the center of the enveloping algebra of the p lie algebras n n n when p divides n abstract let be a reductive lie algebra over an algebraically closed field f with char f p 0 suppose g satisfies jantzen s standard assumptions then the structure of z the center of the enveloping algebra is described by the extended veldkamp s theorem we examine here the deviation of z from this theorem in case or and p n it is shown that veldkamp s description is valid for this implies that friedlander parshall donkin decomposition theorem for holds in case p is good for a semi simple simply connected g excluding if p 2 a 1 factors of g in case or we prove a fiber product theorem for a polynomial extension of z however veldkamp s description mostly fails for and in particular z is not cohen macaulay if n 4 in both cases contrary to a result of kac weisfeiler we show for an odd prime p that and do not generate we also show for that the codimension of the non azumaya locus of z is at least 2 if n 3 and exceeds 2 if n 4 this refutes a conjecture of brown goodearl we then show that z is factorial excluding thus confirming a conjecture of premet tange we also verify humphreys conjecture on the parametrization of blocks in case p is good for g

complex geometry of moment angle manifolds moment angle manifolds provide a wide class of examples of non kahler compact complex manifolds a complex moment angle manifold mathcal z is constructed via certain combinatorial data called a complete simplicial fan in the case of rational fans the manifold mathcal z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori in general a complex moment angle manifold mathcal z is equipped with a canonical holomorphic foliation mathcal f which is equivariant with respect to the mathbb c times m action examples of moment angle manifolds include hopf manifolds of vaisman type calabi eckmann manifolds and their deformations we construct transversely kahler metrics on moment angle manifolds under some restriction on the combinatorial data we prove that any kahler submanifold or more generally a fujiki class mathcal c subvariety in such a moment angle manifold is contained in a leaf of the foliation mathcal f for a generic moment angle manifold mathcal z in its combinatorial class we prove that all subvarieties are moment angle manifolds of smaller dimension and there are only finitely many of them this implies in particular that the algebraic dimension of mathcal z is zero

simple lie algebras of small characteristic vi completion of the classification let l be a finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 it is proved in this paper that if the p envelope of adl in derl contains a torus of maximal dimension whose centralizer in adl acts nontriangulably on l then p 5 and l is isomorphic to one of the melikian algebras m m n in conjunction with a premet h strade simple lie algebras of small characteristic v the non melikian case j algebra 314 2007 664 692 theorem 1 2 this implies that up to isomorphism any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 is either classical or a filtered lie algebra of cartan type or a melikian algebra of characteristic 5 this result finally settles the classification problem for finite dimensional simple lie algebras over algebraically closed fields of characteristic 2 3

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

the hesselink stratification of nullcones and base change let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 we give a case free proof of lusztig s conjectures lusztig in transform groups 10 449 487 2005 on so called unipotent pieces this presents a uniform picture of the unipotent elements of g which can be viewed as an extension of the dynkin kostant theory but is valid without restriction on p we also obtain analogous results for the adjoint action of g on its lie algebra mathfrak g and the coadjoint action of g on mathfrak g

invariants of centralisers in positive characteristic let q be a simple algebraic group of type a or c over a field of good positive characteristic let x q lie q and consider the centraliser qx y q xy 0 we show that the invariant algebra s qx qx is generated by the pth power subalgebra and the mod p reduction of the characteristic zero invariant algebra the latter algebra is known to be polynomial 17 and we show that it remains so after reduction using a theory of symmetrisation in positive characteristic we prove the analogue of this result in the enveloping algebra where the p centre plays the role of the pth power subalgebra in zassenhaus foundational work 30 the invariant theory and representation theory of modular lie algebras were shown to be explicitly intertwined we exploit his theory to give a precise upper bound for the dimensions of simple qx modules an application to the geometry of the zassenhaus variety is given when g is of type a and g k p is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent e k the invariant algebra s pe ke is generated by the pth power subalgebra and s pe ke which is also shown to be polynomial

deformations of special legendrian submanifolds in sasaki einstein manifolds in this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed we apply this deformation theory to special legendrian submanifolds in sasaki einstein manifolds in general special legendrian deformations have the obstruction however we show that the deformation space of special legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types we also prove that any special legendrian submanifold admits smooth deformations which are not special legendrian deformations given by harmonic 1 forms

a modular analogue of morozov s theorem on maximal subalgebras of simple lie algebras abstract let g be a simple algebraic group over an algebraically closed field of characteristic p 0 and suppose that p is a very good prime for g in this paper we prove that any maximal lie subalgebra m of g lie g with rad m 0 has the form m lie p for some maximal parabolic subgroup p of g this means that morozov s theorem on maximal subalgebras is valid under mild assumptions on g we show that such assumptions are necessary by providing a counterexample to morozov s theorem for groups of type e 8 over fields of characteristic 5 our proof relies on the main results and methods of the classification theory of finite dimensional simple lie algebras over fields of prime characteristic

