generalized restricted lie algebras and representations of the zassenhaus algebra abstract letfbe a field of characteristicp 0 la generalized restricted lie algebra overf andp l the primitivep envelope ofl a close relation betweenl representations andp l representations is established in particular the irreducible reduced modules oflfor any l coincide with the irreducible 0 reduced modules ofp l where 0 p l is a trivial extension of from this result the determination of all irreducible representations of the zassenhaus algebra is completed and the dimensions of the corresponding modules are also given

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

bounds on the unstable eigenvalue for the asymmetric renormalization operator for period doubling we establish rigorous bounds for the unstable eigenvalue of the period doubling renormalization operator for asymmetric unimodal maps herglotz function techniques and cone invariance ideas are used our result generalizes an established result for conventional period doubling

restricting invariants of unitary reflection groups suppose that g is a finite unitary reflection group acting on a complex vector space v and x is the fixed point subspace of an element of g define n to be the setwise stabilizer of x in g z to be the pointwise stabilizer and c n z then restriction defines a homomorphism from the algebra of g invariant polynomial functions on v to the algebra of c invariant functions on x extending earlier work by douglass and rohrle for coxeter groups we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups g in terms of the exponents of g and c and their reflection arrangements a consequence of our main result is that the variety of g orbits in the g saturation of x is smooth if and only if it is normal

existence and properties of p tupling fixed points we prove the existence of fixed points of p tupling renormalization operators for interval and circle mappings having a critical point of arbitrary real degree r 1 some properties of the resulting maps are studied analyticity univalence behavior as r tends to infinity

representations of hopf algebras arising fromlie algebras of cartan type c w x 1 1 this is a continuation of 13 in an effort to use algebraic group actions to study the representations of restricted lie algebras of cartan type in this paper we study the representation theory of certain finite diw x mensional hopf algebras that arise from the construction in 13 the representation theory of these hopf algebras closely resembles the representation theory of frobenius kernels of reductive algebraic groups in positive characteristics let g be a connected reductive algebraic group scheme over an algebraically closed field k of characteristic p 0 it is well known that if f g a g is the r th iteration of the frobenius map and g s ker f r r there is a chain of normal subgroup schemes g eg e eg in earlier 1 2 terminology this is equivalent to the existence of a chain of finite dimen sional cocommutative hopf algebras dist g dist g 1 2 dist g where dist g is known as the r th hyperalgebra of the distribur tion algebra dist g of g let g be the lie algebra of g and u g be its restricted universal enveloping algebra there exists an isomorphism of

reflection subgroups of finite complex reflection groups abstract all simple extensions of the reflection subgroups of a finite complex reflection group g are determined up to conjugacy as a consequence it is proved that if the rank of g is n and if g can be generated by n reflections then for every set r of n reflections which generate g every subset of r generates a parabolic subgroup of g

new proofs of the existence of the feigenbaum functions a new proof of the existence of analytic unimodal solutions of the cvitanovic feigenbaum functional equation g x g g x g x 1 const x r at 0 valid for all in 0 1 is given and the existence of the eckmann wittwer functions 8 is recovered the method also provides the existence of solutions for certain given values ofr and in particular forr 2 a proof requiring no computer

on the construction of highly symmetric tight frames and complex polytopes abstract many highly symmetric configurations of vectors in c d such as the vertices of the platonic solids and the regular complex polytopes are equal norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group for nonabelian groups there are uncountably many such tight frames up to unitary equivalence the aim of this paper is to single out those orbits which are particularly nice such as those which are the vertices of a complex polytope this is done by defining a finite class of tight frames of n vectors for c d n and d fixed which we call the highly symmetric tight frames we outline how these frames can be calculated from the representations of abstract groups using a computer algebra package we give numerous examples with a special emphasis on those obtained from the shephard todd finite reflection groups the interrelationships between these frames with complex polytopes harmonic frames equiangular tight frames and heisenberg frames maximal sets of equiangular lines are explored in detail

bounds on the unstable eigenvalue for period doubling bounds are given for the unstable eigenvalue of the period doubling operator for unimodal maps of the interval these bounds hold for all types of behaviour x r of the interval map near its critical point they are obtained by finding cones in function space which are invariant under the tangent map to the doubling operator at its fixed point

