first integrals of difference hamiltonian equations in the present paper the well known noether s identity which represents the connection between symmetries and first integrals of euler lagrange equations is rewritten in terms of the hamiltonian function this approach based on the hamiltonian identity provides a simple and clear way to find first integrals of canonical hamiltonian equations without integration a discrete analog of the hamiltonian identity is developed it leads to a connection between symmetries and first integrals of difference hamiltonian equations that can be used to conserve the structural properties of hamiltonian equations under discretizaton the results are illustrated by a number of examples for both continuous and difference hamiltonian equations

existence of solutions to a class of indefinite stochastic riccati equations an indefinite stochastic riccati equation is a matrix valued highly nonlinear backward stochastic differential equation together with an algebraic matrix positive definiteness constraint we introduce a new approach to solve a class of such equations including the existence of solutions driven by one dimensional brownian motion the idea is to replace the original equation by a system of backward stochastic differential equations without involving any algebraic constraint whose existence of solutions automatically enforces the original algebraic constraint to be satisfied

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

an algebraic slice in the coadjoint space of the borel and the coxeter element abstract let g be a complex simple lie algebra and b a borel subalgebra the algebra y of polynomial semi invariants on the dual b of b is a polynomial algebra on rank g generators grothendieck and dieudonne 1965 1967 16 the analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action that is an affine translate y v of a vector subspace of b such that the restriction map induces an isomorphism of y onto the algebra r y v of regular functions on y v this holds in type a and even extends to all biparabolic subalgebras joseph 2007 20 but the construction fails in general even with respect to the borel moreover already in type c 2 no algebraic slice exists very surprisingly the exception of type c 2 is itself an exception indeed an algebraic slice for the coadjoint action of the borel subalgebra is constructed for all simple lie algebras except those of types b 2 m c n and f 4 outside type a the slice obtained meets an open dense subset of regular orbits even though the special point y of the slice is not itself regular this explains the failure of our previous construction

the adjoint equation method for constructing first integrals of difference equations a new method for finding first integrals of discrete equations is presented it can be used for discrete equations which do not possess a variational lagrangian or hamiltonian formulation the method is based on a newly established identity which links symmetries of the underlying discrete equations solutions of the discrete adjoint equations and first integrals the method is applied to an invariant mapping and to discretizations of second order and third order ordinary differential equations in examples the set of independent first integrals makes it possible to find the general solution of the discrete equations

on the goldie quotient ring of the enveloping algebra of a classical simple lie superalgebra abstract if g is a classical simple lie superalgebra g p n the enveloping algebra u g is a prime ring and hence has a simple artinian ring of quotients q u g by goldie s theorem we show that if g has type i then q u g is a matrix ring over q u g 0 on the other hand if g osp 1 2r then by extending the center of u g we obtain a prime ring whose goldie quotient ring is a matrix ring over the quotient division ring of a weyl algebra this is an analog of a result of gelfand and kirillov

linear exponential quadratic gaussian control in this technical note an optimal control problem for a linear stochastic system with brownian motion and a cost that is an exponential of a quadratic functional of the state and the control is solved by obtaining explicitly an optimal control and the optimal cost while this solution has been previously obtained the approach given here is direct and elementary and does not use the well known solution methods of the hamilton jacobi bellman equation or the stochastic maximum principle the approach given here presents a basic insight in the solution by providing a simple explanation for the additional term in the riccati equation for the optimal control as compared to the riccati equation for the linear quadratic gaussian control problem

observer based controller design for stochastic descriptor systems with brownian motions in the paper a complete proof of the properness and mean square exponential stability for stochastic brownian systems is firstly addressed which lays a solid foundation for system analysis and control synthesis for this kind of stochastic systems observer based controller for the descriptor system with brownian motions is then investigated it is noticed that the system dynamics and observer dynamics are both subjected to state brownian fluctuation which makes it challenging either to design observer gain and controller gain simultaneously or design the observer and control gains by using a complete separation design technique a sequential design technique is proposed to calculate the control and observer gains by solving linear matrix inequalities the proposed design methods are readily extended to the case for stochastic descriptor systems with multiple brownian motions a numerical example is finally given to illustrate the design techniques and simulated results have demonstrated the efficiency

