time dependent invariants of motion for complete sets of non commuting observables abstract using the generalized ehrenfest theorem the dynamics of the mean values of a complete set of non commuting observables csnco associated to a given hamiltonian is expressed we found refined time dependent invariants of motion tdim for the csnco and associated them with different lie algebras we also show that when the csnco are related through an antisymmetric matrix then the characteristic polynomial of the covariance matrix for the csnco is also a tdim examples for the heisemberg su 2 and so 2 groups are outlined

gibbs entropy invariants of motion abstract invariants of motion are constructed departing from the gibbs entropy the method we applied also allows us to find the subspace of a real vector space to which the system is confined to during its temporal evolution we also find that the uncertainty principle is itself an invariant of motion for harmonic oscillator like hamiltonians

on sets in rd with dc distance function abstract we study closed sets f r d whose distance function d f dist f is dc i e is the difference of two convex functions on r d our main result asserts that if f r 2 is a graph of a dc function g r r then f has the above property if d 1 the same holds if g r d 1 r is semiconcave however the case of a general dc function g remains open

riemannian geometry and stability of thermodynamical equilibrium systems a geometrical approach to statistical thermodynamics is proposed it is shown that any r parameter generalised gibbs distribution leads to a riemannian metric of parameter space the components of the metric tensor are represented by second moments of stochastic variables the scalar curvature r as a geometrical invariant is a function of the second and third moments so is strictly connected with fluctuations of the system in the case of a real gas r is positive and tends to infinity as the system approaches the critical point in the case of an ideal gas r 0 the obtained results and the results of the authors previous work suggests that for a wide class of models r tends to infinity near the critical point they treat r as a measure of the stability of the system they propose some sort of statistical principle only such models may be accepted for which r tends to infinity if the system is approaching the critical point it is shown that if this criterion is adopted for a class of models for which the scaling hypothesis holds then they obtain the new inequalities for the critical indices these inequalities are in good agreement with model calculations and experiment

the product on smooth and generalized valuations the product of smooth valuations on manifolds is described in terms of differential forms gelfand transforms and blow up spaces it is shown that the product extends partially to generalized valuations and corresponds geometrically to transversal intersections this result is used to prove a general kinematic formula on compact rank one symmetric spaces

extensions of modules for algebraic groups let g be a connected semi simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p in this paper we show how one can obtain the extensions of any two simple g modules if one knows the socles of all the g modules extg l l where g1 denotes the kernel of the frobenius endomorphism on g and l as well as l are simple g1 modules alternatively we prove how all such g extensions can be found once one knows the extensions of any two simple g modules belonging to a specified finite set of simple modules with small highest weight such information is available in certain low rank cases it should also be pointed out that it is possible from lusztig s conjecture on irreducible characters 11 to derive the dimensions of ext l1 l2 when l1 and l2 are simple with such small highest weights to obtain our results on exth we need to know that there exists no g1 extension of a simple module by itself this turns out to be somewhat

information geometry of the spherical model motivated by the observation that geometrizing statistical mechanics offers an interesting alternative to more standard approaches we calculate the scaling behavior of the curvature mathcal r of the information geometry metric for the spherical model we find that mathcal r ensuremath sim ensuremath epsilon ensuremath 2 where ensuremath epsilon ensuremath beta c ensuremath ensuremath beta is the distance from criticality the discrepancy from the naively expected scaling mathcal r ensuremath sim ensuremath epsilon ensuremath 3 is explained and compared with that for the ising model on planar random graphs which shares the same critical exponents

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

generalized metric phase space for quantum systems and the uncertainty principle abstract we demonstrate that when the gibbs entropy is an invariant of motion following information theory procedures it is possible to define a generalized metric phase space for the temporal evolution of the mean values of a given hamiltonian the metric is positive definite and this fact leads to a metric tensor k t whose properties are well defined working with these properties we shown that a the generalized uncertainty principle gup is always the summation over the principal minors of order 2 belonging to k t b several invariants of motion can be derived from the metric tensor and c particularly under certain conditions the gup itself is also a motion invariant

algebraic group actions in the cohomology theory of lie algebras of cartan type abstract given a lie algebra g of cartan type we construct an infinite dimensional cocommutative hopf algebrad g u g which is the analog of the distribution algebra of a connected reductive algebraic group we show that the simpleu g modules lift to ad g u g structure this additional structure is used to formulate and prove relative projectivity theorems for lie algebras of cartan type the support varieties of certaind g u g modules are computed by using these algebraic group techniques

