extension of the category og and a vanishing theorem for the ext functor for kac moody algebras we extend the equivalence relation z resp the category or defined in dgk from kg to an in general much larger subset pg c b resp to the category 3 where w g stands for weakly good we denote the extended equivalence relation by x we prove that the equivalence classes in pa under z have description exactly similar to that for the equivalence classes in kg under x given by deodhar gabber kac we also prove that their category decomposition theorem for ur can be extended to cy8 we further prove that extz m n 0 if m and ne 0 resp in c yg are of inequivalent resp types now we describe the contents in more detail unless otherwise stated g will denote an arbitrary symmetrizable kac moody lie algebra with its cartan subalgebra fj section 1 fix ii p e h let us say that 2 is related to u if the irreducible module l a is a subquotient of the verma module m p and denote by the equivalence relation thus generated bernsteingelfand gelfand using central character theory gave a necessary and sufficient condition for 2 to be related to p in the case when g is finite dimensional bgg later jantzen j simplified their proof by making use of the shapovalov bilinear form an important aspect specially for the infinite dimensional lie algebraists of jantzen s simplification was that it did not use central character theory later kac kazhdan generalized jantzen s arguments to prove an analogue of the bgg theorem for arbitrary g kk it is clear from the description of kac kazhdan kk sect 4 theorem 21 that the equivalence classes under due to the existence of imaginary roots in the infinite dim case are fairly complicated to describe e g it is no more true

the uniqueness of blow up solution for radially symmetric semilinear elliptic equation abstract in this paper we establish a blow up rate of the large positive solutions of the singular boundary value problem u u b x u p u with a ball domain and radially function b x all previous results in the literature assumed the decay rate of b x to be approximated by a distance function near the boundary obtaining the accurate blow up rate of solutions for general b x requires more subtle mathematical analysis of the problem

irreducible characters of general linear superalgebra and super duality we develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra including all the finite dimensional modules by directly relating the problem to the classical kazhdan lusztig theory furthermore we prove that certain parabolic bgg categories over the general linear algebra and over the general linear superalgebra are equivalent we also verify a parabolic version of a conjecture of brundan on the irreducible characters in the bgg category of the general linear superalgebra

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

remarks on large solutions of a class of semilinear elliptic equations abstract in this paper we show existence uniqueness and exact asymptotic behavior of solutions near the boundary to a class of semilinear elliptic equations u g u b x f u in where is a real number b x 0 in and vanishes on the special feature is to consider g u and f u to be regularly varying at infinity and b x is vanishing on the boundary with a more general rate function the vanishing rate of b x determines the exact blow up rate of the large solutions and the exact blow up rate allows us to obtain the uniqueness result

asymptotic behavior of entire large solutions to semilinear elliptic equations abstract in this paper by using karamata regular variation theory we study the exact asymptotic behavior of entire large solutions to u b x f u x r n n 3 where b c r n is nonnegative and nontrivial in r n f c 0 is positive and non decreasing on 0 which satisfies keller osserman condition and is rapidly varying or regularly varying at infinity with index 1

blow up rates of radially symmetric large solutions abstract this paper adapts a technical device going back to j lopez gomez optimal uniqueness theorems and exact blow up rates of large solutions j differential equations 224 2006 385 439 to ascertain the blow up rate of the unique radially symmetric large solution given through the main theorem of j lopez gomez uniqueness of radially symmetric large solutions discrete contin dyn syst supplement dedicated to the 6th aims conference poitiers france 2007 pp 677 686 the requested underlying estimates are based upon the main theorem of s cano casanova j lopez gomez existence uniqueness and blow up rate of large solutions for a canonical class of one dimensional problems on the half line j differential equations 244 2008 3180 3203 precisely we show that if is a ball or an annulus f c 0 is positive and non decreasing v c 0 c 2 0 satisfies v 0 0 v u 0 v u 0 for every u 0 and v u h u p 1 as u for some h 0 and p 1 then for each 0 u u f dist x v u u possesses a unique positive large solution in l which must be radially symmetric by uniqueness and we can estimate the exact blow up rate of l x at in terms of p h and f see theorem 1 1