a simple proof of indefinite linear quadratic stochastic optimal control with random coefficients we consider the indefinite linear quadratic optimal control problem for stochastic systems with random coefficients where the corresponding cost parameters need not be definite matrices although the solution to this problem was obtained previously via the stochastic maximum principle or by solving the stochastic hamilton jacobi bellman hjb equation our approach is simple and direct specifically we develop a direct approach also known as the completion of squares method to characterize the explicit optimal solution and the optimal cost the corresponding optimal solution is linear characterized in terms of the stochastic riccati differential equation srde and the linear backward stochastic differential equation bsde in our approach the completion of squares method together with the srde and the bsde allows us to construct an equivalent cost functional that is quadratic in the control u provided that an additional positive definiteness condition in terms of the srde holds then the optimal control and the associated cost can be obtained by eliminating the quadratic term of u in the equivalent cost functional the additional positive definiteness condition is induced due to the indefiniteness of the cost parameters and the dependence of the control on the diffusion term we verify the optimal solution by solving the stochastic hjb equation

multidimensional backward stochastic riccati equations and applications backward stochastic riccati differential equations bsrdes for short arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients the existence and uniqueness question of the global adapted solutions has been open since bismut s pioneering research publication in 1978 seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 one distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component in a previous article stochastic process appl 97 2002 pp 255 288 the authors solved the one dimensional case driven by brownian motions in this paper the multidimensional case driven by brownian motions is studied a closeness property for solutions of bsrdes with respect to their coefficients is stated and is proved for general bsrdes which is used to obtain the existence of a global adapted solution to some bsrdes the global existence and uniqueness results are obtained for two classes of bsrdes whose generators contain a quadratic term of l the second unknown component more specifically the two classes of bsrdes are for the regular case n 0 email protected email protected email protected dk displaystyle a k ka q ld n d kd 1 d l dt l dw k t m end array right under the condition d 1 and for the singular case left begin array email protected email protected email protected dk displaystyle a k ka c kc q c l lc displaystyle kb c kd ld d kd 1 kb c kd ld dt displaystyle l dw k t m end array right under the condition d 1 and m n the arguments given in this paper are completely new and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above we make full use of the special structure the nonnegativity of the quadratic term for example of the underlying riccati equation applications in optimal stochastic control are exposed

connected graded hopf algebras we study algebraic and homological properties of two classes of infinite dimensional hopf algebras over an algebraically closed field k of characteristic zero the first class consists of those hopf k algebras that are connected graded as algebras and the second class are those hopf k algebras that are connected as coalgebras for many but not all of the results presented here the hopf algebras are assumed to have finite gel fand kirillov dimension it is shown that if the hopf algebra h is a connected graded hopf algebra of finite gel fand kirillov dimension n then h is a noetherian domain which is cohen macaulay artin schelter regular and auslander regular of global dimension n it has s2 idh and is calabi yau detailed information is also provided about the hilbert series of h our results leave open the possibility that the first class of algebras is properly contained in the second for this second class the hopf k algebras of finite gel fand kirillov dimension n with connected coalgebra the underlying coalgebra is shown to be artin schelter regular of global dimension n both these classes of hopf algebra share many features in common with enveloping algebras of finite dimensional lie algebras for example an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra nevertheless we construct as one of our main results an example of a hopf k algebra h of gel fand kirillov dimension 5 which is connected graded as an algebra and connected as a coalgebra but is not isomorphic as an algebra to u g for any lie algebra g

solution of a q difference noether problem and the quantum gelfand kirillov conjecture for mathfrak gl _n it is shown that the q difference noether problem for all classical weyl groups has a positive solution simultaneously generalizing well known results on multisymmetric functions of mattuck proc am math soc 19 764 765 1968 and miyata nagoya math j 41 69 73 1971 in the case q 1 and q deforming the noncommutative noether problem for the symmetric group futorny et al in adv math 223 773 796 2010 it is also shown that the quantum gelfand kirillov conjecture for mathfrak gl n for a generic q follows from the positive solution of the q difference noether problem for the weyl group of type d n the proof is based on the theory of galois rings futorny and ovsienko in j algebra 324 598 630 2010 from here we obtain a proof of the quantum gelfand kirillov conjecture for mathfrak gl n and for a certain extension of mathfrak sl n previously the case of mathfrak sl n was shown by fauquant millet j algebra 218 93 116 1999 and by alev and dumas j algebra 170 229 265 1994 for n 2 3 moreover we give an explicit description of the skew fields of fractions for u q mathfrak gl n and u q mathrm ext mathfrak sl n which generalizes the results of alev and dumas j algebra 170 229 265 1994