dimensions of the simple restricted modules for the restricted contact lie algebra according to 2 the finite dimensional restricted simple lie algebras over an algebraically closed field of characteristic p 7 are either classical or of cartan type lusztig has a conjecture see 8 p 294 which provides a formula for the dimensions of the simple modules for the classical algebras in this paper modules are assumed to be left finitedimensional and restricted anderson jantzen and soergel 1 have recently shown that for p a 0 the lusztig conjecture follows from an analogous conjecture also due to lusztig for certain quantum groups they have also pointed out that this latter conjecture is implied by recent work of kazhdan and lusztig and a paper by casian although casian s paper has not yet convinced all of its readers the simple modules for the cartan type algebras fall into two classes there is a large class see 11 and 5 consisting of modules that are induced from simple modules for the homogeneous component of degree zero extended trivially to positive components constructed as mixed products in 11 the degree zero homogeneous components of cartan type algebras are either classical or nearly so therefore the lusztig conjecture can be used to get a formula for the dimensions of the modules in this class the small class consists of modules corresponding to exceptional weights in 11 the dimensions of these modules were determined for three of the four cartan type algebras namely the witt special and hamiltonian algebras there the exceptional weights turned out to be fundamental dominant weights in this paper we obtain the dimensions assuming p is not too small in the remaining case of the contact algebra in the process we determine the contragredients of the simple modules

normal cycles and curvature measures of sets with d c boundary abstract we show that for every compact domain in a euclidean space with d c delta convex boundary there exists a unique legendrian cycle such that the associated curvature measures fulfill a local version of the gauss bonnet formula this was known in dimensions two and three and was open in higher dimensions in fact we show this property for a larger class of sets including also lower dimensional sets we also describe the local index function of the legendrian cycles and we show that the associated curvature measures fulfill the crofton formula

cartan matrices for restricted lie algebras consider a finite dimensional restricted lie algebra over a field of prime characteristic each linear form on this lie algebra defines a finite dimensional quotient of its universal enveloping algebra called a reduced enveloping algebra this leads to a cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra in this paper we show how to compare such cartan matrices belonging to distinct linear forms as an application we rederive and generalise the reciprocity formula first discovered by humphreys for lie algebras of reductive groups for simple lie algebras of cartan type we see for example that the cartan matrices for linear forms of non positive height are submatrices of the cartan matrix for the zero linear form

irreducible representations of lie algebras of reductive groups and the kac weisfeiler conjecture letg be the lie algebra of a connected reductive groupg over an algebraically closed field of characteristicp 0 suppose thatg 1 is simply connected andp is good for the root system ofg ifp 2 suppose in addition thatg admits a nondegenerateg invariant trace form letv be an irreducible and faithfulg module withp character g it is proved in the paper that dimv is divisible byp1 2dim where stands for the orbit of under the coadjoint action ofg

brauer type reciprocity for a class of graded associative algebras classical brauer reciprocity can be stated roughly as follows let g be a finite group and let k be an algebraically closed field of characteristic p 0 if s is a simple kg module and p s is its projective cover then the multiplicity of a simple module l as a composition factor of a characteristic zero lift of p s is the same as the multiplicity of s as a composition factor of a modulo p reduction of l all modules in this paper are assumed to be finite dimensional one thinks of the module l as playing an intermediate role between p s and s reciprocities similar to brauer reciprocity here called brauer type reciprocities have subsequently been found to occur in many other settings for instance if g is a classical modular lie algebra definition given in section 2 then there exists a set 9 of g modules with the following property given a simple g module s the projective cover p s of s has a filtration with each successive quotient isomorphic to a module in 5 and for each such filtration the number of times ze 9 occurs is the same as the multiplicity of s as a composition factor of z this was proved for g of type a i by pollack lo and for arbitrary g by humphreys 3 the reciprocity in this setting is often called humphreys reciprocity inspired by humphreys result bernstein gelfand and gelfand sought and found a brauer type reciprocity bgg reciprocity in a certain truncated category their category 8 of modules for a complex semisimple lie algebra see 11 later this was generalized by mirollo and

simple modules with character height at most one for the restricted witt algebras abstract the simple modules with character height at most one for the restricted witt algebras are considered their classification construction and dimension formulas are reduced to the same for the general linear algebra results of chang and shen are recovered in the process

classification of the restricted simple lie algebras soit l une algebre de lie simple restreinte de dimension finie sur un corps algebriquement clos f de caracteristique p 7 alors l est de type classique ou de cartan

information geometry of the ising model on planar random graphs it has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterization of the phase structure particularly in the case where there are two such parameters such as the ising model with inverse temperature ensuremath beta and external field h in various two parameter calculable models the scalar curvature mathcal r of the information metric has been found to diverge at the phase transition point ensuremath beta c and a plausible scaling relation postulated mathcal r ensuremath sim ensuremath beta ensuremath ensuremath beta c ensuremath alpha ensuremath 2 for spin models the necessity of calculating in nonzero field has limited analytic consideration to one dimensional mean field and bethe lattice ising models in this paper we use the solution in field of the ising model on an ensemble of planar random graphs where ensuremath alpha ensuremath 1 ensuremath beta 1 2 ensuremath gamma 2 to evaluate the scaling behavior of the scalar curvature and find mathcal r ensuremath sim ensuremath beta ensuremath ensuremath beta c ensuremath 2 the apparent discrepancy is traced back to the effect of a negative ensuremath alpha

