N=2 structures on solvable Lie algebras: Thec=9 classification

Let be a finite-dimensional Lie algebra (not necessarily semisimple). It is known that if is self-dual (that is, if it possesses an invariant metric) then it admits anN=1 (affine) Sugawara construction. Under certain additional hypotheses, thisN=1 structure admits anN=2 extension. If this is the case, is said to possess anN=2 structure. It is also known that anN=2 structure on a self-dual Lie algebra is equivalent to a vector space decomposition , where are isotropic Lie subalgebras. In other words,N=2 structures on in one-to-one correspondence with Manin triples . In this paper we exploit this correspondence to obtain a classification of thec=9N=2 structures on solvable Lie algebras. In the process we also give some simple proofs for a variety of Lie algebras. In the process we also give some simple proofs for a variety of Lie algebraic results concerning self-dual Lie algebras admitting symplectic or Kähler structures.     

Quantum-algebras and elliptic algebras

We define a quantum-algebra associated to as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary-algebra of , or theq-deformed classical-algebra algebra of . We construct free field realizations of the quantum-algebra and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in.The research of the second author was partially supported by NSF grant DMS-9501414.     

Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials

Theq=0 combinatorics for is studied in connection with solvable lattice models. Crystal bases of highest weight representations of are labelled by paths which were introduced as labels of corner transfer matrix eigenvectors atq=0. It is shown that the crystal graphs for finite tensor products ofl-th symmetric tensor representations of approximate the crystal graphs of levell representations of . The identification is made between restricted paths for the RSOS models and highest weight vectors in the crystal graphs of tensor modules for .Partially supported by NSF grant MDA904-90-H-4039     

Limits of states

Estimates for vector representations of states are used to prove that {C _n C ₀} is strong-operator convergent toC ₀, whereC _n is the universal central support of _n and { _n } is a sequence of states of aC*-algebra converging in norm to ₀. States of of a given type are shown to form a norm-closed convex subset of the (norm) dual of . The pure states of form a norm-closed subset of the dual.With partial support of the National Science Foundation (USA)     

Spinor Representations of $${U}_q (\hat {\mathfrak{o}}(N))$$ )

This Letter concerns an extension of the quantum spinor construction of . We define quantum affine Clifford algebras based on the tensor category and the solutions of q-KZ equations, and construct quantum spinor representations of .     

Derivations and automorphisms of operator algebras

The theorem that each derivation of aC*-algebra extends to an inner derivation of the weak-operator closure ( )^– of in each faithful representation of is proved in sketch and used to study the automorphism group of in its norm topology. It is proved that the connected component of the identity in this group contains the open ball of radius 2 with centerl and that each automorphism in extends to an inner automorphism of ( )^–.Research conducted with the partial support of the NSF and ONR.     

\mathfrak{g}-Relative Star Products

We consider Kontsevich star products on the duals of Lie algebras. Such a star product is relative if, for any Lie algebra, its restriction to invariant polynomial functions is the usual pointwise product. Let be a fixed Lie algebra. We shall say that a Kontsevich star product is -relative if, on ^*, its restriction to invariant polynomial functions is the usual pointwise product. We prove that, if is a semi-simple Lie algebra, the only strict Kontsevich -relative star products are the relative (for every Lie algebras) Kontsevich star products.     

Covariance algebras in field theory and statistical mechanics

Starting from aC*-algebra and a locally compact groupT of automorphisms of we construct a covariance algebra with the property that the corresponding *-representations are in one-to-one correspondence with covariant representations of i.e. *-representations of in which the automorphisms are continuously unitarily implemented. We further construct for relativistic field theory an algebra yielding the *-representations of in which the space time translations have their spectrum contained inV. The problem of denumerable occurence of superselection sectors is formulated as a condition on the spectrum of . Finally we consider the covariance algebra built with space translations alone and show its relevance for the discussion of equilibrium states in statistical mechanics, namely we restore in this framework the equivalence of uniqueness of the vacuum, irreducibility and a weak clustering property.On leave of absence from Istituto di Fisica G. Marconi — Roma.     

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 ^·.     

Double quantization on the coadjoint representation of sl(n)

For we construct a two parametric -invariant family of algebras, , that is a quantization of the function algebra on the coadjoint representation. Along the parameter t the family gives a quantization of the Lie bracket. This family induces a two parametric -invariant quantization on the maximal orbits, which includes a quantization of the Kirillov-Kostant-Souriau bracket. Yet we construct a quantum de Rham complex on .     

