on semi invariants and index for biparabolic seaweed algebras ii abstract in a continuation of i we study the semi invariant subalgebra sy q of s q for q a biparabolic subalgebra of a semisimple or of an affine lie algebra in most cases this is shown to be polynomial and we compute the degrees and weights of the generators we describe explicitly the canonical truncation q e of q for which sy q s q e q e let t denote the nilradical of q and q its derived algebra we show that the subalgebra s t q of s q q sy q is always polynomial and describe the degrees and weight of the generators finally a sum rule is given for the false degrees of s q q these are true degrees in most cases

fractional sobolev hardy inequality in rn abstract in this paper we show that the minimizing problem 1 s inf u h s r n u 0 r n s 2 u x 2 d x r n u x 2 s x d x 2 2 s is achieved by a positive radially symmetric and strictly decreasing function provided 0 s n 2 0 2 s

primitive localizations of an enveloping algebra abstract we prove a number of sufficient conditions for a localization of an enveloping algebra at a multiplicatively closed set of semi invariants to be a primitive ring in case of a central localization these conditions are also necessary

the frobenius semiradical of a lie algebra univ limburg dept math b 3590 diepenbeek belgium ooms ai univ limburg dept math univ campus b 3590 diepenbeek belgium alfons ooms luc ac be

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

exact solutions to a class of feedback systems on so n this paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on so n the closed loop systems are solved exactly for the rotation matrices as explicit functions of time the initial conditions and the gain parameters of the control laws the exact solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance knowledge of the future trajectory of the states can also be used as an alternative to the zero order hold in systems where the attitude is sampled at discrete time instances

on the center and semi center of enveloping algebras in prime characteristic abstract theorems of j dixmier and c moeglin on the semi center of enveloping algebras in zero characteristic and of r p stanley and h nakajima on polynomial relative invariants of finite groups are shown to have analogs for enveloping algebras and for lie algebra polynomial invariants in the prime characteristic case we shall illustrate by examples the extent of these analogies a key result is the realization of the semi center as a fixed ring of the action of a finite set of nilpotent derivations on the center of an enveloping algebra of a related lie algebra

nonexistence results for a class of fractional elliptic boundary value problems abstract in this paper we study a class of fractional elliptic problems of the form s u f x u in u 0 in r n where s 0 1 we prove nonexistence of positive solutions when is star shaped and f is supercritical we also derive a nonexistence result for subcritical f in some unbounded domains the argument relies on the method of moving spheres applied to a reformulated problem using the caffarelli silvestre extension caffarelli and silvestre 2007 11 of a solution of the above problem

cayley kinematics and the cayley form of dynamic equations the cayley transform and the cayleytransform kinematic relationships are an important and fascinating set of results that have relevance in ndimensional orientations and rotations in this paper th

the pohozaev identity for the fractional laplacian in this paper we prove the pohozaev identity for the semilinear dirichlet problem delta s u f u in omega u equiv0 in mathbb r n backslash omega here s in 0 1 s is the fractional laplacian in mathbb r n and is a bounded c1 1 domain to establish the identity we use among other things that if u is a bounded solution then u delta s omega is c up to the boundary where x dist x in the fractional pohozaev identity the function u delta s partial omega plays the role that u plays in the classical one surprisingly from a nonlocal problem we obtain an identity with a boundary term an integral over which is completely local as an application of our identity we deduce the nonexistence of nontrivial solutions in star shaped domains for supercritical nonlinearities

fractional hardy sobolev inequalities on half spaces abstract we investigate the existence of extremals for hardy sobolev inequalities involving the dirichlet fractional laplacian s of order s 0 1 on half spaces

adapted pairs in type a and regular nilpotent elements abstract let g be a simple lie algebra over an algebraically closed field k of characteristic zero and g its adjoint group a biparabolic subalgebra q of g is the intersection of two parabolic subalgebras whose sum is g the algebra sy q of semi invariants on q of a proper biparabolic subalgebra q of g is polynomial in most cases in particular when g is simple of type a or c on the other hand q admits a canonical truncation q such that sy q sy q y q where y q denotes the algebra of invariant functions on q an adapted pair for q is a pair h q q such that is regular in q and a d h in joseph 2008 9 adapted pairs for every truncated biparabolic subalgebra q of a simple lie algebra g of type a were constructed and then provide weierstrass sections for y q in q these weierstrass sections are linear subvarieties v of q such that the restriction map induces an algebra isomorphism of y q onto the algebra of regular functions on v the main result of the present work is to show that for each of the adapted pairs h constructed in joseph 2008 9 one can express not quite uniquely as the image of a regular nilpotent element y of g under the restriction map g q this is a significant extension of joseph and fauquant millet 2011 12 which obtains this result in the rather special case of a truncated biparabolic of index one observe that y must be a g translate of the standard regular nilpotent element defined in terms of the already chosen set of simple roots consequently one may attach to y a unique element of the weyl group w of g ultimately one can then hope to be able to describe adapted pairs in general that is not only for g of type a through the weyl group

