artin hasse exponentials of derivations abstract we show how artin hasse exponentials of nilpotent derivations can be used to produce new cyclic gradings of non associative algebras of prime characteristic from a given one we discuss the connection with the technique of switching tori and consequently the associated root space decompositions in restricted modular lie algebras however differently from toral switching our method can produce gradings over cyclic p groups of order higher than p in particular we apply it to construct cyclic gradings of the zassenhaus lie algebra of dimension p n for p 3

rigorous application of the stochastic functional method to plane wave scattering from a random cylindrical surface the stochastic functional method is applied to plane wave scattering from a random cylindrical surface whereupon the dirichlet boundary condition is rigorously imposed analytical results accurate to second and fourth order in surface roughness are obtained for the coefficients of the wiener hermite expansion of the secondary scattered wave field the validity of approximate solutions is numerically investigated by means of the boundary condition criterion and of the energy consistency criterion the former which is introduced herein states that any approximate solution should be in conformity with the boundary condition whereas the latter pertains to the energy conservation law the numerical investigation indicates that the rigorous application of the stochastic functional method yields more accurate results in terms of both criteria than did previous treatments of the problem under consideration moreover it is suggested that applicability limits should be set through the mean boundary condition criterion instead of the energy consistency criterion the latter may lead to underestimating deficiencies of the approximate solution under test

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

determination of the algebraic relations among special γ values in positive characteristic we devise a new criterion for linear independence over function fields using this tool in the setting of dual t motives we find that all algebraic relations among special values of the geometric r function over f q t are explained by the standard functional equations

the restricted simple lie algebras are of classical or cartan type abstract the classification of the finite dimensional restricted simple lie algebras over an algebraically closed field f of prime characteristic p 7 is announced such an algebra is either of classical type an analogue over f of a finite dimensional simple lie algebra over the complex numbers or of cartan type an analogue over f of one of the infinite lie algebras of cartan over the complex numbers

correction to cartan subalgebras of simple lie algebras trans amer math soc 234 1977 no 2 435 446 mr0480650 58 806

gradings derivations and automorphisms of nearly associative algebras abstract in this paper we examine a class of algebras which includes lie algebras lie color algebras right alternative algebras left alternative algebras antiassociative algebras and associative algebras we call this class of algebras algebras and we examine gradings of these algebras by groups with finite support we generalize various results on associative algebras and finite dimensional lie algebras two of our main results are t heorem 2 2 let a be a g graded left algebra and v g g v g a g graded left a module with finite support where g is a torsion free abelian group if a 0 acts nilpotently on v then a also acts nilpotently on v t heorem 2 12 let a be a g graded algebra with finite support where g t z m and t is a torsion free abelian group if the identity component a 0 0 acts nilpotently on a on both sides then a is solvable these results are used to examine the invariants of automorphisms and derivations one such application is c orollary 3 3 let l g g l g be a lie color algebra over a field k of characteristic 0 and let d be a finite dimensional nilpotent lie algebra of homogeneous derivations of l which are algebraic as k linear transformations of l if l d 0 then l is nilpotent we conclude this paper with counterexamples to various questions which arise naturally in light of our results

varieties of tori and cartan subalgebras of restricted lie algebras this paper investigates varieties of tori and cartan subalgebras of a finite dimensional restricted lie algebra g p defined over an algebraically closed field k of positive characteristic p we begin by showing that schemes of tori may be used as a tool to retrieve results by a premet on regular cartan subalgebras moreover they give rise to principal fibre bundles whose structure groups coincide with the weyl groups in case g lie g is the lie algebra of a smooth group g for solvable lie algebras varieties of tori are full affine spaces while simple lie algebras of classical or cartan type cannot have varieties of this type in the final sections the quasi projective variety of cartan subalgebras of minimal dimension rk g is shown to be irreducible of dimension dim k g rk g with premet s regular cartan subalgebras belonging to the regular locus

