Yangian double

Studying the algebraic structure of the doubleDY(g) of the Yangian Y(g), we present the triangular decomposition ofDY(g) and a factorization for the canonical pairing of the Yangian with its dual inside Y⁰(g). As a consequence, we describe a structure of the universalR-matrixR forDY(g) which is complete forDY(s1₂). We demonstrate how this formula works in evaluation representations of Y(sl₂). We interpret the one-dimensional factor arising in concrete representations ofR as a bilinear form on highest-weight polynomials of irreducible representations of Y(g) and express this form in terms of -functions.Partially supported by ISF grant MBI000 and Russian Foundation for Fundamental Researches     

One-dimensional configuration sums in vertex models and affine Lie algebra characters

We study the local state probabilities of the vertex models in the face formulation associated with the simple Lie algebras X _n =A _n, B _n, C _n, D _n. The corner transfer matrix method expresses them in terms of one-dimensional configuration sums. We show that the latter are the string functions of X _n ⁽¹⁾ modules. We also present similar results for the restricted face models of types B _n ⁽¹⁾, C _n ⁽¹⁾, D _n ⁽¹⁾.     

Construction of cyclic representations of quantum algebras at q p =1 from their regular representations

Cyclic representations of maximal dimension of the quantum algebra U _q L associated with any finite-dimensional simple Lie algebra L are studied from its regular representation at q ^p =1, which is proved to be a quotient module of itself as a left module with respect to some submodules. The general theory is given after an instructive example U _q sl(2) is studied. Another explicit example U _q sl(3) is also presented.This work is supported in part by the National Natural Science Foundation of China. Author Fu is also supported by the Jilin Provincial Science and Technology Foundation of China     

On the hopf structure of U p,q (gl(1∣1)) and the universal T-matrix of fun p,q (GL(1∣1))

The dually conjugate Hopf superalgebras Fun _p,q (GL(11)) and U _p,q (gl(11)) are studied using the Frønsdal-Galindo approach and the full Hopf structure of U _p,q (gl(11)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obtained for Fun _p,q (GL(11)). In a representation with a colour index, the T-matrix assumes a form that satisfies a coloured graded Yang-Baxter equation.     

Classification of all Poisson-Lie structures on an infinite-dimensional jet group

A local classification of all Poisson-Lie structures on an infinite-dimensional group G of formal power series is given. All Lie bialgebra structures on the Lie algebra G of G are also classified.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

The algebra of differential operators on the circle and W KP (q)

Radul has recently introduced a map from the Lie algebra of differential operators on the circle of W _n . In this Letter, we extend this map to W _KP ^(q) , a recently introduced one-parameter deformation of W_KP - the second Hamiltonian structure of the KP hierarchy. We use this to give a short proof that W is the algebra of additional symmetries of the KP equation.     

A natural and rigid model of quantum groups

We introduce a natural (Fréchet-Hopf) algebra A containing all generic Jimbo algebras U _t (sl(2)) (as dense subalgebras). The Hopf structures on A extend (in a continuous way) the Hopf structures of generic U _t (sl(2)). The Universal R-matrices converge in A A. Using the (topological) dual of A, we recover the formalism of functions of noncommutative arguments. In addition, we show that all these Hopf structures on A are isomorphic (as bialgebras), and rigid in the category of bialgebras.     

Symmetries and generalizedW∞ algebra of the modified KP equation

The infinitely many symmetries with arbitrary functions of timet for the potential modified Kadomtsev-Petviashvilli equation are obtained by using a simple direct method. These symmetries constitute a generalization of the well-knownW algebra.     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

Complex parametrizedW-algebras: The gl case

We define deformations of the Poisson bracket algebras of functions on manifolds = { + _i ^/ =1w _i ^-i |w _i C (S ¹)} of pseudodifferential symbols on the circle ( C), arising in the works of Rosly and Khesin-Zakharevich. These deformations have vertex operator algebraic (VOA) counterparts, which have (for =n integer) a quotient isomorphic to theW-algebraW _n associated by Fateev and Lukyanov to gl _n .The product operation of symbols defines a Lie-Poisson structure on _C L (Rosly, Khesin-Zakharevich); we show that this structure has also a VOA counterpart.     

Differential geometry on linear quantum groups

An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group GL _q (N). SL _q (N) is then found by constructing matrices with determnant unity, and the induced calculus is found.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.     

A quantum dual pair \mathfraks\mathfrakl2 ,\mathfrakon \mathfrak{s}\mathfrak{l}_2 ,\mathfrak{o}_n and the associated Capelli identity

A quantum analogue of the dual pair is introduced in terms of the oscillator representation of U _q . Its commutant and the associated identity of Capelli type are discussed.     

A new realization of the basic representation of A n (1)

The basic representation of A _n ⁽¹⁾ is constructed on a space of paths in (n + 1). A q-analogue of this is also presented. Our construction gives a Lie theoretical proof of a combinatorial identity used in the evaluation of the local state probabilities for a class of solvable lattice models.     

Embeddings of Hyperbolic Kac–Moody Algebras into E 10

We show that the rank 10 hyperbolic Kac–Moody algebra E ₁₀ contains every simply laced hyperbolic Kac–Moody algebra as a Lie subalgebra. Our method is based on an extension of earlier work of Feingold and Nicolai.      

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

Fundamental representations of quantum groups at roots of 1

To every finite-dimensional irreducible representation V of the quantum group U(g) where is a primitive lth root of unity (l odd) and g is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C _V in the adjoint group G of g. We describe explicitly, when g is of type A _n , B _n , C _n , or D _n , the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

Local Index Formula on the Equatorial Podleś Sphere

We discuss spectral properties of the equatorial Podleś sphere S _q ². As a preparation we also study the ‘degenerate’ (i.e. q=0) case (related to the quantum disk). Over S _q ² we consider two different spectral triples:one related to the Fock representation of the Toeplitz algebra and the isopectral one given in [7]. After the identification of the smooth pre-C ^*-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the C ^*-algebra (the nontrivial generator of the K ₀-group) as well as the pairing with the q-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.     

Hilbert space cocycles as representations of (3 + 1)-d current algebras

It is proposed that instead of normal representations, one should look at cocycles of group extensions valued in certain groups of unitary operators acting in a Hilbert space (e.g. the Fock space of chiral fermions), when dealing with groups associated to current algebras in gauge theories in 3 + 1 spacetime dimensions. The appropriate cocycle is evaluated in the case of the group of smooth maps from the physical three-space to a compact Lie group.The cocyclic representation of a componentX of the current is obtained through two regularizations, (1) a conjugation by a background potential dependent unitary operatorh _A, (2) by a subtraction-h _A ^-1 _xhA, where _x is a derivative along a gauge orbit. It is only the total operatorh _A ^-1 Xh _A -h _A ^-1 _xhA which is quantizable in the Fock space using the usual normal ordering subtraction.Supported by the Alexander von Humboldt Foundation