fractals without anomalous diffusion by an alternative analytical technique we solve the random walk problem on a particular class of fractal trees calculating the exact value of their spectral dimension the result shows that for all these structures the spectral dimension and the fractal dimension are equal providing an example of fractal structures with nonanomalous diffusion as well as an example of fractals with a noninteger spectral dimension greater than 2

a mackey imprimitivity theory for algebraic groups let g be an affine algebraic group over an algebraically closed field k and let h be a closed subgroup of g if v is a rational h module a comodule for the coordinate ring of h there is a now well known notion of an induced module vi g for g defined as the space morph g v of all h equivariant morphisms from g to a finite dimensional subspace of v with obvious g action the question arises given a rational g module m how can one recognize m as an induced module vig for a finite group g the answer is the mackey imprimitivity theorem the module m is induced if and only if it is a direct sum of subspaces permuted transitively by g with h the stabilizer of one of these subspaces called v one uses this result for example in proving the famous mackey decomposition theorem which describes the restriction of any induced module to a second subgroup l as a direct sum of suitable induced modules given the imprimitivity theorem the proof is just a matter of grouping the summands permuted by g into their orbits under l in the case of algebraic groups the situation is quite different for some subgroups h all g modules are induced this occurs for example if g is connected and k g h k e g if h is parabolic also if g is a connected unipotent group then a rational g module m is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule e for the conjugation action of g with e k 1 and such that k l is precisely the subspace annihilated by the action of the lie algebra of g on e 27 cf also 5 5 below the latter is an application of a general criterion in case g h is affine a rational g module m is induced if and only if there is an action of the coordinate ring a of g h compatible with the action of g on both a and m note that this generalizes the imprimitivity theorem in the case of finite groups since then a has a k basis of g h orthogonal idempotents permuted transitively by g

a criterion for irreducibility of parabolic baby verma modules of reductive lie algebras abstract let g be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p and g lie g in this paper we study representations of g with a p character of standard levi form when g is of type a n b n c n or d n a sufficient condition for the irreducibility of standard parabolic baby verma g modules is obtained this partially answers a question raised by friedlander and parshall 1990 in 2 moreover as an application in the special case that g is of type a n or b n and lies in the sub regular nilpotent orbit we recover a result of jantzen 1999 in 6

on the block structure of supersolvable restricted lie algebras block theory is an important tool in the modular representation theory w x w x of finite groups cf 18 apart from a few papers e g 17 13 16 dealing with restricted simple lie algebras there apparently has been no effort to do the same for other classes of restricted lie algebras despite a good w x knowledge of the simple modules cf e g 21 for the solvable case the aim of this paper is to develop the block theory for reduced universal enveloping algebras of a finite dimensional supersolvable restricted lie algebra as far as possible in close analogy to modular group algebras in order to organize the paper in a concise way we include some open questions on tensor products of simple modules under certain conditions either on the lie algebra or on the character they have affirmative answers which are decisively used in the proofs of our main results but unfortunately this is not true in general as we show by an example in the following we are going to describe the contents of the paper in more detail the second section provides the necessary background from the block theory of associative algebras which perhaps is not as well known to lie theorists in particular we stress that the block decomposition induces an equivalence relation linkage relation on the finite set of isomorphism classes of simple modules and indicate the proof of a cohomological characterization of the linkage relation which is fundamental for some of the following results the latter can be used to give a combinatorial description of the linkage relation via the gabriel quiver we include some results on the number of blocks resp the principal block that hold for any restricted lie algebra

on the gorenstein property for modular invariants let g gl v be a finite group where v is a finite dimensional vector space over a field f of arbitrary characteristic let s v be the symmetric algebra of v and s v g the ring of g invariants we prove here the following results theorem suppose that g contains no pseudo reflection of any kind 1 if s v g is gorenstein then g sl v 2 if g sl v then the cohen macaulay locus of s v g coincides with its gorenstein locus in particular if s v g is cohen macaulay then it is also gorenstein this extends well known results of k watanabe in case charf g 1 it also confirms a special case of a conjecture due to g kemper e kording g malle b h matzat d vogel and g wiese a similar extension is given to d benson s theorem about the gorenstein property of s v v g the polynomial tensor exterior algebra invariants our proof uses non commutative algebra methods in an essential way

representations of the witt superalgebra w 2 abstract in this paper we study representations of the lie superalgebra w 2 of witt type over an algebraically closed field of characteristic p 2 all irreducible modules and their dimensions are determined up to isomorphisms thus all irreducible modules and their dimensions of osp 2 2 are determined in prime characteristic up to isomorphisms

