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.     

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.     

On the purification of factor states

Let be aC*-algebra and be an opposite algebra. Notions of exact andj-positive states of are introduced. It is shown, that any factor state of can be extended to a pure exactj-positive state of . The correspondence generalizes the notion of the purifications map introduced by Powers and Størmer. The factor states ₁ and ₂ are quasi-equivalent if and only if their purifications and are equivalent.     

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.     

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.     

Fields,observables and gauge transformations I

Starting from an algebra of fields and a compact gauge group of the first kind , the observable algebra is defined as the gauge invariant part of . A gauge group of the first kind is shown to be automatically compact if the scattering states are complete and the mass and spin multiplets have finite multiplicity. Under reasonable assumptions about the structure of it is shown that the inequivalent irreducible representations of (sectors) which occur are in one-to-one correspondence with the inequivalent irreducible representations of and that all of them are strongly locally equivalent. An irreducible representation of satisfies the duality property only if the sector corresponds to a 1-dimensional representation of . If is Abelian the sectors are connected to each other by localized automorphisms.On leave of absence from Instituto di Fisica G. Marconi, Università di Roma.     

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).     

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.     

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.     

An algebraic spectrum condition

It is shown that a translationally invariant algebra of local observables (see [1]) admits a representation in a Hilbert space having a vacuum state. Furthermore an algebraic criterion is given which is necessary and sufficient for the existence of at least one representation of in which the usual spectral condition for the energy-momentum operators holds.     

Quantum principal commutative subalgebra in the nilpotent part ofU_q \widehat{sl}_2 and lattice KdV variables

We propose a quantum lattice version of B. Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of . We construct an action of the nilpotent part of on their lattice counterparts, and embed the lattice variables in a , coinduced from a quantum version of the principal commutative subalgebra, which is defined using the identification of with its dual algebra.     

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.     

Representations of empirical set theories

Any manual of Boolean locales in the strong sense, namely a subcategory of the opposite category of the category of complete Boolean algebras and complete Boolean homomorphisms satisfying not only conditions (3.1)–(3.10) of our previous paper [International Journal of Theoretical Physics,32, 1293 (1993b)], but also conditions (4.1)–(4.4) of that paper, is shown to be representable as the second-class orthomodular manual of Boolean locales on an orthomodular poset In this sense the study on manuals of Boolean locales in the strong sense is tantamount to the study on a special class of orthomodular posets, though our viewpoint is radically different from the conventional one in the traditional approach to orthomodular posets. Then the notion of a manual of Hilbert spaces or exactly what is called a manual of Hilbert locales is introduced, over which a variant of the celebrated Gelfand-Naimark-Segal theorem for a manual of Boolean locales in the strong sense is established.     

A Construction of Representations and Quantum Homogeneous Spaces

A simplified construction of representations is presented for the quantized enveloping algebra q ( ), with being a simple complex Lie algebra belonging to one of the four principal series A\ell, B\ell, C\ell or D\ell. The carrier representation space is the quantized algebra of polynomials in antiholomorphic coordinate functions on the big cell of a coadjoint orbit of K where K is the compact simple Lie group with the Lie algebra – the compact form of .     

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.     

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.     

On the support of the Ashtekar-Lewandowski measure

We show that the Ashtekar-Isham extension of the configuration space of Yang-Mills theories is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices.These results are then used to prove that is contained in a zero measure subset of with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure on . Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the extended configuration space .     

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.     

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)     

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If , and is a finite (nonabelian) group, then is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of . We characterize when MCA are group endomorphisms of , and show that MCA on inherit a natural structure theory from the structure of . We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.