realizations of conformal current type lie algebras in this paper we obtain the realizations of some infinite dimensional lie algebras named conformal current type lie algebras in terms of a two dimensional novikov algebra and its deformations furthermore ovsienko and roger s loop cotangent virasoro algebra which can be regarded as a nice generalization of the virasoro algebra with two space variables is naturally realized as an affinization of the tensor product of a deformation of the two dimensional novikov algebra and the laurent polynomial algebra these realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite dimensional lie algebras

an explicit approach to the λ operator and the h theorem in the prigogine theory of irreversibility abstract a reformulation of the prigogine theory of irreversible dynamical systems is given in contrast to the previous works a stochastic integral with respect operator valued martingales is employed here this approach provides a mathematically rigorous and simple construction of the operator for dynamical flows it has an explicit random character we derive also the boltzmann h theorem explicitly from the properties of the semigroup only finally the form of the boltzmann kinetic equation for the irreversible dynamics is found

the deformed lie algebra of differential operators of order at most one in this paper we study the algebraic properties of the deformed lie algebra l of differential operators of order at most one we determine all the derivations central extensions and automorphism group of l furthermore the reducibility criterion for the verma modules of the universal central extension of l is achieved through a study of the shapovalov form

deformations of some infinite dimensional lie algebras the concept of a versal deformation of a lie algebra is investigated and obstructions to extending an infinitesimal deformation to a higher order one are described the rigidity of the witt algebra and the virasoro algebra is deduced from cohomology computations for certain lie algebras of vector fields on the real line the lie algebra of vector fields on the line that vanish at the origin also turns out to be rigid all the affine lie algebras are rigid this is derived from the cohomology of their maximal nilpotent subalgebra on the other hand the maximal nilpotent subalgebras in both the virasoro and affine cases are not rigid and have interesting nontrivial deformations in fact most vector field lie algebras are not rigid

on an infinite dimensional lie algebra of virasoro type in this paper we study an infinite dimensional lie algebra of virasoro type which is realized as an affinization of a two dimensional novikov algebra it is a special deformation of the lie algebra of differential operators on a circle of order at most 1 there is an explicit construction of a vertex algebra associated with the lie algebra we determine all derivations of this lie algebra in terms of some derivations and centroids of the corresponding novikov algebra the universal central extension of this lie algebra is also determined

characterizations of intrinsically random dynamical systems abstract we show that intrinsically random dynamical systems with the prigogine operator of the form of a random laplace transform can be characterized as kolmogorov flows k flows we also obtain a spectral characterization in the language of the weyl commutation relation as a consequence we conclude that the dynamical system is intrinsically random if and only if its liouvillian and time operators form a schrodinger couple

on jordan holder series of some locally analytic representations let g be a split reductive p adic group this paper is about the jordan holder series of locally analytic g representations which are induced from locally algebraic representations of a parabolic subgroup p g we construct for every representation m of lie g in the bgg category o which is equipped with an algebraic p action and for every smooth p representation v a locally analytic representation fg p m v of g this gives rise to a bi functor to the category of locally analytic representations we prove that it is exact and give a criterion for the topological irreducibility of fg p m v in terms of m and v

minimal models of u 1 conformal current algebra a detailed study of the degenerate representations of the u 1 u 1 conformal current algebra and field theoretical models based on it is presented the main peculiarity of these models is that they contain an infinite set of fields with unfixed anomalous dimensions simple fusion rules for these fields are found and their four point functions are calculated explicitly the case of thirring like two dimensional models is discussed in detail

a nonequilibrium entropy for dynamical systems it is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure these are chosen by considering the totality of future possibilities for observation on the system it is shown that this entropy is necessarily a nondecreasing function of the timet there is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal under suitable conditions which are stated explicitly this entropy approaches the equilibrium entropy ast and the fine grained entropy ast in particular the conditions can always be satisfied if the system is ak system as in the sinai billiard models some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition and at the same time elucidate the connection between our concept of entropy and the entropy invariant of kolmogorov and sinai

the classification of irreducible admissible mod p representations of a p adic gln let f be a finite extension of p using the mod p satake transform we define what it means for an irreducible admissible smooth representation of an f split p adic reductive group over overline mathbb f p to be supersingular we then give the classification of irreducible admissible smooth gln f representations over overline mathbb f p in terms of supersingular representations as a consequence we deduce that supersingular is the same as supercuspidal these results generalise the work of barthel livne for n 2 for general split reductive groups we obtain similar results under stronger hypotheses

highest weight theory for truncated current lie algebras abstract in this paper a highest weight theory for the truncated current lie algebra g g g k k t t n 1 k t is developed when the underlying lie algebra g possesses a triangular decomposition the principal result is the reducibility criterion for the verma modules of g for a wide class of lie algebras g including the symmetrizable kac moody lie algebras the heisenberg algebra and the virasoro algebra this is achieved through a study of the shapovalov form

harish chandra modules over the twisted heisenberg virasoro algebra in this paper we classify all indecomposable harish chandra modules of the intermediate series over the twisted heisenberg virasoro algebra meanwhile some bosonic modules are also studied

on the structure of verma modules over the w algebra w 2 2 in this paper we describe the structure of verma modules over the w algebra w 2 2 we show that either a verma module over w 2 2 is irreducible or its maximal submodule is cyclic