Algebras of observables with continuous representations of symmetry groups

Concrete C*-algebras, interpreted physically as algebras of observables, are defined for quantum mechanics and local quantum field theory.Aquantum mechanical system is characterized formally by a continuous unitary representation up to a factorU _g of a symmetry group in Hilbert space and a von Neumann algebra on invariant with respect toU _g . The set of all operatorsX such thatU _g X U _g ^–1 , as a function ofg , is continuous with respect to the uniform operator topology, is aC*-algebra called thealgebra of observables. The algebra is shown to be the weak (or strong) closure of .Infield theory, a unitary representation up to a factorU(a, ) of the proper inhomogeneous Lorentz group and local von Neumann algebras _C for finite open space-time regionsC are assumed, with the usual transformation properties of underU(a, ). The collection of allX_C giving uniformly continuous functionsU (a, )X U ^–1 (a, ) on is then a localC*-algebra , called thealgebra of local observables. The algebra is again weakly (or strongly) dense in _c . The norm-closed union of the for allC is calledalgebra of quasilocal observables (or quasilocal algebra).In either case, the group is represented by automorphisms V _g resp. V(a, ) — with V _g X=U _g X U _g ^–1 — of theC*-algebra , and this is astrongly continuous representation of on the Banach space . Conditions for V (a, ) can then be formulated which correspond to the usualspectrum condition forU (a, ) in field theory.Work supported in part by the Deutsche Forschungsgemeinschaft.     

Unbounded derivations commuting with compact group actions

Let be a closed * derivation in aC* algebra which commutes with an ergodic action of a compact group on . Then generates aC* dynamics of . Similar results are obtained for non-ergodic actions on abelianC* algebras and on the algebra of compact operators.Research supported by N.S.F.     

Anyonic realization of the quantum affine Lie algebras

We give a realization of the quantum affine Lie algebras and in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter q being related to the statistical parameter of the anyons by q = eⁱ. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel [1] fermionic construction of undeformed affine Lie algebras.     

Space-time symmetries of superstring and Jordan algebras

The sequence of Jordan algebras , whose elements are the 3×3 Hermitian matrices over the division algebras , , , and , is considered. These algebras are naturally related to supersymmetric structures in space-time dimensions of 3, 4, 6, and 10, as the Lorentz groups in these dimensions can be expressed in a unified way as a subgroup of the structure group of the Jordan algebras . The generators of the complete structure group and the automorphism group can be separated into bosonic and fermionic generators, depending on their transformation properties under the Lorentz subgroup. A peculiar connection between these fermionic generators and the supersymmetry generators of the superstring action is introduced and discussed.     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

Quantum affine algebras and holonomic difference equations

We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the face formulation for any type of Lie algebra and arbitrary finite-dimensional representations of. We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq1 these solutions degenerate again into solutions with . We also study the simples examples of solutions of our holonomic difference equations associated to and find their expressions in terms of basic (orq–)-hypergeometric series. In the special case of spin –1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.     

Ergodic states in a non commutative ergodic theory

Using the Godement mean of positive-type functions over a groupG, we study -abelian systems { , } of aC*-algebra and a homomorphic mapping of a groupG into the homomorphism group of . Consideration of the Godement mean off(g)U _g withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) ( _g (A)) withA and a covariant representation of the system { , } for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over . Finally we investigate the discrete spectrum of covariant representations of { , } (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).On leave of absence from Istituto di Fisica G. Marconi Piazzale delle Scienze 5 — Roma.     

On a new derivation of the Navier-Stokes equation

Given a net of finite-dimensional real Lie algebras contracting into a Lie algebra, a representation of is constructed explicitly as limit of a net ( _l ) of representations, each _l being a representation of on a complex Hilbert space _l . Conditions are imposed on the net ( _l ) implying that the carrier space of contain a-stable set of vectors which are analytic for all, where is a basis of. As a corollary, the corresponding result for contractions of representations of simply connected finite-dimensional real Lie groups is derived.Supported by the Swiss National Science Foundation     

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     

Universal Drinfeld-Sokolov reduction and matrices of complex size

We construct affinization of the algebra of complex size matrices, that contains the algebras for integral values of the parameter. The Drinfeld-Sokolov Hamiltonian reduction of the algebra results in the quadratic Gelfand-Dickey structure on the Poisson-Lie group of all pseudodifferential operators of complex order.This construction is extended to the simultaneous deformation of orthogonal and symplectic algebras which produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.Partially supported by NSF grant DMS 9307086.Partially supported by NSF grant DMS 9401215.