new algorithms for finding irreducible polynomials over finite fields we present a new algorithm for finding an irreducible polynomial of specified degree over a finite field our algorithm is deterministic and it runs in polynomial time for fields of small characteristic we in fact prove the stronger result that the problem of finding irreducible polynomials of specified degree over a finite field is deterministic polynomial time reducible to the problem of factoring polynomials over the prime field

on the finite w algebra for the lie superalgebra q n in the non regular case in this paper we study the finite w algebra for the queer lie superalgebra q n associated with the non regular even nilpotent coadjoint orbits in the case when n nl and the corresponding nilpotent element has jordan blocks each of size l we prove that this finite w algebra is isomorphic to a quotient of the super yangian of q n

centers of reduced enveloping algebras we compute the inverse image of a functional in the zassenhaus variety we apply this computation to describe the category of representations for a regular functional

a note on verma modules for finite w algebras abstract a finite w algebra u g e is a certain finitely generated algebra associated to a nilpotent element e of a complex reductive lie algebra g there is a loop filtration on u g e such that the associated graded algebra is isomorphic to u g e where g e is the centralizer of e in g in this short note we show that verma modules for finite w algebras as defined in brundan et al 2008 bgk are filtered deformations of verma modules for u g e

on symmetric invariants of centralisers in reductive lie algebras let g be a finite dimensional simple lie algebra of rank l over an algebraically closed field of characteristic 0 let e be a nilpotent element of g and let ge be the centraliser of e in g in this paper we study the algebra s ge ge of symmetric invariants of ge we prove that if g is of type a or c then s ge ge is always a graded polynomial algebra in l variables and we show that this continues to hold for some nilpotent elements in the lie algebras of other types in type a we prove that the invariant algebra s ge ge is freely generated by a regular sequence in s ge and describe the tangent cone at e to the nilpotent variety of g

a class of algebras similar to the enveloping algebra of sl 2 fix f e c x define r c a b h subject to the relations ha ah a hb bh b ab ba f h we study these algebras for different f and in particular show how they are similar to and different from u sl 2 the enveloping algebra of sl 2 c there is a notion of highest weight modules and a category a for such r for each n 0 if f x x 1 n l xn 1 then r has precisely n simple modules in each finite dimension and every finite dimensional r module is semisimple

whittaker coinvariants for gl m n abstract let w m n be the finite w algebra attached to the principal nilpotent orbit in the general linear lie superalgebra gl m n c in this paper we study the whittaker coinvariants functor which is an exact functor from category o for gl m n c to a certain category of finite dimensional modules over w m n we show that this functor has properties similar to soergel s functor v in the setting of category o for a semisimple lie algebra we also use it to compute the center of w m n explicitly and deduce consequences for the classification of blocks of o up to morita derived equivalence

weight distribution and list decoding size of reed muller codes the weight distribution and list decoding size of reed muller codes are studied in this work given a weight parameter we are interested in bounding the number of reed muller codewords with weight up to the given parameter and given a received word and a distance parameter we are interested in bounding the size of the list of reed muller codewords that are within that distance from the received word obtaining tight bounds for the weight distribution of reed muller codes has been a long standing open problem in coding theory dating back to 1976 in this work we make a new connection between computer science techniques used to study low degree polynomials and these coding theory questions this allows us to resolve the weight distribution and list decoding size of reed muller codes for all distances previous results could only handle bounded distances azumi kasami and tokura gave bounds on the weight distribution which hold up to 2 5 times the minimal distance of the code and gopalan klivans and zuckerman gave bounds on the list decoding size which hold up to the johnson bound

a simple deterministic reduction for the gap minimum distance of code problem we present a simple deterministic gap preserving reduction from sat to the minimum distance of code problem over f 2 we also show how to extend the reduction to work over any fixed finite field previously a randomized reduction was known due to dumer micciancio and sudan which was recently derandomized by cheng and wan these reductions rely on highly nontrivial coding theoretic constructions whereas our reduction is elementary as an additional feature our reduction gives hardness within a constant factor even for asymptotically good codes i e having constant positive rate and relative distance previously it was not known how to achieve a deterministic reduction for such codes