algebra of derivations of lie algebras abstract we show a method to determine the space of derivations of any lie algebra and in particular we apply this method to a special class of lie algebras those nilpotent with low nilindex most calculations have been supported by the software mathematica 3 0

modular lie algebras and the gelfand kirillov conjecture let mathfrak g be a finite dimensional simple lie algebra over an algebraically closed field mathbb k of characteristic 0 let mathfrak g mathbb z be a chevalley form of mathfrak g and mathfrak g bbbk mathfrak g mathbb z otimes mathbb z bbbk where bbbk is the algebraic closure of mathbb f p let g bbbk be a simple simply connected algebraic bbbk group with operatorname lie g bbbk mathfrak g bbbk in this paper we apply recent results of rudolf tange on the fraction field of the centre of the universal enveloping algebra u mathfrak g bbbk to show that if the gelfand kirillov conjecture from 1966 holds for mathfrak g then for all p 0 the field of rational functions bbbk mathfrak g bbbk is purely transcendental over its subfield bbbk mathfrak g bbbk g bbbk very recently it was proved by colliot thelene kunyavski popov and reichstein that the field of rational functions mathbb k mathfrak g is not purely transcendental over its subfield mathbb k mathfrak g mathfrak g if mathfrak g is of type b n n 3 d n n 4 e6 e7 e8 or f4 we prove a modular version of this result valid for p 0 and use it to show that in characteristic 0 the gelfand kirillov conjecture fails for the simple lie algebras of the above types in other words if mathfrak g is of type b n n 3 d n n 4 e6 e7 e8 or f4 then the lie field of mathfrak g is more complicated than expected

the polynomiality of the poisson center and semi center of a lie algebra and dixmier s fourth problem abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we provide necessary and also some sufficient conditions in order for its poisson center and semi center to be polynomial algebras over k this occurs for instance if g is quadratic of index 2 with g g g and also if g is nilpotent of index at most 2 the converse holds for filiform lie algebras of type l n q n r n and w n we show how dixmier s fourth problem for an algebraic lie algebra g can be reduced to that of its canonical truncation g moreover dixmier s statement holds for all lie algebras of dimension at most eight the nonsolvable ones among them possess a polynomial poisson center and semi center

enveloping algebras of simple three dimensional lie algebras abstract the killing form of a simple three dimensional lie algebra is shown to classify the lie algebra and its enveloping algebra the lie algebra is isomorphic to the two by two trace zero matrices the so called split case if and only if the form is isotropic in the sense of quadratic form theory in characteristic zero the skew field of quotients of the enveloping algebra is shown to be isomorphic to a weyl skew field if and only if the lie algebra is split automorphisms of the lie algebra and its enveloping algebra are induced by isometries of the killing form in the nonsplit case

filtrations on simple artinian rings abstract filtrations on simple artinian rings are related to pseudo valuations by conditions on the associated graded rings

the gelfand kirillov conjecture for semi direct products of lie algebras let g be an n dimensional lie algebra over a field k of characteristic zero and let w be a g module such that dimw n sufficient conditions are given in order for the semi direct product g w to satisfy the gelfand kirillov conjecture this implies that this conjecture holds for an important class of frobenius lie algebras special attention is devoted to the case where g sl 2 k

discrete variational hamiltonian mechanics the main contribution of this paper is to present a canonical choice of a hamiltonian theory corresponding to the theory of discrete lagrangian mechanics we make use of lagrange duality and follow a path parallel to that used for construction of the pontryagin principle in optimal control theory we use duality results regarding sensitivity and separability to show the relationship between generating functions and symplectic integrators we also discuss connections to optimal control theory and numerical algorithms

general linear quadratic optimal stochastic control problems with random coefficients linear stochastic hamilton systems and backward stochastic riccati equations consider the minimization of the following quadratic cost functional j u e langle mx t x t rangle e int 0 t langle q sx s x s rangle langle n su s u s rangle ds where x is the solution of the following linear stochastic control system eq dx t a tx t b tu t dt sum i 1 d c t ix t d t iu t dw t i cr x 0 h cr u is a square integrable adapted process the problem is conventionally called the stochastic lq the abbreviation of linear quadratic problem we are concerned with the following general case the coefficients a b ci di q n and m are allowed to be adapted processes or random matrices we prove the existence and uniqueness result for the associated riccati equation which in our general case is a backward stochastic differential equation with the generator the drift term being highly nonlinear in the two unknown variables this solves bismut and peng s long standing open problem for the case of a brownian filtration which was initially proposed by the french mathematician j m bismut in seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 we also provide a rigorous derivation of the riccati equation from the stochastic hamilton system this completes the interrelationship between the riccati equation and the stochastic hamilton system as two different but equivalent tools for the stochastic lq problem there are two key points in our arguments the first one is to connect the existence of the solution of the riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system actually we establish their equivalence as a consequence we can construct solutions to a sequence of suitably modified riccati equations in terms of the associated stochastic hamilton systems and the optimal controls the second key point is to establish a new type of a priori estimate for solutions of riccati equations with which we show that the sequence of constructed solutions has a limit which is a solution to the original riccati equation