a weyl group for the virasoro and n 1 super virasoro algebras we introduce a weyl group for the highest weight modules over the virasoro algebra and the neveu schwarz and ramond superalgebras using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical weyl character formula for the finite dimensional irreducible representations of semi simple lie algebras and also of the weyl kac character formula for the integrable highest weight modules over affine kac moody algebras this is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine kac moody algebraa 1 1

uniqueness and blow up rate of large solutions for elliptic equation δu λu b x h u abstract in this paper we establish the blow up rate of the large positive solution of the singular boundary value problem u u b x h u in u on where is a smooth bounded domain in r n the weight function b x is a non negative continuous function in the domain h u is locally lipschitz continuous and h u u is increasing on 0 and h u h u p for sufficiently large u with h 0 and p 1 naturally the blow up rate of the problem equals its blow up rate for the very special but important case when h u h u p we distinguish two cases i is a ball domain and b is a radially symmetric function on the domain in theorem 1 1 ii is a smooth bounded domain and b satisfies some local condition on each boundary normal section assumed in theorem 1 2 the blow up rate is explicitly determined by functions b and h in case i the singular boundary value problem has a unique solution u satisfying lim d x 0 u x k h b x x 0 1 where d x dist x b r and k are defined by b r r r s r b t d t d s k 1 c 0 1 1 p 1 1 p 1 in case ii the blow up rates of the solutions to the boundary value problem are established and the uniqueness is proved

pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs in this paper we analyze the uniqueness and the pointwise growth of the positive solutions of a nonlinear elliptic boundary value problem of general sublinear type with a weight function multiplying the nonlinearity when this function vanishes on some subdomain the problem exhibits a bifurcation from infinity in this case almost nothing is known about the pointwise growth of the positive solutions as the parameter approaches the critical value where the bifurcation from infinity occurs in this work we show that the positive solutions grow to infinity in the region where the weight function vanishes and that on its support they stabilize to the minimal positive solution of the original equation subject to infinite dirichlet boundary conditions this behavior provides us with the uniqueness of the positive solution near the value of the parameter where the bifurcation from infinity occurs also we solve the problem using spectral collocation methods coupled with path following techniques to show how the main uniqueness result is optimal throughout the paper the mathematical analysis aids the numerical study and the numerical study confirms and illuminates the analysis

finite dimensional representations of invariant differential operators ii abstract in part i of this paper g w schwarz finite dimensional representations of invariant differential operators j algebra 258 2002 160 204 we considered the representation theory of the algebra b d g g where g sl 3 c and d g g denotes the algebra of g invariant polynomial differential operators on the lie algebra g of g we also considered the representation theory of the subalgebra a of b where a is generated by the invariant functions o g g b and the invariant constant coefficient differential operators s g g b among other things we found that the finite dimensional representations of a and b are completely reducible and we could reduce the study of the finite dimensional irreducible representations of b to those of a irreducible finite dimensional representations of a are quotients of verma modules we found sufficient conditions for the irreducible quotients of verma modules to be finite dimensional and we conjectured that these sufficient conditions are also necessary in this paper we establish the conjecture giving a complete classification of the finite dimensional representations of a and b

categories and functors in the representation theory of lie algebras the fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring these include of course hom ext and tor as well as the more specialized examples of completion and localization used in the setting of representation theory of a semisimple lie algebra in this article we let a be a lie subalgebra of a lie algebra 0 and r be a functor on some category of a modules we then consider the following general question for a g module e what hypotheses on r and e are sufficient to insure that f e admits a canonical structure as a 0 module the article offers an answer through the introduction of the notion of s categories and a functors the last section of the article treats various examples of this theory

affine jacquet functors and harish chandra categories abstract we define an affine jacquet functor and use it to describe the structure of induced affine harish chandra modules at noncritical levels extending the theorem of kac and kazhdan on the structure of verma modules in the bernstein gelfand gelfand categories o for kac moody algebras this is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine harish chandra modules of lian and zuckerman the latter provides an extension of the works of rocha caridi and wallach a rocha caridi n r wallach projective modules over infinite dimensional graded lie algebras math z 180 1982 151 177 and deodhar gabber and kac v deodhar o gabber v kac structure of some categories of representations of infinite dimensional lie algebras adv math 45 1982 92 116 on block decompositions of bgg categories for kac moody algebras we also derive a compatibility relation between the affine jacquet functor and the kazhdan lusztig tensor product and apply it to prove that the affine harish chandra category is stable under fusion tensoring with the kazhdan lusztig category this compatibility will be further applied in studying translation functors for the affine harish chandra category based on the fusion tensor product