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

approximation resistance from pairwise independent subgroups we show optimal up to a constant factor np hardness for a maximum constraint satisfaction problem with k variables per constraint max kcsp whenever k is larger than the domain size this follows from our main result concerning csps given by a predicate a csp is approximation resistant if its predicate contains a subgroup that is balanced pairwise independent our main result is analogous to austrin and mossel s bypassing their unique games conjecture assumption whenever the predicate is an abelian subgroup our main ingredient is a new gap amplification technique inspired by xor lemmas using this technique we also improve the np hardness of approximating independent set on bounded degree graphs almost coloring label cover and various other problems

on the variety of 1 dimensional representations of finite w algebras in low rank abstract let g be a simple lie algebra over c and let e g be nilpotent we consider the finite w algebra u g e associated to e and the problem of determining the variety e g e of 1 dimensional representations of u g e for g of low rank we report on computer calculations that have been used to determine the structure of e g e and the action of the component group e of the centralizer of e on e g e as a consequence we provide two examples where the nilpotent orbit of e is induced but there is a 1 dimensional e stable u g e module which is not induced via losev s parabolic induction functor in turn this gives examples where there is a non induced multiplicity free primitive ideal of u g

isomorphism problems of noncommutative deformations of type d kleinian singularities we construct all possible noncommutative deformations of a kleinian singularity c 2 gamma of type d n in terms of generators and relations and solve the isomorphism problem for the associative algebras thus constructed we prove that in our parametrization all isomorphisms arise from the action of the normalizer n sl 2 gamma on c gamma we deduce that the moduli space of isomorphism classes of noncommutative deformations in type d n is isomorphic to a vector space of dimension n

a morita theorem for modular finite w algebras we consider the lie algebra mathfrak g of a simple simply connected algebraic group over a field of large positive characteristic for each nilpotent orbit mathcal o subseteq mathfrak g we choose a representative e in mathcal o and attach a certain filtered associative algebra widehat u mathfrak g e known as a finite w algebra defined to be the opposite endomorphism ring of the generalised gelfand graev module associated to mathfrak g e this is shown to be morita equivalent to a certain central reduction of the enveloping algebra of u mathfrak g the result may be seen as a modular version of skryabin s equivalence

pseudorandom bits for polynomials we present a new approach to constructing pseudorandom generators that fool low degree polynomials over finite fields based on the gowers norm using this approach we obtain the following main constructions of explicitly computable generators g mathbb f s to mathbb f n that fool polynomials over a finite field mathbb f we stress that the results in 1 and 2 are unconditional i e do not rely on any unproven assumption moreover the results in 3 rely on a special case of the conjecture which may be easier to prove our generator for degree d polynomials is the componentwise sum of d generators for degree 1 polynomials on independent seeds prior to our work generators with logarithmic seed length were only known for degree 1 i e linear polynomials j naor and m naor siam j comput 22 1993 pp 838 856 in fact over small fields such as mathbb f 2 0 1 our results constitute the first progress on these problems since the long standing generator by luby velickovic and wigderson deterministic approximate counting of depth 2 circuits in proceedings of the 2nd israeli symposium on theoretical computer science istcs 1993 pp 18 24 whose seed length is much bigger s exp left omega left sqrt log n right right even for the case of degree 2 polynomials over mathbb f 2

quantized symplectic actions and w algebras with a nilpotent element in a semisimple lie algebra g one associates a finitely generated associative algebra w called a w algebra of finite type this algebra is obtained from the universal enveloping algebra u g by a certain hamiltonian reduction we observe that w is the invariant algebra for an action of a reductive group g with lie algebra g on a quantized symplectic affine variety and use this observation to study w our results include an alternative definition of w a relation between the sets of prime ideals of w and of the corresponding universal enveloping algebra the existence of a one dimensional representation of w in the case of classical g and the separation of elements of w by finite dimensional representations