deformations of lie algebra representations let g be a reductive algebraic group defined over an algebraically closed field k of positive characteristic p in this paper we continue our previous investigation 7 of the representation theory of the lie algebra f of g a key ingredient in our earlier work involved the consideration of a family a x s of finite dimensional algebras parameterized by the set 0 qs 1homka1g y b k of characters on a certain central subalgebra 0 6 of the universal enveloping algebra u 6 of g each irreducible module is naturally an irreducible module for exactly one such ax with the special case x 0 corresponding to restricted modules we introduce a so called deformation algebra s4 intermediate between the universal enveloping algebra u and certain of the algebras ax here sl is a quotient of u q such that each ax containing the restricted enveloping algebra v a of a fixed borel subalgebra 2a is in turn a quotient of s4 given a module mx for such an a we introduce the important notion of a deformation family of mx in definition 1 3 the hypothesis that the behavior of mx is related to that of the corresponding modules for near by algebras at motivates the present study a basic result 7 theorem 3 2 reduces the representation theory of ax for x 0 to that of an associated a for a subalgebra of to which the restriction t of x is nilpotent in the sense of 121 thus a study of the representation theory of the algebras ax for x a nilpotent character subsumes the classical problem of determining the irreducible representations of the lie algebra each g conjugacy class in the nullcone x of nilpotent elements of 0 y f defines a family of isomorphic algebras ax presumably the restricted enveloping algebra v ao corresponding to the conjugacy class o is the most complicated algebra of this kind its representation theory has been extensively investigated in recent years because of its intimate connections

extensions for finite chevalley groups iii rational and generic cohomology abstract let g be a connected reductive algebraic group and b be a borel subgroup defined over an algebraically closed field of characteristic p 0 in this paper the authors study the existence of generic g cohomology and its stability with rational g cohomology groups via the use of methods from the authors earlier work new results on the vanishing of g and b cohomology groups are presented furthermore vanishing ranges for the associated finite group cohomology of g f q are established which generalize earlier work of hiller in addition to stability ranges for generic cohomology which improve on seminal work of cline parshall scott and van der kallen

contact measures in isotropic spaces abstract we revisit the contact measures introduced by firey and further developed by schneider and teufel from the perspective of the theory of valuations on manifolds this reveals a link between the kinematic formulas for area measures studied by wannerer and the integral geometry of curved isotropic spaces as an application we find explicitly the kinematic formula for the surface area measure in hermitian space

representations of cartan type lie algebras in characteristic p in this paper we discuss the representations of cartan type lie algebras in characteristic p 2 from the viewpoint of reducing rank when the character is regular semisimple for generalized witt algebras we can essentially reduce higher rank representations to lower rank representations

kinematic formulas for sets defined by differences of convex functions abstract the class wdc m consists of all subsets of a smooth manifold m that may be expressed in local coordinates as sufficiently regular sublevel sets of dc differences of convex functions if m is riemannian and g is a group of isometries acting transitively on the sphere bundle sm we define the invariant curvature measures of compact wdc subsets of m and show that pairs of such subsets are subject to the array of kinematic formulas known to apply to smoother sets restricting to the case m g r d s o d this extends and subsumes federer s theory of sets with positive reach in an essential way the key technical point is equivalent to a sharpening of a classical theorem of ewald larman and rogers characterizing the dimension of the set of directions of line segments lying in the boundary of a given convex body

modular representation theory of lie algebras on etudie les representations non necessairement restreintes de l algebre universelle enveloppante u 7b g pour une algebre de lie restreinte 7b g

representation theory of the witt algebra 1 1 let g be a finite dimensional restricted lie algebra over an algebraically closed field of characteristic p 0 every simple g module is u w finite dimensional and therefore admits a character x g g see sf x u theorem 5 2 5 conversely for any such linear form x g g there exists a finite dimensional algebra u g x which is a quotient of the universal enveloping algebra u g and whose simple modules are exactly the simple modules for g with character x because of this fact these algebras play an important role in studying the representations of g in particular the restricted g representations coincide with the u g 0 modules every finite dimensional restricted simple lie algebra over an algebraically closed field of characteristic p 7 is either classical or of cartan w x type bw for a classical lie algebra g the representation theory of w x u g x was first studied by kac and weisfeiler wd kw and further

ext 1 quivers for the witt algebra w 1 1 let g be the finite dimensional witt algebra w 1 1 over an algebraically closed field of characteristic p 3 it is well known that all simple w 1 1 modules are finite dimensional each simple module admits a character g given g one can form the finite dimensional reduced enveloping algebra u g the simple modules for u g are precisely those simple w 1 1 modules admitting the character in this paper the authors compute ext1 between pairs of simple modules for u g