representation theory of neveu schwarz and ramond algebras i verma modules abstract in this article we study the structure of verma modules of n 1 super virasoro algebras as applications we construct bernstein gel fand gel fand type resolutions this article is the detailed and expanded version of iohara and koga c r acad sci paris ser i 328 1999 381

verma type modules of level zero for affine lie algebras we study the structure of verma type modules of level zero induced from non standard borel subalgebras of an affine kac moody algebra for such modules in general position we describe the unique irreducible quotients construct a bgg type resolution and prove the bgg duality in certain categories all results are extended to generalized verma type modules of zero level

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

ergodic theory on moduli spaces let m be a compact surface with x m 0 and let g be a compact lie group whose levi factor is a product of groups locally isomorphic to su 2 for example su 2 itself then the mapping class group rm of m acts on the moduli space x m of flat g bundles over m possibly twisted by a fixed central element of g when m is closed then fm preserves a symplectic structure on x m which has finite total volume on m more generally the subspace of x m corresponding to flat bundles with fixed behavior over am carries a rm invariant symplectic structure the main result is that rm acts ergodically on x m with respect to the measure induced by the symplectic structure

general uniqueness results and variation speed for blow up solutions of elliptic equations let omega be a smooth bounded domain in r n we prove general uniqueness results for equations of the form delta u au b x f u in omega subject to u infty on partial omega our uniqueness theorem is established in a setting involving karamata s theory on regularly varying functions which is used to relate the blow up behavior of u x with f u and b x where b equiv 0 on partial omega and a certain ratio involving b is bounded near partial omega a key step in our proof of uniqueness uses a modification of an iteration technique due to safonov

nonsymmetric koornwinder polynomials and duality in the fundamental work of lusztig l on affine hecke algebras a special role is played by the root system of type cn the affine hecke algebra is a deformation of the group algebra of an affine weyl group which usually depends on as many parameters as there are distinct root lengths i e one or two for an irreducible root system however in the cn case the hecke algebra h has three parameters corresponding to the fact that there is a simple coroot which is divisible by 2 recently cherednik c1 c3 has introduced the notion of a double affine hecke algebra and has used it to prove several conjectures on macdonald polynomials these polynomials and cherednik s double affine hecke algebra involve two or three parameters i e one more than the number of root lengths

skein modules and the noncommutative torus we prove that the kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus the proof is based on chebyshev polynomials as an application we describe the structure of the kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus and recover a result of hoste and przytycki about the skein module of a lens space we establish simple formulas for jones wenzl idempotents in the skein algebra of a cylinder over a torus and give a straightforward computation of the n th colored kauffman bracket of a torus knot evaluated in the plane or in an annulus

the exact boundary blow up rate of large solutions for semilinear elliptic problems in this paper we establish the blow up rate of the large positive solution of the singular boundary value problem u u a x up u where is a bounded smooth domain in rn the weight function a x in front of the nonlinearity can vanish on the boundary of the domain at different rates according to the point x0 of the boundary the decay rate of the weight function a x may not be approximated by a power function of distance near the boundary we combine the localization method of j lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 with some previous radially symmetric results of t ouyang z xie the uniqueness of blow up solution for radially symmetric semilinear elliptic equation nonlinear anal 64 9 2006 2129 2142 to prove that any large solution u x must satisfy limx x0u x k bx0 dist x 1for each x0 where bx0 r 0r 0sbx0 t dtds k 1 c0 1 1p 1 1p 1 c0 limr 0 0rbx0 t dt 2bx0 r bx0 r and bx0 r is the boundary normal section of a x at x0 i e bx0 r a x0 rnx0 r 0 r 0 and nx0 stands for the outward unit normal vector at x0

on arkhipov s and enright s functors we give a description of arkhipov s and joseph s and deodhar mathieu s versions of enright s endofunctors on the category open image in new window associated with a fixed triangular decomposition of a complex finite dimensional semi simple lie algebra in terms of co approximation functors with respect to suitably chosen injective resp projective modules we establish some new connections between these functors for example we show that arkhipov s and joseph s functors are adjoint to each other we also give several proofs of braid relations for arkhipov s and enright s functors