conjugacy classes in weyl groups and q w algebras abstract we define noncommutative deformations w q s g of algebras of regular functions on certain transversal slices to the set of conjugacy classes in an algebraic group g which play the role of slodowy slices in algebraic group theory the algebras w q s g called q w algebras are labeled by conjugacy classes of elements s of the weyl group of g the algebra w q s g is a quantization of a poisson structure defined on the corresponding transversal slice in g with the help of poisson reduction of a poisson bracket associated to a poisson lie group g dual to a quasitriangular poisson lie group we also define a quantum group counterpart of the category of generalized gelfand graev representations and establish an equivalence between this category and the category of representations of the corresponding q w algebra the algebras w q s g can be regarded as quantum group counterparts of w algebras however in general they are not deformations of the usual w algebras

noncommutative deformations of type a kleinian singularities abstract the type a kleinian singularities are the surfaces of the form c 2 g where g is a finite cyclic subgroup of sl2 c the standard noncommutative analog is the fixed ring of the weyl algebra a1 c under the induced action of g these fixed rings are however only part of a large family of algebras whose associated graded ring is the coordinate ring of such a singularity the structure of these algebras is studied with reference to the structure of the associated singularity and to the structure of primitive factor rings of u s l 2 c

structure of classical affine and classical affine fractional w algebras we introduce a classical brst complex see definition 3 2 and show that one can construct a classical affine w algebra via the complex this definition clarifies that classical affine w algebras can be considered as quasi classical limits of quantum affine w algebras we also give a definition of a classical affine fractional w algebra as a poisson vertex algebra as in the classical affine case a classical affine fractional w algebra has two compatible brackets and is isomorphic to an algebra of differential polynomials as a differential algebra when a classical affine fractional w algebra is associated to a minimal nilpotent we describe explicit forms of free generators and compute brackets between them provided some assumptions on a classical affine fractional w algebra we find an infinite sequence of integrable systems related to the algebra using the generalized drinfel d and sokolov reduction

1 dimensional representations and parabolic induction for w algebras a w algebra is an associative algebra constructed from a reductive lie algebra and its nilpotent element this paper concentrates on the study of 1 dimensional representations of w algebras under some conditions on a nilpotent element satisfied by all rigid elements we obtain a criterium for a finite dimensional module to have dimension 1 it is stated in terms of the brundan goodwin kleshchev highest weight theory this criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras we make an explicit computation in a special case in type e8 our second principal result is a version of a parabolic induction for w algebras in this case the parabolic induction is an exact functor between the categories of finite dimensional modules for two different w algebras the most important feature of the functor is that it preserves dimensions in particular it preserves one dimensional representations a closely related result was obtained previously by premet we also establish some other properties of the parabolic induction functor

finite w algebras for gln abstract we study the quantum finite w algebras w gl n f associated to the lie algebra gl n and its arbitrary nilpotent element f we construct for such an algebra an r 1 r 1 matrix l z of yangian type where r 1 is the number of maximal parts of the partition corresponding to f the matrix l z is the quantum finite analogue of the operator of adler type which we introduced in the classical affine setup as in the latter case the matrix l z is obtained as a generalized quasideterminant it should encode the whole structure of w gl n f including explicit formulas for generators and the commutation relations among them we describe in all detail the examples of principal rectangular and minimal nilpotent elements

finite w superalgebras for basic lie superalgebras abstract we consider the finite w superalgebra u g f e for a basic lie superalgebra g f g f 0 g f 1 associated with a nilpotent element e g f 0 both over the field of complex numbers f c and over f k an algebraically closed field of positive characteristic in this paper we mainly present the pbw theorem for u g f e then the construction of u g f e can be understood well which in contrast with finite w algebras is divided into two cases in virtue of the parity of dim g f 1 1 this observation will be a basis of our sequent work on the dimensional lower bounds in the super kac weisfeiler property of modular representations of basic lie superalgebras cf 43 7 9