moduli spaces of curves and representation theory we establish a canonical isomorphism between the second cohomology of the lie algebra of regular differential operators on x of degree 1 and the second singular cohomology of the moduli space hat f g 1 of quintuples c p z l wherec is a smooth genusg riemann surface p a point onc z a local parameter atp l a degreeg 1 line bundle onc and a class of local trivializations ofl atp which differ by a non zero factor the construction uses an interplay between various infinite dimensional manifolds based on the topological spaceh of germs of holomorphic functions in a neighborhood of 0 in x and related topological spaces the basic tool is a canonical map from hat f g 1 to the infinite dimensional grassmannian of subspaces ofh which is the orbit of the subspaceh of holomorphic functions on x vanishing at under the group auth as an application we give a lie algebraic proof of the mumford formula n 6n2 6n 1 1 where n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg whose fiber overc is the space of differentials of degreen onc

intrinsic irreversibility of kolmogorov dynamical systems we give the mathematical details and various extensions of the results stated in previous work of courbage and prigogine intrinsic random systems are deterministic and conservative dynamical systems for which we can associate two dissipative markov processes through a one to one change of representation the first leading to equilibrium for t and the second for t the microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t the set of admitted initial conditions d is then characterized by a non equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions thus rejected initial states correspond to an infinite amount of information to realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to markov processes to such measures owing to the non invariance of d under the time inversion the evolution of these states in the new representation can only be given by one of the two possible markov processes

equivariant vector bundles on drinfeld s upper half space let mathcal x subset mathbb p k d be drinfeld s upper half space over a finite extension k of p we construct for every gl d 1 equivariant vector bundle mathcal f on d k a gl d 1 k equivariant filtration by closed subspaces on the k frechet h 0 mathcal x mathcal f this gives rise by duality to a filtration by locally analytic gl d 1 k representations on the strong dual h 0 mathcal x mathcal f prime the graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups this paper generalizes the cases of the canonical bundle due to schneider and teitelbaum st1 and that of the structure sheaf by pohlkamp p

evolution semigroups and time operators on banach spaces we present new general methods to obtain shift representation of evolution semigroups defined on banach spaces we introduce the notion of time operator associated with a generalized shift on a banach space and give some conditions under which time operators can be defined on an arbitrary banach space we also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on banach spaces satisfying some geometric conditions the last part of the paper contains some examples of explicit constructions of time operators on function spaces

homological vanishing theorems for locally analytic representations let g be the group of rational points of a split connected reductive group over a p adic local field and let be a discrete and cocompact subgroup of g motivated by questions on the cohomology of p adic symmetric spaces we investigate the homology of with coefficients in locally analytic principal series and related representations of g the vanishing and finiteness results that we find partially rely on the compactness of certain banach hecke operators we also give a new construction of p schneider s reduced hodge de rham spectral sequence and show that the induced filtration is the hodge de rham filtration in a previously unknown case our vanishing theorems then also imply two other of p schneider s conjectures

nonunitary transformations and hydrodynamic limit in the one dimensional hard point gas abstract the ergodic theoretical formalism of nonunitary transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit the one dimensional hard point system is used as an illustration both methods rest on the k mixing property of the physical systems considered we show that the specific form of the operator is important only on the microscopic time scale on the macroscopic scale it acts like a projector with broken time reversal symmetry we also discuss the connection between the kolmogorov sinai k s entropy of the model dynamical system the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate

quasifinite representations of classical lie subalgebras of w1 we show that there are precisely two up to conjugation anti involutions of the algebra of differential operators on the circle preserving the principal gradation we classify the irreducible quasifinite highest weight representations of the central extension d of the lie subalgebra of this algebra fixed by and find the unitary ones we realize them in terms of highest weight representations of the central extension of the lie algebra of infinite matrices with finitely many non zero diagonals over the algebra c u um 1 and its classical lie subalgebras ofb canddtypes character formulas forpositive primitiverepresentations of d including all the unitary ones are obtained we also realize a class of primitive representations of d in terms of free fields and establish a number of duality results between these primitive representations and finite dimensional irreducible representations of finite dimensional lie groups and supergroups we show that the vacuum modulevcof d carries a vertex algebra structure and establish a relationship betweenvcforc 12z and w algebras

derivations and central extensions of finitely generated graded lie algebras abstract some general results concerning derivations of finitely generated lie algebras are established these are employed in order to determine the derivations and central extensions of kac moody lie algebras

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

whittaker modules for the twisted heisenberg virasoro algebra we define whittaker modules for the twisted heisenberg virasoro algebra and obtain several results from the classical setting including a classification of simple whittaker modules by central characters

on intrinsic randomness of dynamical systems we discuss the problem of nonunitary equivalence via positivity preserving similarity transformations between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes dynamical systems admitting such nonunitary equivalence with stochastic markov processes are said to beintrinsically random in a previous work it was found that the so called bernoulli systems discrete time are intrinsically random in this sense this result is extended here by showing that a more general class of dynamical systems the so calledk systems andk flows are intrinsically random the connection of intrinsic randomness with local instability of motion is briefly discussed we also show that markov processes associated through nonunitary equivalence tononisomorphic k flows are necessarily non isomorphic