n , D _n , and we calculate a classical r-matrix for the elliptic Gaudin system with spin.      

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Existence and equivalence of twisted products on a symplectic manifold

The twisted products play an important role in Quantum Mechanics [1, 2]. We introduce here a distinction between Vey *_ν-products and strong Vey *_ν-products and prove that each *_ν-product is equivalent to a Vey *_ν-product. If b ₃(W)=0, the symplectic manifold (W, F) admits strong Vey *_ν-products. If b ₂(W)=0, all *_ν-products are equivalent as well as the Vey Lie algebras. In the general case, we characterize the formal Lie algebras which are generated by a *_ν-product and we prove that the existence of a *_ν-product is equivalent to the existence of a formal Lie algebra infinitesimally equivalent to a Vey Lie algebra at the first order.     

Symmetries of WDVV equations

We say that a function F(τ) obeys WDVV equations, if for a given invertible symmetric matrix η^αβ and all , the expressions can be considered as structure constants of commutative associative algebra; the matrix η_αβ inverse to η^αβ determines an invariant scalar product on this algebra. A function x^α(z,τ) obeying is called a calibration of a solution of WDVV equations. We show that there exists an infinite-dimensional group acting on the space of calibrated solutions of WDVV equations (in different form such a group was constructed in [A. Givental, math.AG/0305409]). We describe the action of Lie algebra of this group.     

A nonlinear superposition principle admitted by coupled Riccati equations of the projective type

A definite theorem due to Lie which group theoretically characterizes those systems of ordinary differential equations which possess nonlinear superposition principles is employed along with an observation by Lie on the exponentiated form of a fibered Lie algebra to obtain an explicit expression for the Vessiot-Guldberg-Lie nonlinear superposition principle admitted by n-coupled Riccati equations of the projective type. This also, immediately, yields an explicit expression for the generalized cross-ratio for the projective group in n-dimensions.Reported at the Georgia Workshop in Mathematical Physics, November 26–28, 1979, UGA, Athens, Georgia.     

Soliton equations and fundamental representations of A 2l (2)

Fundamental representations of the Euclidean Lie algebra A _2l ⁽²⁾ is constructed by decomposing the vertex representations of gI(∞). For l=1 the multiplicities of highest weights are determined. Soliton equations associated with each of these representations are also discussed.     

The Principal SO(1,2) Subalgebra of a Hyperbolic Kac–Moody Algebra

The analog of the principal SO(3) subalgebra of a finite-dimensional simple Lie algebra can be defined for any hyperbolic Kac–Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact algebra SO(1,2). We exhibit the decomposition of g(A) into representations of SO(1,2). With the exception of the adjoint SO(1,2) algebra itself, all of these representations are unitary. We compute the Casimir eigenvalues; the associated exponents are complex and noninteger.     

The sh Lie Structure of Poisson Brackets in Field Theory

A general construction of an sh Lie algebra (L _∞-algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket. Received: 5 March 1997 / Accepted: 21 May 1997     

An interesting representation of lie algebras of linear groups

I have presented a means of getting a representation space of a general linear group ofn dimensions in terms of homogeneous functions ofn,n-dimensional vectors. Except in particular cases, the representation is of the Lie algebra, rather than the group. A general formalism is set up to evaluate the Casimir operators of the Lie algebra of the group in terms of the degrees of homogeneity of the functions (which are eigenfunctions of the Casimir operators) in then variables. It is noticed that the Casimir operators exhibit certain symmetries in these degrees of homogeneity which relate different representations having the same eigenvalues for the Casimir operators. Contour integral formulas that enable one to pass from one such representation to another are presented. An expression for the eigenvalues of a general Casimir operator in terms of the degree of homogeneity is presented.     

Vertex Operators Arising from the Homogeneous Realization for

We use the underlying Fock space for the homogeneous vertex operator representation of the affine Lie algebra to construct a family of vertex operators. As an application, an irreducible module for an extended affine Lie algebra of type A _{N −1} coordinatized by a quantum torus ℂ _q of 2 variables (or 3 variables) is obtained. Moreover, this module turns out to be a highest weight module which is an analog of the basic module for affine Lie algebras. Received: 16 August 1999 / Accepted: 18 January 2000     

The hopf algebra of vector fields on complex quantum groups

We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A _n , B _n , C _n , and D _n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.Humboldt Fellow.     

Algebraic deformation program on minimal nilpotent orbit

We construct an algebraic star product on the minimal nilpotent coadjoint orbit of a simple complex Lie group with a Lie algebra which is not of typeA _n. According to the deformation program, we study the representations of the Lie algebra associated to this orbit.     

<Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-Algebras from Multisymplectic Geometry

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n + 1. In previous work with Baez and Hoffnung, we described how the ‘higher analogs’ of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L _∞-algebras: graded vector spaces equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras. We also mention a possible connection with the work of Barnich, Fulp, Lada, and Stasheff on the Gelfand–Dickey–Dorfman formalism.     

Twisted Traces of Quantum Intertwiners¶and Quantum Dynamical R-Matrices¶Corresponding to Generalized Belavin–Drinfeld Triples

We consider weighted traces of products of intertwining operators for quantum groups U _q (?), suitably twisted by a “generalized Belavin–Drinfeld triple”. We derive two commuting sets of difference equations – the (twisted) Macdonald–Ruijsenaars system and the (twisted) quantum Knizhnik–Zamolodchikov–Bernard (qKZB) system. These systems involve the nonstandard quantum R-matrices defined in a previous joint work with T. Schedler ([ESS]). When the generalized Belavin–Drinfeld triple comes from an automorphism of the Lie algebra ?, we also derive two additional sets of difference equations, the dual Macdonald–Ruijsenaars system and the \textit{dual} qKZB equations. Received: 20 March 2000 / Accepted: 11 December 2000