dimensions of irreducible modules over w algebras and goldie ranks the main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple lie algebra the first one very classical is the goldie rank of an ideal the second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite w algebra we concentrate on the integral central character case we prove modulo a conjecture that in this case the two are equal our conjecture asserts that there is a one dimensional module over the w algebra with certain additional properties the conjecture is proved for the classical types also modulo the same conjecture we compute certain scale factors introduced by joseph this allows to compute the goldie ranks of the algebras of locally finite endomorphisms of simples in the bgg category mathcal o this completes a program of computing goldie ranks proposed by joseph in the 80 s for integral central characters and modulo our conjecture we also provide an essentially kazhdan lusztig type formula for computing the characters of the irreducibles in the brundan goodwin kleshchev category mathcal o for a w algebra again under the assumption that the central character is integral in particular this allows to compute the dimensions of the finite dimensional irreducible modules the formula is based on a certain functor from an appropriate parabolic category mathcal o to the w algebra category mathcal o this functor can be regarded as a generalization of functors previously constructed by soergel and by brundan kleshchev we prove a number of properties of this functor including the quotient property and the double centralizer property we develop several side topics related to our generalized soergel functor for example we discuss its analog for the category of harish chandra modules we also discuss generalizations to the case of categories mathcal o over dixmier algebras the most interesting example of this situation comes from the theory of quantum groups we prove that an algebra that is a mild quotient of luszitg s form of a quantum group at a root of unity is a dixmier algebra for this we check that the quantum frobenius epimorphism splits

shifted yangians and finite w algebras we give a presentation for the finite w algebra associated to a nilpotent matrix in the general linear lie algebra over c in the special case that the nilpotent matrix consists of n jordan blocks each of the same size l the presentation is that of the yangian of level l associated to gln as was first observed by ragoucy and sorba in the general case we are lead to introduce some generalizations of the yangian which we call the shifted yangians

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

bounded independence fools halfspaces we show that any distribution on 1 1 n that is k wise independent fools any halfspace or linear threshold function h 1 1 n to 1 1 i e any function of the form h x operatorname sign sum i 1 n w i x i theta where the w 1 dots w n and theta are arbitrary real numbers with error epsilon for k o epsilon 2 log 2 1 epsilon our result is tight up to log 1 epsilon factors using standard constructions of k wise independent distributions we obtain the first explicit pseudorandom generators g 1 1 s to 1 1 n that fool halfspaces specifically we fool halfspaces with error epsilon and seed length s k cdot log n o log n cdot epsilon 2 log 2 1 epsilon our approach combines classical tools from real approximation theory with structural results on halfspaces by servedio comput complexity 16 2007 pp 180 209

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

block representation type of reduced enveloping algebras let k be an algebraically closed field of characteristic p g a connected reductive k group g lie g x e g and u x g the reduced enveloping algebra of g associated with x assume that g 1 is simply connected p is good for g and g has a non degenerate g invariant bilinear form all blocks of u x g having finite and tame representation type are determined

good grading polytopes in a recent paper of bennett and the author it was shown that the elliptic curve defined by y 2 x 3 ax b where a and b are integers has no rational points of finite order if a is sufficiently large relative to b at least if one assumes the abc conjecture of masser and oesterle in the present article we show perhaps surprisingly that the rational torsion on the above curve is also quite restricted if b is sufficiently large relative to a in particular we demonstrate that for any epsilon 0 there is a constant c epsilon such that if a and b are integers satisfying b c epsilon a 6 epsilon then the elliptic curve defined above has no rational torsion points other than a possible point of order 2 again making use of the abc conjecture in some cases we then extend this by proving similar results for elliptic curves admitting non trivial mathbb q isogenies elliptic curves written in other forms and elliptic curves over certain number fields curiously the results on isogenies lead to two unexpected irrationality measures for certain algebraic numbers