on the coadjoint representation of z2 contractions of reductive lie algebras abstract we study the coadjoint representation of contractions of reductive lie algebras associated with symmetric decompositions let g g 0 g 1 be a symmetric decomposition of a reductive lie algebra g then the semi direct product of g 0 and the g 0 module g 1 is a contraction of g we conjecture that these contractions have many properties in common with reductive lie algebras in particular it is proved that in many cases the algebra of invariants is polynomial we also discuss the so called codim 2 property for coadjoint representations and its relationship with the structure of algebra of invariants

on the sobolev and hardy constants for the fractional navier laplacian abstract we prove the coincidence of the sobolev and hardy constants relative to the dirichlet and navier fractional laplacians of any real order m 0 n 2 over bounded domains in r n

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

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

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

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

the poisson center and polynomial maximal poisson commutative subalgebras especially for nilpotent lie algebras of dimension at most seven abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we collect some general results on the poisson center of s g including some simple criteria regarding its polynomiality and also on certain poisson commutative subalgebras of s g these facts are then used to complete our previous work on the subject ooms 2009 o5 5 i e to give an explicit description for the poisson center of all indecomposable nilpotent lie algebras of dimension at most seven among other things we also provide a polynomial maximal poisson commutative subalgebra of s g enjoying additional properties as a by product we show that a conjecture by milovanov is valid in this situation these results easily carry over to the enveloping algebra u g

strong maximum principles for fractional laplacians we give a unified approach to strong maximum principles for a large class of nonlocal operators of order s 0 1 that includes the dirichlet the neumann restricted or regional and the neumann semirestricted laplacians

on the semi center of a universal enveloping algebra throughout this paper l will be a nonzero finite dimensional lie algebra over a field k of characteristic zero let u l be the universal enveloping algebra of l with center z u l d l the division ring of quotients of u l with center z d l for each 2 e l we denote by d l the set of those ued l such that ad x u n x u for all xe l its elements are called the semi invariants of d l relative to 1 clearly d l l c al a for all i p e l the sum of the d l is direct and is a subalgebra sz d l of d l which is called the semi center of d l we put u l u l n d l the sum sz u l of the u l is called the semi center of u l the semi center sz s l of the symmetric algebra s l of l can be defined in a similar fashion the semi center is a commutative subalgebra of u l which never reduces to k 7 and which is stable under all automorphisms of u l 16 from the fact that it is a unique factorization domain in the algebraically closed case 131 we are able to deduce that in the general case sz u l is contained in a factorial subalgebra of u l theorem 1 2 in particular any nonzero semi invariant of u l can be factored in a unique way as the product of irreducible semiinvariants proposition 1 5 this result proves to be extremely useful throughout the paper we also obtain some characterizations of the semiinvariants of d l proposition 1 8 let a l resp a l be the set of all z e l such that u l 0 resp d l o a l is a semi group which is not always finitely generated 9 the group n l which is isomorphic to kharchenko s group of x inner automorphisms of u l 16 turns out to be the additive subgroup of l generated by a l corollary 1 9 next the centralizer of sz u l in u l coincides with u l where l is the intersection of ker 2 ac l corollary 1 16 l 324 0021 8693 85 3 00

a degree inequality for lie algebras with a regular poisson semi center for lie algebras whose poisson semi center is a polynomial ring we give a bound for the sum of the degrees of the generating semi invariants this bound was previously known in many special cases

hamel coefficients for the rotational motion of an n dimensional rigid body many of the kinematic and dynamic concepts relating to rotational motion have been generalized for ndimensional rigid bodies in this paper a new derivation of the generalized euler equations is pr

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

minimal parameter solution of the orthogonal matrix differential equation the straightforward solution of the first order differential equation satisfied by all nth order orthogonal matrices requires n sup 2 integrations to obtain the matrix elements there are however only n n 1 2 independent parameters which determine an orthogonal matrix the questions of choosing them finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered in the present work several possibilities which are based on attitude determination in three dimensions are examined it is concluded that not all 3 d methods have useful extensions to other dimensions and that the 3 d gibbs vector or cayley parameters provide the most useful extension an algorithm is developed using the resulting parameters which are termed extended rodrigues parameters and numerical results are presented of the application of the algorithm to a fourth order matrix