dynamic programming for general linear quadratic optimal stochastic control with random coefficients we are concerned with the linear quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable but essentially bounded processes and the terminal state weighting matrix in the cost functional is allowed to be random under suitable conditions we prove that the value field mathbb v t x omega t x omega in 0 t times r n times omega is quadratic in x and has the following form mathbb v t x langle mathbb k tx x rangle where mathbb k is an essentially bounded nonnegative symmetric matrix valued adapted processes using the dynamic programming principle dpp we prove that mathbb k is a continuous semimartingale of the form mathbb k t mathbb k 0 int 0 t dk s int 0 t sum i 1 d l s i dw s i t in 0 t with k being a continuous process of bounded variation and the stochastic integral int 0 cdot sum i 1 d l i s dw i s being a bound

the gelfand kirillov conjecture for lie algebras of dimension at most eight univ reims dept math f 51687 reims 2 france univ limburg dept math b 3590 diepenbeek belgium alev j univ reims dept math bp 1039 f 51687 reims 2 france

computing invariants and semi invariants by means of frobenius lie algebras abstract let u be the enveloping algebra of a finite dimensional lie algebra over a field k of characteristic zero z u its center and sz u its semi center a sufficient condition is given in order for sz u to be a polynomial algebra over k surprisingly this condition holds for many lie algebras especially among those for which the radical is nilpotent in which case sz u z u in particular it allows the explicit description of z u for more than half of all complex indecomposable nilpotent lie algebras of dimension at most 7

a class of counterexamples to the gel fand kirillov conjecture limburgs univ centrum dept wni b 3500 diepenbeek belgium alev j univ reims ufr sci dept math moulin housse bp 347 f 51062 reims france

construction of representations of poincare group using lie fields in this paper we give an explicit construction of the unitary irreducible representations of the poincare groups in 2 3 and 4 space time dimensions on hilbert spaces associated with the schrodinger representation of the weyl algebra for n 1 2 and 3 respectively our method of constructing the representations uses extension and localization of the enveloping algebras associated with these weyl algebras and the poincare algebras

the gelfand kirillov conjecture and gelfand tsetlin modules for finite w algebras abstract we address two problems with the structure and representation theory of finite w algebras associated with general linear lie algebras finite w algebras can be defined using either kostant s whittaker modules or a quantum hamiltonian reduction our first main result is a proof of the gelfand kirillov conjecture for the skew fields of fractions of finite w algebras the second main result is a parameterization of finite families of irreducible gelfand tsetlin modules using gelfand tsetlin subalgebra as a corollary we obtain a complete classification of generic irreducible gelfand tsetlin modules for finite w algebras

on certain maximal subfields in the quotient division ring of an enveloping algebra univ limburg dept math b 3590 diepenbeek belgium ooms ai univ limburg dept math b 3590 diepenbeek belgium

continuous symmetries of lagrangians and exact solutions of discrete equations one of the difficulties encountered when studying physical theories in discrete space time is that of describing the underlying continuous symmetries like lorentz or galilei invariance one of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices in a previous article we have classified ordinary difference schemes invariant under lie groups of point transformations the present article is devoted to an invariant lagrangian formalism for scalar single variable difference schemes the formalism is used to obtain first integrals and explicit exact solutions of the schemes equations invariant under two and three dimensional groups of lagrangian symmetries are considered