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

classical affine w algebras associated to lie superalgebras in this paper we prove classical affine w algebras associated to lie superalgebras w superalgebras which can be constructed in two different ways via affine classical hamiltonian reductions and via taking quasi classical limits of quantum affine w superalgebras also we show that a classical finite w superalgebra can be obtained by a zhu algebra of a classical affine w superalgebra using the definition by hamiltonian reductions we find free generators of a classical w superalgebra associated to a minimal nilpotent moreover we compute generators of the classical w algebra associated to spo 2 3 and its principal nilpotent in the last part of this paper we introduce a generalization of classical affine w superalgebras called classical affine fractional w superalgebras we show these have poisson vertex algebra structures and find generators of a fractional w superalgebra associated to a minimal nilpotent

special transverse slices and their enveloping algebras let g be a simple simply connected algebraic group over image image lie g image image the nilpotent cone of image and e h f an imageimage2 triple in image let s e ker ad f the special transverse slice to the adjoint orbit of e and s0 s image image the coordinate ring image s0 is naturally graded see slodowy simple singularities and simple algebraic groups lecture notes in mathematics vol 815 springer verlag berlin heidelberg new york 1980 let z image be the centre of the enveloping algebra u image and z image image an algebra homomorphism identify image with image via a killing isomorphism and let denote the linear function on image corresponding to e following kawanaka generalized gelfand graev representations and ennola duality in algebraic groups and related topics advanced studies in pure mathematics vol 6 pp 175 206 north holland amsterdam new york oxford 1985 moeglin c r acad sci paris ser i 303 no 17 1986 845 848 and premet invent math 121 1995 79 117 we attach to a nilpotent subalgebra image subset ofimage of dimension dim 2 and a 1 dimensional image module image let imageh height 14 width 14 denote the algebra opposite to endimage u image multiply sign in circleu image image and imageh height 14 width 14 imageh height 14 width 14 multiply sign in circlez image image it is proved in the paper that the algebra imageh height 14 width 14 has a natural filtration such that gr imageh height 14 width 14 the associated graded algebra is isomorphic to image s0 this construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of image

finite dimensional representations of w algebras w algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple lie algebras in this paper we prove a conjecture of premet that gives an almost complete classification of finite dimensional irreducible modules for w algebras also we study a relation between harish chandra bimodules and bimodules over w 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 principal finite w algebras for the lie superalgebra d 2 1 α we study finite w algebras associated to even regular principal nilpotent elements for the family of simple exceptional lie superalgebras d 2 1 and for the universal central extension of x1d52d x1d530 x1d529 2 2 we give a complete description of these finite w algebras in terms of generators and relations

transfer results for frobenius extensions abstract we study frobenius extensions which are free filtered by a totally ordered finitely generated abelian group and their free graded counterparts first we show that the frobenius property passes up from a free graded extension to a free filtered extension then also from a free filtered extension to the extension of their rees algebras our main theorem states that under some natural hypotheses a free filtered extension of algebras is frobenius if and only if the associated graded extension is frobenius in the final section we apply this theorem to provide new examples and non examples of frobenius extensions

finite dimensional irreducible representations of finite w algebras associated to even multiplicity nilpotent orbits in classical lie algebras we consider finite w algebras u mathfrak g e associated to even multiplicity nilpotent elements in classical lie algebras we give a classification of finite dimensional irreducible u mathfrak g e modules with integral central character in terms of the highest weight theory from brundan et al int math res notices 15 art id rnn051 2008 as a corollary we obtain a parametrization of primitive ideals of u mathfrak g with associated variety the closure of the adjoint orbit of e and integral central character

reed muller codes for random erasures and errors this paper studies the parameters for which binary reed muller rm codes can be decoded successfully on the binary erasure channel and binary symmetry channel and in particular when can they achieve capacity for these two classical channels necessarily this paper also studies the properties of evaluations of multivariate gf 2 polynomials on the random sets of inputs for erasures we prove that rm codes achieve capacity both for very high rate and very low rate regimes for errors we prove that rm codes achieve capacity for very low rate regimes and for very high rates we show that they can uniquely decode at about the square root of the number of errors at capacity the proofs of these four results are based on different techniques which we find interesting in their own right in particular we study the following questions about e m r the matrix whose rows are the truth tables of all the monomials of degree leq r in m variables what is the most resp least number of random columns in e m r that define a submatrix having full column rank resp full row rank with high probability we obtain tight bounds for very small resp very large degrees r which we use to show that rm codes achieve capacity for erasures in these regimes our decoding from random errors follows from the following novel reduction for every linear code c of sufficiently high rate we construct a new code c obtained by tensorizing c such that for every subset s of coordinates if c can recover from erasures in s then c can recover from errors in s specializing this to the rm codes and using our results for erasures imply our result on the unique decoding of the rm codes at high rate finally two of our capacity achieving results require tight bounds on the weight distribution of rm codes we obtain such bounds extending the recent bounds from constant degree to linear degree polynomials

