quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

sugawara operators and kac kazhdan conjecture soit g une algebre de lie de kac moody affine de type x l 1 soit l le module de poids le plus eleve unique irreductible pour chaque poids de kac kazhdan on a chl exp re 1 exp 1

azumaya loci and discriminant ideals of pi algebras abstract we prove that under mild assumptions for all positive integers l the zero set of the discriminant ideal d l r z r tr of a prime affine polynomial identity pi algebra r coincides with the zero set of the modified discriminant ideal m d l r z r tr of r and give an explicit description of this set in terms of the dimensions of the irreducible representations of r furthermore we prove that when l is the square of the pi degree of r this zero set is precisely the complement of the azumaya locus of r this description is used to classify the azumaya loci of the multiparameter quantized weyl algebras at roots of unity as another application we prove that the zero set of the top discriminant ideal of a prime affine pi algebra r coincides with the singular locus of the center of r provided that the discriminant ideal has height at least 2 r has finite global dimension and r is a cohen macaulay module over its center

techniques for the efficient evaluation of two dimensional autocorrelation functions the evaluation of the autocorrelation of multidimensional sequences using the fft is investigated the bidimensional case is treated in detail not only because it is of special interest for the applications but also because the reasoning used for this case can be applied to higher dimension signals the analysis confirms that the use of the proposed frequency domain techniques leads to significant computation time savings

cherednik hecke and quantum algebras as free frobenius and calabi yau extensions we show how the existence of a pbw basis and a large enough central subalgebra can be used to deduce that an algebra is frobenius we apply this to rational cherednik algebras hecke algebras quantised universal enveloping algebras quantum borels and quantised function algebras in particular we give a positive answer to r rouquier representations of rational cherednik algebras in infinite dimensional aspects of representation theory and applications amer math soc 2005 pp 103 131 stating that the restricted rational cherednik algebra at the value t 0 is symmetric

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

poisson spectra in polynomial algebras abstract a significant class of poisson brackets on the polynomial algebra c x 1 x n is studied and for this class of poisson brackets the poisson prime ideals poisson primitive ideals and symplectic cores are determined moreover it is established that these poisson algebras satisfy the poisson dixmier moeglin equivalence

the ramification of centres lie algebras in positive characteristic and quantised enveloping algebras let h be a hopf algebra over the field k which is a finite module over a central affine sub hopf algebra r examples include enveloping algebras u mathfrak g of finite dimensional k lie algebras mathfrak g in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity the ramification behaviour of the maximal ideals of z h with respect to the subalgebra r is studied and the conclusions are then applied to the cases of classical and quantised enveloping algebras in the case of u mathfrak g for mathfrak g semisimple a conjecture of humphreys 28 on the block structure of u mathfrak g is confirmed in the case of u epsilon mathfrak g for mathfrak g semisimple and epsilon an odd root of unity we obtain a quantum analogue of a result of mirkovic and rumynin 35 and we fully describe the factor algebras lying over the regular sheet 9 the blocks of u epsilon mathfrak g are determined and a necessary condition which may also be sufficient for a baby verma u epsilon mathfrak g module to be simple is obtained

unruffled extensions and flatness over central subalgebras abstract a condition on an affine central subalgebra z of a noetherian algebra a of finite gelfand kirillov dimension which we call here unruffledness is shown to be equivalent in some circumstances to the flatness of a as a z module unruffledness was studied by borho and joseph in work on enveloping algebras of complex semisimple lie algebras and we discuss applications of our result to enveloping algebras as well as beginning the study of this condition for more general algebras

the calogero moser partition and rouquier families for complex reflection groups let w be a complex reflection group we formulate a conjecture relating blocks of the corresponding restricted rational cherednik algebras and rouquier families for cyclotomic hecke algebras we verify part of the conjecture in the case that w is a wreath product of a symmetric group with a cyclic group of order l

the weyl algebra as a fixed ring abstract we prove that the weyl algebra over c cannot be a fixed ring of any domain under a nontrivial action of a finite group by algebra automorphisms thus settling a 30 year old problem in fact we prove the following much more general result let x be a smooth affine variety over c let d x denote the ring of algebraic differential operators on x and let be a finite group if d x is isomorphic to the ring of invariants of a c domain r on which acts faithfully by c algebra automorphisms then r is isomorphic to the ring of differential operators on a galois covering of x