a chiral borel weil bott theorem abstract we compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold this is applied to 1 finding the character of g integrable irreducible highest weight modules over the affine lie algebra at the critical level and 2 computing a certain elliptic genus of the flag manifold the main tool is a result that interprets the drinfeld sokolov reduction as a derived functor

kac moody groups and integrability of soliton equations a new approach to integrability of affine toda field theories and closely related to them kdv hierarchies is proposed the flows of a hierarchy are explicity identified with infinitesimal action of the principal abelian subalgebra of the corresponding affine kac moody algebra on a homogeneous space

blow up rate and uniqueness of singular radial solutions for a class of quasi linear elliptic equations we establish the uniqueness and the blow up rate of the large positive solution of the quasi linear elliptic problem pu up 1 b x h u in br x0 with boundary condition u on br x0 where br x0 is a ball centered at x0 rn with radius r n 3 2 p 0 are constants and the weight function b is a positive radially symmetrical function we only require h u to be a locally lipschitz function with h u up 1 increasing on 0 and h u uq 1 for large u with q p 1 our results extend the previous work z xie uniqueness and blow up rate of large solutions for elliptic equation u u b x h u j differential equations 247 2009 344 363 from case p 2 to case 2 p

the thomas fermi von weizsacker theory of atoms and molecules we place the thomas fermi von weizsacker model of atoms on a firm mathematical footing we prove existence and uniqueness of solutions of the thomas fermi von weizsacker equation as well as the fact that they minimize the thomas fermi von weizsacker energy functional moreover we prove the existence of binding for two very dissimilar atoms in the frame of this model

quantum group structure in the fock space resolutions of widehat sl n representations we describe a complex of wakimoto type fock space modules for the affine kac moody algebra widehat sl n the intertwining operators that build the complex are obtained from contour integrals of so called screening operators we show that a quantum group structure underlies the algebra of screening operators this observation greatly facilitates the explicit determination of the intertwiners we conjecture that the complex provides a resolution of an irreducible highest weight module in terms of fock spaces

affine cubic surfaces and character varieties of knots it is known that the fundamental group homomorphism 1 t 2 1 s 3 k 1 t 2 1 s 3 k induced by the inclusion of the boundary torus into the complement of a knot k in s 3 s 3 is a complete knot invariant many classical invariants of knots arise from the natural restriction map induced by the above homomorphism on the sl 2 sl 2 character varieties of the corresponding fundamental groups in our earlier work 3 we proposed a conjecture that the classical restriction map admits a canonical deformation into a two parameter family of affine cubic surfaces in c 3 c 3 in this paper we show that modulo some mild technical conditions our conjecture follows from a known conjecture of brumfiel and hilden 1 on the algebraic structure of the peripheral system of a knot we then confirm the brumfiel hilden conjecture for an infinite class of knots including all torus knots 2 bridge knots and certain pretzel knots we also show the class of knots for which the brumfiel hilden conjecture holds is closed under taking connect sums and knot coverings

existence uniqueness and blow up rate of large solutions for a canonical class of one dimensional problems on the half line this paper shows the existence and the uniqueness of the positive solution l t of the singular boundary value problem u t f t h u t t 0 u 0 u 0 where f is a continuous non decreasing function such that f 0 0 and h is a non negative function satisfying the keller osserman condition moreover it also ascertains the exact blow up rate of l t at t 0 in the special case when there exist h 0 and p 1 such that h u hup for sufficiently large u naturally the blow up rate of the problem in such a case equals its blow up rate for the very special but important case when h u hup for all u 0 so our results are substantial improvements of some previous findings of j lopez gomez uniqueness of large solutions for a class of radially symmetric elliptic equations in s cano casanova j lopez gomez c mora corral eds spectral theory and nonlinear analysis with applications to spatial ecology world scientific 2005 pp 75 110 and j lopez gomez optimal uniqueness theorems and exact blow up rates of large solutions j differential equations 224 2006 385 439

generalised verma modules for the orthosymplectic lie superalgebra ospk 2 abstract the composition factors with their multiplicities and the submodule lattices are determined for generalised verma modules over the orthosymplectic lie superalgebra osp k 2 the results enable us to obtain explicit formulae for the formal characters and dimensions of the finite dimensional irreducible modules applying these results we also compute the first and second cohomology groups of the lie superalgebra with coefficients in finite dimensional kac modules and irreducible modules