representations of quantum groups at roots of unity whittaker vectors and q w algebras abstract let u e g be the standard simply connected version of the drinfeld jumbo quantum group at an odd m th root of unity e the center of u e g contains a huge commutative subalgebra isomorphic to the algebra z g of regular functions on a finite covering of a big cell in a complex connected simply connected algebraic group g with lie algebra g let v be a finite dimensional representation of u e g on which z g acts according to a non trivial character g given by evaluation of regular functions at g g then v is a representation of the finite dimensional algebra u g u e g u e g ker g we show that in this case under certain restrictions on m u g contains a subalgebra u g m of dimension m 1 2 dim o where o is the conjugacy class of g and u g m has a one dimensional representation c g we also prove that if v is not trivial then the space of whittaker vectors hom u g m c g v is not trivial and the algebra w g end u g u g u g m c g naturally acts on it which gives rise to a schur type duality between representations of the algebra u g and of the algebra w g called a q w algebra

isomorphisms of quantizations via quantization of resolutions abstract in this paper we prove the existence of isomorphisms between certain non commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain poisson algebras we show that quantizations of kleinian singularities obtained by three different constructions are isomorphic to each other the constructions are via symplectic reflection algebras quantum hamiltonian reduction and w algebras next we prove that parabolic w algebras in type a are isomorphic to quantum hamiltonian reductions associated to quivers of type a finally we show that the symplectic reflection algebras for wreath products of the symmetric group and a kleinian group are isomorphic to certain quantum hamiltonian reductions our results involving w algebras are new while for those dealing with symplectic reflection algebras we just find new proofs a key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate poisson varieties

quantization and representation theory of finite w algebras in this paper we study the finitely generated algebras underlyingw algebras these so called finitew algebras are constructed as poisson reductions of kirillov poisson structures on simple lie algebras the inequivalent reductions are labeled by the inequivalent embeddings ofsl 2 into the simple lie algebra in question for arbitrary embeddings a coordinate free formula for the reduced poisson structure is derived we also prove that any finitew algebra can be embedded into the kirillov poisson algebra of a semi simple lie algebra generalized miura map furthermore it is shown that generalized finite toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finitew symmetry in the second part we brst quantize the finitew algebras the brst cohomology is calculated using a spectral sequence which is different from the one used by feigin and frenkel this allows us to quantize all finitew algebras in one stroke examples are given in the last part of the paper we study the representation theory of finitew algebras it is shown using a quantum version of the generalized miura transformation that the representations of finitew algebras can be constructed from the representations of a certain lie subalgebra of the original simple lie algebra as a byproduct of this we are able to construct the fock realizations of arbitrary finitew algebras

modular representation theory of lie algebras on etudie les representations non necessairement restreintes de l algebre universelle enveloppante u 7b g pour une algebre de lie restreinte 7b g

representation theory of rectangular finite w algebras abstract we classify the finite dimensional irreducible representations of rectangular finite w algebras i e the finite w algebras u g e where g is a symplectic or orthogonal lie algebra and e g is a nilpotent element with jordan blocks all the same size