deformations of w1 n a and modular semisimple lie algebras with a solvable maximal subalgebra abstract in one of his last papers boris weisfeiler proved that if modular semisimple lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration then associated graded algebra is isomorphic to one constructed from the zassenhaus algebra tensored with the divided powers algebra we completely determine such class of algebras calculating in process low dimensional cohomology groups of zassenhaus algebra tensored with any associative commutative algebra

lie algebras graded by finite root systems and intersection matrix algebras this paper classifies the lie algebras graded by doubly laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized cartan matrices of types b c f and g this completes the determination of the lie algebras graded by finite root systems initiated by berman and moody who studied the simply laced finite root systems of rank 2

infinitesimal deformations of restricted simple lie algebras i abstract we compute the infinitesimal deformations of two families of restricted simple modular lie algebras of cartan type the witt jacobson and the special lie algebras

simple lie algebras of small characteristic ii exceptional roots abstract letlbe a finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristicp 3 andta 2 dimensional torus in the semisimplep envelope ofl suppose thatlis not isomorphic to a melikian algebra it is proved in this paper that for every root l t the subalgebrak generated by i f pki whereki x li x l i 0 acts triangulably onl in particular this implies that in the terminology of r e block and r l wilson 1988 j algebra114 115 259 all roots of l t are nonexceptional

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

simple lie algebras of small characteristic iv solvable and classical roots abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 and t a torus of maximal dimension in the p envelope of l in der l in this paper we describe the t semisimple quotients of the 2 sections of l relative to t and prove that if all 1 sections of l relative to t are compositionally classical or solvable then l is either classical or a block algebra or a filtered lie algebra of type s

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

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

rank one lie algebras let l denote a finite dimensional lie algebra over an algebraically closed field such an algebra is called a rank one lie algebra provided l has a one dimensional cartan subalgebra when d is of characteristic zero a fundamental result in the theory of simple lie algebras over d is that the only rank one algebra is the 3 dimensional lie algebra b t 2 of traceless 2 x 2 matrices the representation theory of t t 2 plays a critical role in the structure and classification of simple algebras over as well as in the study of their representations when the field d is of prime characteristic p 3 in addition to b 2 the albert zassenhaus lie algebras 1 see also 3 are known to be rank one algebras by an albert zassenhaus lie algebra is meant an algebra over d with basis gala e g where g is a finite additive subgroup of d and with multiplication defined by

scattering of a scalar wave from a slightly random surface scalar wave scattering from a slightly random surface is analyzed by a probabilistic method we make use of the homogeneity of an infinite random surface that is the shift invariance property of the strictly homogeneous random field by the group theoretic consideration of such a shift invariance property the wave solution proves to be a homogeneous random field multiplied by an exponential function then such a homogeneous random field is approximately solved for a slightly random surface to yield a wave solution involving multiple scattering several statistical properties of the scattering are calculated and shown in the figures the accuracy of the approximate solution is examined in terms of the error of the boundary value equation

a family of simple lie algebras in characteristic two abstract in this paper a new two parameter family of simple lie algebras defined over fields of characteristic two is described they are used to construct loop algebras which are central to the classification of graded lie algebras with maximal class and characteristic two

a formula on the wiener hermite expansion two mathematical formulas are given in an explicit form one develops a product of two different multiple wiener integrals in a series of wiener hermite expansions the other one develops a product of two random variables each of which is described by a wiener hermite expansions in a series of another wiener hermite expansion

approximate solution of a mixed nonlinear stochastic oscillator in this paper nonlinear oscillators under mixed quadratic and cubic nonlinearities with stochastic inputs are considered different methods are used to obtain second order approximations namely the wiener hermite and perturbation whep technique and the homotopy perturbation method hpm some statistical moments are computed for the different methods using mathematica 5 comparisons are illustrated through figures for different case studies

simple lie algebras of small characteristic v the non melikian case abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 we prove in this paper that if for every torus t of maximal dimension in the p envelope of adl in derl the centralizer of t in adl acts triangulably on l then l is either classical or of cartan type as a consequence we obtain that any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 5 is either classical or of cartan type this settles the last remaining case of the generalized kostrikin shafarevich conjecture the case where p 7

determination of algebraic relations among special zeta values in positive characteristic abstract as an analogue to special values at positive integers of the riemann zeta function we consider carlitz zeta values c n at positive integers n by constructing t motives after papanikolas we prove that the only algebraic relations among these characteristic p zeta values are those coming from the euler carlitz relations and the frobenius pth power relations