feigenbaum theory for unimodal maps with asymmetric critical point rigorous results we apply the methods of h epstein to prove the existence of a line of period 2 solutions of the feigenbaum period doubling renormalisation transformation these solutions govern the universal behaviour of families of unimodal maps with asymmetric critical points of degree d for which the d th derivative has differing left and right limits

arrangements defined by unitary reflection groups let v be a complex vector space of dimension i an arrangement in v is a finite set d of hyperplanes all containing the origin let l l d be the set of intersections of elements of r partially order l by reverse inclusion so that l has v as its minimal element and d as its set of atoms the poset l is a finite geometric lattice with rank function r x dim v x xe l without loss of generality we assume that n h h 0 is the maximal element of l and thus l has rank i the characteristic polynomial z l t of l is defined by

the generalized restricted representations of graded lie algebras of cartan type abstract let f be an algebraically closed field of characteristic p 2 in this paper the concepts of generalized restricted lie algebras and their generalized restricted representations over f are introduced any graded lie algebra of cartan type l x m n 2 x w s h or k is a generalized restricted lie algebra associated with a standard basis e of l and s a positive integer max 1 i m n i the irreducible generalized restricted representations of l are determined which are in a one to one correspondence with p sd weight functions here d is the dimension of the standard torus in l 1 x k e m 0 the descriptions of the blocks and a brauer type reciprocity are obtained

bgg algebras and the bgg reciprocity principle the bgg reciprocity principle is a famous result in the representation theary of complex semisimple lie algebras proved by bernstein gelfand and gelfand in bgg 21 it may be viewed as an analogue for the category 6 of the brauer reciprocity principle in the modular representation theory of finite groups and of humphreys reciprocity principle in the representation theory of restricted lie algebras indeed the result of humphreys served as motivation for the bgg result similar results have since been proved in a number of other settings by various approaches for all these results one works in a suitable category of modules in which each simple module l has a projective cover p in addition there is a family of intermediate modules mi such that each p has a filtration with m s as quotients then the appropriate reciprocity principle states that m occurs as a filtration quotient in p as often as l occurs as a composition factor in mi in the case of the category 0 the simple modules are the simple highest weight modules the category is introduced as a setting for projective covers to exist and the intermediate modules are the verma modules the proof of bgg reciprocity in bgg 21 is fairly straightforward once the basic facts about the category 0 are developed the crucial point is the construction of enough projective modules in 0 once these are constructed the existence of projective covers the existence of suitable filtrations on these and the reciprocity principle are ail proved with reference to the construction however another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules this proof can be used in some of the other settings as well although in any given setting no drastic simplifications result i believe it is still worthwhile to draw attention to this more conceptual approach the properties of the category 0 which are needed for the proof may be

dimensions of the simple restricted modules for the restricted contact lie algebra according to 2 the finite dimensional restricted simple lie algebras over an algebraically closed field of characteristic p 7 are either classical or of cartan type lusztig has a conjecture see 8 p 294 which provides a formula for the dimensions of the simple modules for the classical algebras in this paper modules are assumed to be left finitedimensional and restricted anderson jantzen and soergel 1 have recently shown that for p a 0 the lusztig conjecture follows from an analogous conjecture also due to lusztig for certain quantum groups they have also pointed out that this latter conjecture is implied by recent work of kazhdan and lusztig and a paper by casian although casian s paper has not yet convinced all of its readers the simple modules for the cartan type algebras fall into two classes there is a large class see 11 and 5 consisting of modules that are induced from simple modules for the homogeneous component of degree zero extended trivially to positive components constructed as mixed products in 11 the degree zero homogeneous components of cartan type algebras are either classical or nearly so therefore the lusztig conjecture can be used to get a formula for the dimensions of the modules in this class the small class consists of modules corresponding to exceptional weights in 11 the dimensions of these modules were determined for three of the four cartan type algebras namely the witt special and hamiltonian algebras there the exceptional weights turned out to be fundamental dominant weights in this paper we obtain the dimensions assuming p is not too small in the remaining case of the contact algebra in the process we determine the contragredients of the simple modules