generalized gelfand graev representations of exceptional simple algebraic groups over a finite field i introduction let ffi be an adjoint simple algebraic group of type e n 6 7 8 f 4 or g2 over an algebraically closed field k of characteristic p 0 we assume that 5 has a fixed fq rational structure for a finite subfield fq of k the purpose of this paper is to determine explicitly the values of the irreducible characters of g ffi fq at the unipotent elements when p is good 38 i 4 3 in this part i we shall do this only for large p and the full result will be proved in the forthcoming part ii after submitting the first version of the present paper the author was informed by lusztig that he solved analogous problems for classical groups using his theory of character sheaves and that he can also recover the result of this paper mentioned above our method depends upon 1 a parametrization of the irreducible characters of g and the determination of their multiplicities in deligne lusztig virtual representations lusztig 25 2 the computation of the green functions of g in large characteristic springer 36 7 1 shoji 33 beynon and spaltenstein 3 4 3 the determination of the values of generalized gelfand graev characters 20 of g in good characteristic see section 3 by 1 and 2 the uniform parts of the irreducible characters are already known explicitly in large characteristic this implies that in large characteristic our problem for groups of type e 6 o r e 7 has already been solved it should be mentioned here that the whole character table of the g2 group is also known by chang and ree 8 hence in the g2 case this paper gives just another approach to a part of the result of chang and ree if g is of type g2 f4 or e 8 the restriction of an irreducible character r of g to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

super formal darboux weinstein theorem and finite w superalgebras abstract let v v 0 v 1 be a z 2 graded super vector space with an even c action and v 0 be a fixed point of the induced action in this paper we prove an equivariant darboux weinstein theorem for the formal polynomial algebras a s v 0 v 1 we also give a quantum version of it let g g 0 g 1 be a basic lie superalgebra and e g 0 be a nilpotent element we use the equivariant quantum darboux weinstein theorem to give a poisson geometric realization of the finite w superalgebra u g e in the sense of losev an indirect relation between finite w super algebras u g e and u g 0 e is presented finally we use such a realization to study the finite dimensional irreducible modules over u g e

localization of affine w algebras we introduce the notion of an asymptotic algebra of chiral differential operators we then construct via a chiral hamiltonian reduction one such algebra over a resolution of the intersection of the slodowy slice with the nilpotent cone we compute the space of global sections of this algebra thereby proving a localization theorem for affine w algebras at the critical level

localization of quantum biequivariant ɗ modules and q w algebras the author is grateful to y kremnizer for useful discussions and to the referee for careful reading of the manuscript

pseudorandom bits for constant depth circuits with few arbitrary symmetric gates we exhibit an explicitly computable pseudorandom generator stretching l bits into m l l omega log l bits that look random to constant depth circuits of size m l with log m l arbitrary symmetric gates e g parity majority this improves on a generator by luby velickovic and wigderson proceedings of the second israel symposium on theory of computing systems 1993 pp 18 24 that achieves the same stretch but fools only circuits of depth 2 with one arbitrary symmetric gate at the top our generator fools a strictly richer class of circuits than nisan s generator for constant depth circuits but nisan s generator has a much bigger stretch combinatorica 11 1991 pp 63 70 in particular we conclude that every function computable by uniform poly n size probabilistic constant depth circuits with o log n arbitrary symmetric gates is in mathit time 2 n o 1 this seems to be the richest probabilistic circuit class known to admit a subexponential derandomization our generator is obtained by constructing an explicit function f zo n to zo that is very hard on average for constant depth circuits of size s n n omega log n with log s n arbitrary symmetric gates and plugging it into the nisan wigderson pseudorandom generator construction j comput system sci 49 1994 pp 149 167 the proof of the average case hardness of this function is a modification of arguments by razborov and wigderson inform process lett 45 1993 pp 303 307 and hansen and miltersen proceedings of the 29th international symposium on mathematical foundations of computer science lecture notes in comput sci 3153 springer verlag berlin 2004 pp 334 345 and combines harstad s switching lemma computational limitations of small depth circuits mit press cambridge ma 1987 with a multiparty communication complexity lower bound by babai nisan and szegedy j comput system sci 45 1992 pp 204 232

on kostant s theorem for the lie superalgebra q n abstract in this paper we study finite w algebras for basic superalgebras and q n associated with the regular even nilpotent coadjoint orbits we prove that this algebra satisfies the amitsur levitzki identity and therefore all its irreducible representations are finite dimensional in the case of q n we give an explicit description of the w algebra in terms of generators and relations and realize it as a quotient of the super yangian of q 1