a new geometric construction of compact complex manifolds in any dimension we consider holomorphic linear foliations of dimension m of mathbb c n with n 2m fulfilling a so called weak hyperbolicity condition and equip the projectivization of the leaf space for the foliation restricted to an adequate open dense subset with a structure of compact complex manifold of dimension n m 1 we show that except for the limit case n 2m 1 where we obtain any complex torus of any dimension this construction gives non symplectic manifolds including the previous examples of hopf calabi eckmann haefliger linear case loeb nicolau linear case and lopez de medrano verjovsky we study some properties of these manifolds that is to say meromorphic functions holomorphic vector fields forms and submanifolds for each manifold we construct an analytic space of deformations of dimension m n m 1 and show that under some additional conditions it is universal lastly we give explicit examples of new compact complex manifolds in particular of connected sums of products of spheres and show the existence of a momentum like map which classifies these manifolds up to diffeomorphism

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

the zassenhaus variety of a reductive lie algebra in positive characteristic let g be the lie algebra of a connected reductive group g over an algebraically closed field k of characteristic p 0 let z be the centre of the universal enveloping algebra u u g of g its maximal spectrum is called the zassenhaus variety of g we show that under certain mild assumptions on g the field of fractions frac z of z is g equivariantly isomorphic to the function field of the dual space g with twisted g action in particular frac z is rational this confirms a conjecture of j alev furthermore we show that z is a unique factorisation domain confirming a conjecture of a braun and c hajarnavis recently a premet used the above result about frac z a result of colliot thelene kunyavskii popov and reichstein and reduction mod p arguments to show that the gelfand kirillov conjecture cannot hold for simple complex lie algebras that are not of type a c or g2

an analogue of the jacobson morozov theorem for lie algebras of reductive groups of good characteristics let g be the lie algebra of a connected reductive group g over an algebraically closed field of characteristic p 0 suppose that g 1 is simply connected and p is good for the root system of g given a one dimensional torus a c g let g a i denote the weight component of ad i corresponding to weight i e x i z it is proved in the paper that for any nonzero nilpotent element e e g there is a one dimentional torus ae c g such that e e g ae 2 and kerade c fi og ae i

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

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

commutative quotients of finite w algebras abstract in this paper we reduce the problem of 1 dimensional representations for the finite w algebras and humphreys conjecture on small representations of reduced enveloping algebras to the case of rigid nilpotent elements in exceptional lie algebras we use katsylo s results on sections of sheets to determine the krull dimension of the largest commutative quotient of the finite w algebra u g e

real quadrics in cn complex manifolds and convex polytopes in this paper we investigate the topology of a class of non kahler compact complex manifolds generalizing that of hopf and calabi eckmann manifolds these manifolds are diffeomorphic to special systems of real quadrics cn which are invariant with respect to the natural action of the real torus s1 n onto cn the quotient space is a simple convex polytope the problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes we prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich we also resolve an associated wall crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope finally as a nice consequence we obtain that affine non kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups contrasting with the kahler situation

complex manifolds with maximal torus actions we say that an effective action of a compact torus g on a connected smooth manifold m is maximal if there exists a point x m such that dimg dimgx dimm in this talk we give a one to one correspondence between the family of connected closed complex manifolds with maximal torus actions and the family of certain combinatorial objects which is a generalization of the correspondence between complete nonsingular toric varieties and complete nonsingular fans research institute for mathematical science kyoto university e mail address ishida kurims kyoto u ac jp date june 16 2014 1

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

nilpotent coadjoint orbits in small characteristic abstract we show that the numbers of nilpotent coadjoint orbits in the dual of exceptional lie algebra g 2 in characteristic 3 and in the dual of exceptional lie algebra f 4 in characteristic 2 are finite we determine the closure relation among nilpotent coadjoint orbits in the dual of lie algebras of type b c f 4 in characteristic 2 and in the dual of lie algebra of type g 2 in characteristic 3 in each case we give an explicit description of the nilpotent pieces in the dual defined in 2 which are in general unions of nilpotent coadjoint orbits coincide with the earlier case by case definition in 11 26 in the case of classical groups and have nice properties independent of the characteristic of the base field this completes the classification of nilpotent coadjoint orbits in the dual of lie algebras of reductive algebraic groups and the determination of closure relation among such orbits in all characteristic

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

non kahler manifolds and git quotients bosio generalized the construction by lopez de medrano verjovsky meersseman lvm of a family of non algebraic compact complex manifolds of any dimension we describe how to construct the generalized family from certain geometric invariant theory git quotients we show that bosio s generalization parallels exactly the extension from mumford s git to the more general git developed by bialynicki birula and wi cicka this point of view yields new results on the geometry of lvm and bosio s manifolds

lvmb manifolds and quotients of toric varieties in this article we study a class of manifolds introduced by bosio called mathrm lvmb manifolds we provide an interpretation of his construction in terms of quotient of toric manifolds by complex lie groups furthermore mathrm lvmb manifolds extend a class of manifolds obtained by meersseman called mathrm lvm manifolds and we give a characterization of these manifolds using our toric description finally we give an answer to a question asked by cupit foutou and zaffran