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

extensions of modules for algebraic groups let g be a connected semi simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p in this paper we show how one can obtain the extensions of any two simple g modules if one knows the socles of all the g modules extg l l where g1 denotes the kernel of the frobenius endomorphism on g and l as well as l are simple g1 modules alternatively we prove how all such g extensions can be found once one knows the extensions of any two simple g modules belonging to a specified finite set of simple modules with small highest weight such information is available in certain low rank cases it should also be pointed out that it is possible from lusztig s conjecture on irreducible characters 11 to derive the dimensions of ext l1 l2 when l1 and l2 are simple with such small highest weights to obtain our results on exth we need to know that there exists no g1 extension of a simple module by itself this turns out to be somewhat

restricted representations of the witt superalgebras abstract let k be an algebraically closed field of characteristic p 3 and w n the superversion of the witt algebra over k i e the lie superalgebra of superderivations of the grassmann algebra of rank n over k the simple modules and projective indecomposable modules in the restricted supermodule category for w n are studied the character formulas for such simple modules are given and the cartan invariants for this category are presented

complexity of representations of quantised function algebras and representation type abstract let g be a simply connected connected semisimple algebraic group over c b be opposite borel subgroups of g and b lie b let o g be the quantised function algebra at a root of unity and let u 0 be the quantised enveloping algebra of b at a root of unity we study the finite dimensional factor algebras o g g and u 0 b for g g and b b which were introduced by de concini kac and procesi 1992 in geometry and analysis pp 41 65 tata inst fund res bombay and de concini and lyubashenko 1994 adv math 108 205 262 in particular we describe the complexity of these factor algebras in terms of the weyl group of g and deduce results on their representation type finally we completely describe those factor algebras of finite representation type

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

on cocommutative hopf algebras of finite representation type abstract let g be a finite algebraic group defined over an algebraically closed field k of characteristic p 0 such a group decomposes into a semidirect product g g 0 g red with a constant group g red and a normal infinitesimal subgroup g 0 if the principal block b 0 g of the group algebra h g has finite representation type then both constituents have the same property with at least one of them being semisimple we determine the structure of the infinitesimal constituent g 0 up to the classification of v uniserial groups

elementary lie algebras and lie a algebras a finite dimensional lie algebra l over a field f is called elementary if each of its subalgebras has trivial frattini ideal it is an a algebra if every nilpotent subalgebra is abelian the present paper is primarily concerned with the classification of elementary lie algebras in particular we provide a complete list in the case when f is algebraically closed and of characteristic different from 2 3 reduce the classification over fields of characteristic 0 to the description of elementary semisimple lie algebras and identify the latter in the case when f is the real field additionally it is shown that over fields of characteristic 0 every elementary lie algebra is almost algebraic in fact if l has no non zero semisimple ideals then it is elementary if and only if it is an almost algebraic a algebra

representations of finite lie algebras abstract we develop an approach to investigate representations of finite lie algebras g f over a finite field f q through representations of lie algebras g with frobenius morphisms f over the algebraic closure k f q as an application we first show that frobenius morphisms on classical simple lie algebras can be used to determine easily their f q forms and hence reobtain a classical result given in g b seligman modular lie algebras springer verlag berlin 1967 we then investigate representations of finite restricted lie algebras g f regarded as the fixed point algebra of a restricted lie algebra g with restricted frobenius morphism f by introducing the f orbital reduced enveloping algebras u g associated with a reduced enveloping algebra u g we partition simple g f modules via f orbits of their p characters we further investigate certain relations between the categories of g modules with p character g f modules with p character and g f modules with p character for an automorphism of g finally we illustrate the theory with the example of sl 2 f q

cartan matrices for restricted lie algebras consider a finite dimensional restricted lie algebra over a field of prime characteristic each linear form on this lie algebra defines a finite dimensional quotient of its universal enveloping algebra called a reduced enveloping algebra this leads to a cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra in this paper we show how to compare such cartan matrices belonging to distinct linear forms as an application we rederive and generalise the reciprocity formula first discovered by humphreys for lie algebras of reductive groups for simple lie algebras of cartan type we see for example that the cartan matrices for linear forms of non positive height are submatrices of the cartan matrix for the zero linear form

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

on the representation theory of lie triple systems in this paper we take a new look at the representation theory of lie triple systems we consider both ordinary lie triple systems and restricted lie triple systems in the sense of hodge 2001 in a final section we begin a study of the cohomology of lie triple systems