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

harish chandra bimodules over rational cherednik algebras abstract we study harish chandra bimodules over the rational cherednik algebra h c w associated to a complex reflection group w with parameter c our results allow us to partially reduce the study of these bimodules to smaller algebras we classify those pairs of parameters c c for which there exist fully supported harish chandra bimodules and give a description of the category of all harish chandra bimodules modulo those without full support when w is a symmetric group we are able to classify all irreducible harish chandra bimodules our proofs are based on localization techniques the action of the namikawa weyl group on the set of parameters and the study of partial kz functors

an analogue of the kac weisfeiler conjecture abstract in this paper we discuss an analogue of the kac weisfeiler conjecture for a certain class of almost commutative algebras in particular we prove a kac weisfeiler type statement for rational cherednik algebras

random affine iterated function systems curve generation and wavelets it is shown how refinement methods for smooth curve generation can be carried out efficiently through iterated function systems ifs affine transformations are constructed so that when composed randomly they generate the desired smooth curve underlying this random algorithm is the tree traversal property of ifs under a refinement method the points on the curve correspond to leaves on some n tree ifs theory enables one to generate all of these leaves through a single orbit of an appropriate markov chain applications include bezier curves splines wavelets and various interpolants

a new linear quotient of c 4 admitting a symplectic resolution we show that the quotient c4 g admits a symplectic resolution for g q 8 times bf z 2 d 8 sf sp 4 bf c here q8 is the quaternionic group of order eight and d8 is the dihedral group of order eight and g is the quotient of their direct product which identifies the nontrivial central elements id of each it is equipped with the tensor product representation bf c 2 boxtimes bf c 2 cong bf c 4 this group is also naturally a subgroup of the wreath product group q 8 2 rtimes s 2 sf sp 4 bf c we compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming c4 g we also discuss preliminary investigations on the more general question of classifying linear quotients v g admitting symplectic resolutions

stratifications of marsden weinstein reductions for representations of quivers and deformations of symplectic quotient singularities we investigate the poisson geometry of the marsden weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver we show that the stratification by representation type equals the stratification by symplectic leaves the deformed symplectic quotient singularities spectra of centres of symplectic reflection algebras associated to a wreath product are shown to be isomorphic to reductions of a certain quiver this establishes a method to calculate when these deformations are smooth furthermore the isomorphism identifies symplectic leaves and so one can give a description of their symplectic leaves in terms of roots of the quiver

cuspidal representations of rational cherednik algebras at t 0 we study those finite dimensional quotients of the rational cherednik algebra at t 0 that are supported at a point of the centre it is shown that each such quotient is morita equivalent to a certain cuspidal quotient of a rational cherednik algebra associated to a parabolic subgroup of w

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

cuspidal calogero moser and lusztig families for coxeter groups the goal of this paper is to compute the cuspidal calogero moser families for all infinite families of finite coxeter groups at all parameters we do this by first computing the symplectic leaves of the associated calogero moser space and then by classifying certain rigid modules numerical evidence suggests that there is a very close relationship between calogero moser families and lusztig families our classification shows that additionally the cuspidal calogero moser families equal cuspidal lusztig families for the infinite families of coxeter groups

primitive ideals of c q sl 3 the primitive ideals of the hopf algebrac q sl 3 are classified in particular it is shown that the orbits in primc q sl 3 under the action of the representation grouph c c are parameterized naturally byw w wherew is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals ofc q sl 3 and symplectic leaves of the associated poisson algebraic groupsl 3 c

on the dixmier moeglin equivalence for poisson hopf algebras abstract we prove that the poisson version of the dixmier moeglin equivalence holds for cocommutative affine poisson hopf algebras this is a first step towards understanding the symplectic foliation and the representation theory of cocommutative affine poisson hopf algebras our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations as an application we show that the symmetric algebra of a finite dimensional lie algebra equipped with its natural poisson structure satisfies the poisson dixmier moeglin equivalence