minimization of risk and linear quadratic optimal control theory this article is concerned with the optimal control problem for the linear stochastic system xt x t 0 asxs bsus fs ds t 0 d i 1 ci s xs di s us gi s dwi s with the convex risk functional j u em xt et 0 g t xt u t dt in order to guarantee the existence of an optimal control without any weak compactness assumption on the admissible control set we assume that the risk function m is coercive and thatd 1 d di is uniformlypositive rather than to assume like in the control literature that the running risk function g is coercive with respect to the control variable in this new setting the running risk function g maybe independent of the control variable and therefore the so called singular linear quadratic lq stochastic control problem is included a rigorous theoryis developed for the general stochastic lq problem with random coefficients and the bounded mean oscillation martingale theoryis used to account for the concerned integrability it plays a crucial role in the following exposition a to connect the stochastic lq problem to two associated backward stochastic differential equations bsdes one is an n n symmetric matrix valued nonlinear riccati bsde and the other is an n dimensional linear bsde with unbounded coefficients b to show that the latter bsde has an adapted solution pair of the suitablynecessaryregularity this seems to be the first application in a stochastic lq theory of the bmo martingale theory which roots in harmonic analysis furthermore with the help of an a priori estimate on the risk functional existence and uniqueness of the solutions of backward stochastic riccati differential equations bsrdes in the singular case is reduced to the regular case via a perturbation method and then a new existence and uniqueness result on bsrdes is obtained for the singular case

noether s theorem and lie symmetries for time dependent hamilton lagrange systems noether and lie symmetry analyses based on point transformations that depend on time and spatial coordinates will be reviewed for a general class of time dependent hamiltonian systems the resulting symmetries are expressed in the form of generators whose time dependent coefficients follow as solutions of sets of ordinary differential auxiliary equations the interrelation between the noether and lie sets of auxiliary equations will be elucidated the auxiliary equations of the noether approach will be shown to admit invariants for a much broader class of potentials compared to earlier studies as an example we work out the noether and lie symmetries for the time dependent kepler system the runge lenz vector of the time independent kepler system will be shown to emerge as a noether invariant if we adequately interpret the pertaining auxiliary equation furthermore additional nonlocal invariants and symmetries of the kepler system will be isolated by identifying further solutions of the auxiliary equations that depend on the explicitly known solution path of the equations of motion showing that the invariants remain unchanged under the action of different symmetry operators we demonstrate that a unique correlation between a symmetry transformation and an invariant does not exist

algebras of invariant differential operators abstract we prove that an invariant subalgebra a n w of the weyl algebra a n is a galois order over an adequate commutative subalgebra when w is a two parameters irreducible unitary reflection group g m 1 n m 1 n 1 including the weyl group of type b n or alternating group a n or the product of n copies of a cyclic group of fixed finite order earlier this was established for the symmetric group in 15 in each of the cases above except for the alternating groups we show that a n w is free as a right left module similar results are established for the algebra of w invariant differential operators on the n dimensional torus where w is a symmetric group s n or orthogonal group of type b n or d n as an application of our technique we prove the quantum gelfand kirillov conjecture for u q s l 2 the first witten deformation and the woronowicz deformation

indefinite stochastic riccati equations this paper is concerned with stochastic riccati equations sres which are a class of matrix valued nonlinear backward stochastic differential equations bsdes the sres under consideration are in general indefinite in the sense that certain parameter matrices are indefinite this kind of equations arises from the stochastic linear quadratic lq optimal control problem with random coefficients and indefinite state and control weighting costs the latter having profound implications in both theory and applications while the solvability of the sres is the key to solving the indefinite stochastic lq control it remains in general an extremely difficult open problem this paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite sres for a number of special yet important cases

a quadratic poisson gel fand kirillov problem in prime characteristic the quadratic poisson gel fand kirillov problem asks whether the field of fractions of a poisson algebra is poisson birationally equivalent to a poisson affine space i e to a polynomial algebra k x1 xn with poisson bracket defined by xi xj ijxixj for some skew symmetric matrix ij mn k this problem was studied in 9 over a field of characteristic 0 by using a poisson version of the deleting derivation homomorphism of cauchon in this paper we study the quadratic poisson gel fand kirillov problem over a field of arbitrary characteristic in particular we prove that the quadratic poisson gel fand kirillov problem is satisfied for a large class of poisson algebras arising as semiclassical limits of quantised coordinate rings for we introduce the concept of higher poisson derivation which allows us to extend the poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic when a torus is acting rationally by poisson automorphisms on a poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring we prove under some technical assumptions that quotients by poisson prime torus invariant ideals also satisfy the quadratic poisson gel fand kirillov problem in particular we show that coordinate rings of determinantal varieties satisfy the quadratic poisson gel fand kirillov problem