asymptotics of scaling parameters for period doubling in unimodal maps with asymmetric critical points the universal period doubling scaling of a unimodal map with an asymmetric critical point is governed by a period 2 point of a renormalization operator the period 2 point is parametrized by the degree of the critical point and the asymmetry modulus in this paper we study the asymptotics of period 2 points and their associated scaling parameters in the singular limit of degree tending to 1

principal indecomposable representations for restricted lie algebras of cartan type abstract let l be any one of w n 1 s n 1 h n 1 and k n 1 over an algebraically closed held f of characteristic p 3 in this paper we extend the results concerning modular representations of classical lie algebras and semisimple groups to the case of l and obtain some properties of principal indecomposable modules of u l which parallel closely those of classical lie algebras

brauer type reciprocity for a class of graded associative algebras classical brauer reciprocity can be stated roughly as follows let g be a finite group and let k be an algebraically closed field of characteristic p 0 if s is a simple kg module and p s is its projective cover then the multiplicity of a simple module l as a composition factor of a characteristic zero lift of p s is the same as the multiplicity of s as a composition factor of a modulo p reduction of l all modules in this paper are assumed to be finite dimensional one thinks of the module l as playing an intermediate role between p s and s reciprocities similar to brauer reciprocity here called brauer type reciprocities have subsequently been found to occur in many other settings for instance if g is a classical modular lie algebra definition given in section 2 then there exists a set 9 of g modules with the following property given a simple g module s the projective cover p s of s has a filtration with each successive quotient isomorphic to a module in 5 and for each such filtration the number of times ze 9 occurs is the same as the multiplicity of s as a composition factor of z this was proved for g of type a i by pollack lo and for arbitrary g by humphreys 3 the reciprocity in this setting is often called humphreys reciprocity inspired by humphreys result bernstein gelfand and gelfand sought and found a brauer type reciprocity bgg reciprocity in a certain truncated category their category 8 of modules for a complex semisimple lie algebra see 11 later this was generalized by mirollo and

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

representations of cartan type lie algebras in characteristic p in this paper we discuss the representations of cartan type lie algebras in characteristic p 2 from the viewpoint of reducing rank when the character is regular semisimple for generalized witt algebras we can essentially reduce higher rank representations to lower rank representations

highest weight theory for finite dimensional graded algebras with triangular decomposition abstract we show that the category of graded modules over a finite dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category when the algebra is self injective we show furthermore that this highest weight category has tilting modules in the sense of ringel this provides a new perspective on the representation theory of such algebras and leads to several new structures attached to them there are a wide variety of examples in algebraic lie theory to which this applies restricted enveloping algebras lusztig s small quantum groups hyperalgebras finite quantum groups and restricted rational cherednik 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

generalized restricted representations of the zassenhaus superalgebras abstract let f be an algebraically closed field of prime characteristic p 2 and n n let z n be the zassenhaus superalgebra defined over f which as the simplest non restricted simple lie superalgebra is the superversion of the zassenhaus algebra more precisely z n is the lie superalgebra of the special super derivations of the superalgebra n here n is the tensor product of the divided power algebra of one variable and the grassmann superalgebra of one variable in this paper we study generalized restricted simple modules over the zassenhaus superalgebra z n classification of isomorphism classes of generalized restricted simple modules and their dimensions are precisely determined a sufficient and necessary condition for irreducibility of generalized restricted kac modules is provided

macdonald polynomials as characters of cherednik algebra modules we prove that macdonald polynomials are characters of irreducible cherednik algebra modules

normalisers of parabolic subgroups in finite unitary reflection groups abstract it is well known that the normaliser of a parabolic subgroup of a finite coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots we show that a similar result holds for all finite unitary reflection groups namely the normaliser of a parabolic subgroup is a semidirect product and in many but not all cases the complement can be obtained as the stabiliser of a set of roots in addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup

feigenbaum theory for unimodal maps with asymmetric critical point we extend the renormalization analysis for period doubling in unimodal maps to the case of asymmetric critical points universal scaling phenomena are governed by period two points of a renormalization operator