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

simple modules with character height at most one for the restricted witt algebras abstract the simple modules with character height at most one for the restricted witt algebras are considered their classification construction and dimension formulas are reduced to the same for the general linear algebra results of chang and shen are recovered in the process

finiteness of physics and its possible consequences the modular analog of representations of the so 1 4 algebra for a system of two spinless particles is considered in the framework of approach proposed by the author earlier in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space the eigenvalues of the free mass operator and corresponding eigenvectors are found it is shown that if the finite field under consideration satisfies some properties then the particles necessarily interact with each other the corresponding interaction is universal since it is fully defined by the masses of particles and characteristics of the finite field the formula for the kernel of the interaction operator is derived but since the kernel is defined by the sum over the galois field and this sum cannot be calculated explicitly then it cannot be determined how far the above interaction manifests itself in the usual conditions

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

deformations of lie algebra representations let g be a reductive algebraic group defined over an algebraically closed field k of positive characteristic p in this paper we continue our previous investigation 7 of the representation theory of the lie algebra f of g a key ingredient in our earlier work involved the consideration of a family a x s of finite dimensional algebras parameterized by the set 0 qs 1homka1g y b k of characters on a certain central subalgebra 0 6 of the universal enveloping algebra u 6 of g each irreducible module is naturally an irreducible module for exactly one such ax with the special case x 0 corresponding to restricted modules we introduce a so called deformation algebra s4 intermediate between the universal enveloping algebra u and certain of the algebras ax here sl is a quotient of u q such that each ax containing the restricted enveloping algebra v a of a fixed borel subalgebra 2a is in turn a quotient of s4 given a module mx for such an a we introduce the important notion of a deformation family of mx in definition 1 3 the hypothesis that the behavior of mx is related to that of the corresponding modules for near by algebras at motivates the present study a basic result 7 theorem 3 2 reduces the representation theory of ax for x 0 to that of an associated a for a subalgebra of to which the restriction t of x is nilpotent in the sense of 121 thus a study of the representation theory of the algebras ax for x a nilpotent character subsumes the classical problem of determining the irreducible representations of the lie algebra each g conjugacy class in the nullcone x of nilpotent elements of 0 y f defines a family of isomorphic algebras ax presumably the restricted enveloping algebra v ao corresponding to the conjugacy class o is the most complicated algebra of this kind its representation theory has been extensively investigated in recent years because of its intimate connections

varieties of modules for z 2z z 2z let k be an algebraically closed field of characteristic 2 we prove that the restricted nilpotent commuting variety c that is the set of pairs of n n matrices a b such that a2 b2 a b 0 is equidimensional c can be identified with the variety of n dimensional modules for z 2z z 2z or equivalently for k x y x2 y2 on the other hand we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic 2 we also prove that if e2 0 then the set of elements of the centralizer of e whose square is zero is equidimensional finally we express each irreducible component of c as a direct sum of indecomposable components of varieties of z 2z z 2z modules

mod preduction for quantum groups abstract let u g be the quantized enveloping algebra associated to the lie algebra g sl n 1 at a p th root of unity e and assume that p is a prime which does not divide n 1 it is known that the irreducible finite dimensional representations of u e g are parametrized up to isomorphisms by the conjugacy classes of sl n 1 in the paper we prove that the dimension of any u e g module m parametrized by a conjugacy class o is divided by p 1 2 dim o this result was conjectured by c de concini v g kac and c procesi j amer math soc 5 1992 151 190

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

polynomial level set method for polynomial system reachable set estimation in this paper we present a polynomial level set method for advecting a semi algebraic set for polynomial systems this method uses the sub level representation of sets the problem of flowing these sets under the advection map of a dynamical system is converted to a semi definite program which is then used to compute the coefficients of the polynomials the method presented in this paper does not require either the sets being positively invariant or star shaped hence the proposed algorithm can describe the behavior of system states both inside and outside the domain of attraction and can also be used to describe more complex shapes of sets we further address the related problems of constraining the degree of the polynomials various numerical examples are presented to show the effectiveness of advection approach

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

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

infinitesimal hecke algebras of sl2 in positive characteristic abstract we describe centers of infinitesimal hecke algebra of sl 2 in positive characteristic in particular we show that these algebras are finitely generated modules over their centers and the azumaya and smooth loci of the centers coincide

generalized reduced enveloping algebras for restricted lie algebras abstract we observe that any finite dimensional indecomposable module for a restricted lie algebra over an algebraically closed field is a module for a finite dimensional quotient of the universal enveloping algebra these algebras form a two parameter family which generalizes the notion of a reduced enveloping algebra we identify each such algebra as a reduced enveloping algebra for an associated lie algebra and use this to compute support varieties and obtain some representation theoretic consequences