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

bernstein gelfand gelfand resolutions for basic classical lie superalgebras abstract we study kostant cohomology and bernstein gelfand gelfand resolutions for finite dimensional representations of basic classical lie superalgebras for each choice of parabolic subalgebra and representation of such a lie superalgebra there is a natural definition of the boundary and coboundary operators which define co homology of the nilradical of the parabolic subalgebra we prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of generalised verma modules every such a resolution is then given by modules induced by these homology groups we also prove that if these homology groups are completely reducible a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the kostant quabla operator which is equivalent with disjointness of the boundary and coboundary operators then we use these results to derive very explicit conditions under which bgg resolutions exist which are particularly useful for the superalgebras of type i for the unitarisable representations of gl m n and osp 2 2 n we derive conditions on the parabolic subalgebra under which the bgg resolutions exist we also apply the obtained theory to construct specific examples of bgg resolutions for osp m 2 n finally we state some results for typical modules of gl m n and osp 2 2 n

cyclotomic double affine hecke algebras and affine parabolic category o using the orbifold kz connection we construct a functor from an affine parabolic category o of type a to the category o of a cyclotomic rational double affine hecke algebra h then we give several results concerning this functor

representations unitaires des super algebres de ramond et de neveu schwartz using r p langlands method we give an analytic proof of the classification theorem of the unitary representations of ramond and neveu schwartz superalgebras first proved by d friedan z qiu and s shenker by a numerical argument

representation theory of n 2 super virasoro algebra twisted sector abstract in this article we study the structure of verma modules and fock modules over the twisted sector of the n 2 super virasoro algebras

the semi infinite cohomology of affine lie algebras we study the semi infinite or brst cohomology of affine lie algebras in detail this cohomology is relevant in the brst approach to gauged wznw models our main result is to prove necessary and sufficient conditions on ghost numbers and weights for non trivial elements in the cohomology in particular we prove the existence of an infinite sequence of elements in the cohomology for non zero ghost numbers this will imply that the brst approach to the topological wznw model admits many more states than a conventional coset construction this conclusion also applies to some non topological models

characters of irreducible representations of the virasoro algebra on obtient les caracteres des modules irreductibles non integrables de plus haut poids l h 0 on determine les caracteres des familles l t 25 et l t 26 duales de l h 1 et de l h 0

the boundary blow up rate of large solutions abstract in this paper we ascertain the blow up rate of the large solutions of a class of sublinear elliptic boundary value problems with a weight function in front of the nonlinearity that vanishes on the boundary of the underlying domain at different rates according to the point of the boundary x all previous results in the literature assumed the decay rate of the underlying weight function to be the same at any point of this hypothesis substantially simplified the mathematical analysis of the problem as it allowed constructing global sub and supersolutions in an open neighborhood of obtaining general results requires localizing at each particular point of the boundary making particularly involved the mathematical analysis of the problem

the askey wilson algebra and its avatars the original askey wilson algebra introduced by zhedanov encodes the bispectrality properties of the eponym polynomials the name askey wilson algebra is currently used to refer to a variety of related structures that appear in a large number of contexts we review these versions sort them out and establish the relations between them we focus on two specific avatars the first is a quotient of the original zhedanov algebra it is shown to be invariant under the weyl group of type d 4 and to have a reflection algebra presentation the second is a universal analogue of the first one it is isomorphic to the kauffman bracket skein algebra kbsa of the four punctured sphere and to a subalgebra of the universal double affine hecke algebra c 1 vee c 1 this second algebra emerges from the racah problem of u q mathfrak sl 2 and is related via an injective homomorphism to the centralizer of u q mathfrak sl 2 in its threefold tensor product how the artin braid group acts on the incarnations of this second avatar through conjugation by r matrices in the racah problem or half dehn twists in the diagrammatic kbsa picture is also highlighted attempts at defining higher rank askey wilson algebras are briefly discussed and summarized in a diagrammatic fashion

category o for the schrodinger algebra we study category o for the centrally extended schrodinger algebra we determine the quivers for all blocks and relations for blocks of nonzero central charge we also describe the quiver and rel