Bethe subalgebras in twisted Yangians

We study analogues of the Yangian of the Lie algebra for the other classical Lie algebras and . We call them twisted Yangians. They are coideal subalgebras in the Yangian of and admit homomorphisms onto the universal enveloping algebras U( ) and U( ) respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in U( ) and U( ) of these subalgebras are also maximal commutative.     

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.     

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.     

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     

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.     

Central decomposition of invariant states applications to the groups of time translations and of euclidean transformations in algebraic field theory

With aC*-algebra with unit andgG _g a homomorphic map of a groupG into the automorphism group ofG, the central measure of a state of is invariant under the action ofG (in the state space of ) iff is -invariant. Furthermore if the pair { ,G} is asymptotically abelian, is ergodic iff is ergodic. Transitive ergodic states (corresponding to transitive central measures) are centrally decomposed into primary states whose isotropy groups form a conjugacy class of subgroups. IfG is locally compact and acts continuously on , the associated covariant representations of { , } are those induced by such subgroups. Transitive states under time-translations must be primary if required to be stable. The last section offers a complete classification of the isotropy groups of the primary states occurring in the central decomposition of euclidean transitive ergodic invariant states.     

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.     

On the equilibrium states in quantum statistical mechanics

Representations of theC*-algebra of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of onto its commutant. This means that there is an equivalent anti-linear representation of in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.     

Modules over\mathfrak{U}_q (\mathfrak{s}\mathfrak{l}_2 )

The restricted quantum universal enveloping algebra decomposes in a canonical way into a direct sum of indecomposable left (or right) ideals. They are useful for determining the direct summands which occur in the tensor product of two simple . The indecomposable finite-dimensional are classified and located in the Auslander-Reiten quiver.     

Canonical quantization

The dynamical variables of a classical system form a Lie algebra , where the Lie multiplication is given by the Poisson bracket. Following the ideas ofSouriau andSegal, but with some modifications, we show that it is possible to realize as a concrete algebra of smooth transformations of the functionals on the manifold of smooth solutions to the classical equations of motion. It is even possible to do this in such a way that the action of a chosen dynamical variable, say the Hamiltonian, is given by the classical motion on the manifold, so that the quantum and classical motions coincide. In this realization, constant functionals are realized by multiples of the identity operator. For a finite number of degrees of freedom,n, the space of functionals can be made into a Hilbert space using the invariant Liouville volume element; the dynamical variablesF become operators in this space. We prove that for any hamiltonianH quadratic in the canonical variablesq ₁...q _n ,p ₁...p _n there exists a subspace ₁ which is invariant under the action of and , and such that the restriction of to ₁ form an irreducible set of operators. Therefore,Souriau's quantization rule agrees with the usual one for quadratic hamiltonians. In fact, it gives the Bargmann-Segal holomorphic function realization. For non-linear problems in general, however, the operators form a reducible set on any subspace of invariant under the action of the Hamiltonian. In particular this happens for . Therefore,Souriau's rule cannot agree with the usual quantization procedure for general non-linear systems.The method can be applied to the quantization of a non-linear wave equation and differs from the usual attempts in that (1) at any fixed time the field and its conjugate momentum may form a reducible set (2) the theory is less singular than usual.For a particular wave equation , we show heuristically that the interacting field may be defined as a first order differential operator acting onc -functions on the manifold of solutions. In order to make this space into a Hilbert space, one must define a suitable method of functional integration on the manifold; this problem is discussed, without, however, arriving at a satisfactory conclusion.On leave from Physics Department, Imperial College, London SW7.Work partly supported by the Office of Scientific Research, U.S. Air Force.     

Duality for free bose fields

An elementary proof of Araki's duality theorem for free fields is presented. The theorem says that for a certain class of regionsO in Minkowski space, the commutant of (O), the von Neumann algebra generated by all observables belonging to measurements withinO, is exactly (O), whereO is the spacelike separated complement ofO.Supported by the National Science Foundation under grants GP 24003, GP 30819X and GP 31239X.     

Regularized,continuum Yang-Mills process and Feynman-Kac functional integral

Giving an ultraviolet regularization and volume cut off we construct a nuclear Riemannian structure on the Hilbert manifold of gauge orbits. This permits us to define a regularized Laplace-Beltrami operator on and an associated global diffusion in governed by . This enables us to define, via a Feynman-Kac integral, a Euclidean, continuum regularized Yang-Mills process corresponding to a suitable regularization (of the kinetic term) of the classical Yang-Mills Lagrangian onT .On leave of absence from Zaragoza University (Spain)Laboratoire associé au CNRS     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

Statistical mechanics of quantum spin systems. III

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over . With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.     

Minimal affinizations of representations of quantum groups: the irregular case

Let be a finite-dimensional complex simple Lie algebra and U_q( ) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of U_q( ), an affinization ofV is an irreducible representationVV of the quantum affine algebra U_q( ) which containsV with multiplicity one and is such that all other irreducible U_q( )-components ofV have highest weight strictly smaller than the highest weight ofV. There is a natural partial order on the set of U_q( ) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if is of typeA, B, C, F orG, the minimal affinization is unique up to U_q( )-isomorphism; (ii) if is of typeD orE and is not orthogonal to the triple node of the Dynkin diagram of , there are either one or three minimal affinizations (depending on ). In this paper, we show, in contrast to the regular case, that if U_q( ) is of typeD ₄ and is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of .As a by-product of our methods, we disprove a conjecture according to which, if is of typeA _n,every affinization is isomorphic to a tensor product of representations of U_q( ) which are irreducible under U_q( ) (in an earlier paper, we proved this conjecture whenn=1).Both authors were partially supported by the NSF, DMS-9207701.     

Three-hadron form factor via quark-triangle loop

An explicit construction in a manifestly Lorentz-invariant and cyclically symmetric form is given for a three-hadron (H) coupling through their respective (q ) constituents interacting via a quark triangle loop under the most general conditions of unequal mass kinematics atboth the elementary quark and composite-hadron levels. The only ansatz employed is that theHq vertex function _H at each vertexi is a function of _i , the relative momentumtransverse to the corresponding hadron momentumP _i, an assumption which can be justified from several independent angles. The resulting structure , withD( ) being a model-independent function, has proved crucial for an explicit integration over the time-like component of the loop momentum, and thus provided an analytical, cyclically symmetric structure free of overlapping pole singularities. The applicational potential of this quantity is outlined.     

Classical Dynamical r-Matrices,Poisson Homogeneous Spaces,and Lagrangian Subalgebras

Lu has shown that any dynamical r-matrix for the pair ( , ) naturally induces a Poisson homogeneous structure on G/U. She also proved that if is complex simple, is its Cartan subalgebra and r is quasitriangular, then this correspondence is in fact one-to-one. In this Letter we find some general conditions under which the Lu correspondence is one-to-one. Then we apply this result to describe all triangular Poisson homogeneous structures on G/U for a simple complex group G and its reductive subgroup U containing a Cartan subgroup.     

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).     

Finite-dimensional coadjoint orbits in loop algebras

We give a classification of the finite dimensional coadjoint orbits in the dual of the algebra ₊ of polynomials in one variable with values in a semi-simple Lie algebra , and generalise this result to algebras defined over an arbitrary Riemann surface.During the preparation of this work the author was supported by NSERC grant A8361 and FCAR grant EQ3518.     

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.