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

simple lie algebras of small characteristic iii the toral rank 2 case abstract it is proved that every finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristic p 3 is of classical or cartan type or a melikian algebra

scattering of waves from a random spherical surface mie scattering the stochastic theory developed by the authors for the scattering from a random planar surface is extended to the case of a random spherical surface which is assumed to be a homogeneous random field on the sphere homogeneous with respect to spherical rotations based on the group theoretical analogies between the two the formulation of the theory is closely connected to the representation theory of the rotation group the concept of the stochastic spherical harmonics associated with the rotation group and their several formulas are introduced and discussed at the beginning for the plane wave incident on a random spherical surface the scattered random wave field can be expanded systematically in terms of the stochastic spherical harmonics in much the same way as the nonrandom case and several formulas are derived for the coherent scattering amplitude the coherent and incoherent power flows and the coherent and incoherent scattering cross sections the power flow conservation law is cast into the stochastic version of the optical theorem stating that the total scattering cross section consisting of the coherent and incoherent power flow is equal to the imaginary part of the coherent forward scattering amplitude approximate solutions are obtained for the mie scattering with a slightly random spherical surface where the single scattering approximation is valid due to the absence of a real resonance as shown in the previous work on the two dimensional case some numerical calculations are made for the coherent and incoherent scattering cross sections

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

representation theory of the witt algebra 1 1 let g be a finite dimensional restricted lie algebra over an algebraically closed field of characteristic p 0 every simple g module is u w finite dimensional and therefore admits a character x g g see sf x u theorem 5 2 5 conversely for any such linear form x g g there exists a finite dimensional algebra u g x which is a quotient of the universal enveloping algebra u g and whose simple modules are exactly the simple modules for g with character x because of this fact these algebras play an important role in studying the representations of g in particular the restricted g representations coincide with the u g 0 modules every finite dimensional restricted simple lie algebra over an algebraically closed field of characteristic p 7 is either classical or of cartan w x type bw for a classical lie algebra g the representation theory of w x u g x was first studied by kac and weisfeiler wd kw and further

the homotopy perturbation method applied for solving complex valued differential equations with strong cubic nonlinearity abstract in this paper the homotopy perturbation method is adopted for solving a complex valued second order strongly nonlinear differential equation homotopy with an imbedding parameter p 0 1 is constructed the perturbation procedure with parameter p transforms the strongly nonlinear differential equation into a system of linear complex valued differential equations whose solutions give the approximate solution of the initial differential equation to illustrate the effectiveness and convenience of the suggested procedure a duffing equation with strong cubic nonlinearity is considered the periodic solution in the first approximation is obtained the solution is compared with the exact one and shows good agreement

shtukas and jacobi sums we show how the analogues of jacobi sums in the context of function fields introduced and studied in t1 t2 t3 can be obtained from shtukas introduced and studied in d2 d3 m we apply this to obtain some results on the prime factorization of analogues of gauss sums and to prove an analogue of the gross koblitz formula for general function field generalizing the results in t2 for this purpose we also introduce and interpolate a new analogue of gamma function

tannakian duality for anderson drinfeld motives and algebraic independence of carlitz logarithms we develop a theory of tannakian galois groups for t motives and relate this to the theory of frobenius semilinear difference equations we show that the transcendence degree of the period matrix associated to a given t motive is equal to the dimension of its galois group using this result we prove that carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent

wiener analysis of a binary hysteresis system nonlinear transformation of an independent gaussian sequence by a binary hysteresis is studied by wiener s method nonlinear problems in random theory mit cambrige ma 1958 the output sequence of the hysteresis is represented by a series of wiener hermite functionals which are random and orthogonal uncorrelated to each other from the hysteresis equation a set of linear equations for the wiener hermite functionals is derived without approximation the equations directly show that the hysteresis is decomposed into equivalent subsystems consisting of static nonlinear elements multipliers adders and linear filters they are solved by use of random numbers generated in a computer in order to get a sample output sequence approximated by a finite sum of the wiener hermite functionals another set of equations is derived for the mean square values of the wiener hermite functionals in terms of its numerical solutions the convergence property of the wiener hermite functional series is discussed

left symmetric structures on simple modular lie algebras abstract this paper investigates left symmetric structures on finite dimensional simple lie algebras g over a field k if k is of characteristic 0 then g does not admit any left symmetric structure this is known in the theory of affine manifolds in the modular case however such structures may exist the main purpose of this paper is to show that classical simple lie algebras of characteristic p 3 admit left symmetric structures only in case p divides dim g the proof involves the computation of the first cohomology groups of classical lie algebras for certain g modules of small dimension here g is regarded as the lie algebra of a connected semisimple algebraic group over an algebraically closed field of characteristic p 0 most of these computations are due to jantzen for nonrestricted simple lie algebras of cartan type it is shown that many more left symmetric structures can be found one studies so called adjoint structures induced by nonsingular derivations of g the simple algebra l g of block of dimension pn 1 for example admits adjoint structures for every p 0 if p 2 the results are more complicated

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

on the structure of the graded lie algebra associated with a noncontractible filtration the largest r 0 such that g 0 is called the depth of g any such yr g is semisimple and contains a unique minimal homogeneous ideal which will be denoted as g9 the structure of g9 can be described in terms of derivation simple commutative associative graded algebras and simple w x graded lie algebras 23 there are two cases nondegenerate and degenerate in the terminology of weisfeiler which expose different features these two are distinguished by whether g9 has a nonzero positive part or

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

classification of the restricted simple lie algebras soit l une algebre de lie simple restreinte de dimension finie sur un corps algebriquement clos f de caracteristique p 7 alors l est de type classique ou de cartan

solving the random legendre differential equation mean square power series solution and its statistical functions in this paper we construct by means of random power series the solution of second order linear differential equations of legendre type containing uncertainty through its coefficients and initial conditions by assuming appropriate hypotheses on the data we prove that the constructed random power series solution is mean square convergent in addition the main statistical functions of the approximate solution stochastic process generated by truncation of the exact power series solution are given finally we apply the proposed method to some illustrative examples to compare the numerical results for the average and the variance with respect to those obtained by the monte carlo approach

simple lie algebras of small characteristic i sandwich elements abstract let l be a finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 of absolute toral rank 2 and t a 2 dimensional torus in the semisimple p envelope of l it is proved that l is either classical or a block algebra or contains sandwich elements which are homogeneous with respect to t or a conjugate to t in addition several dimension estimates are given

tori in the melikian algebra this paper is concerned with the tori in the restricted melikian algebra g g 1 1 according to the latest classification result of premet and strade 16 g is the unique simple lie algebra of toral rank 2 over an algebraically closed field of characteristic p 5 distinct from the classical algebras and algebras of cartan type here we determine the one dimensional subtori in g up to conjugation by automorphisms and compute their centralizers in g one consequence of this investigation is that all maximal tori in g are two dimensional for each two dimensional torus we find the number of its roots of any possible type classical witt hamiltonian proper the answer depends only on the position of the torus relative to the invariant filtration of g the conjugacy classes of two dimensional tori are not determined in this paper it can be shown that their number is infinite the classification of simple lie algebras requires detailed information about the toral structure of low rank lie algebras see e g 3 19 one of the key points is the possibility of finding a torus of maximal dimension in the p envelope of a simple lie algebra l having all roots proper the classical argument is based on checking that when switching an improper root to a proper one the number of proper roots in every 2 section of l increases in the case of the melikian algebra we encounter a situation when the number of proper roots remains unchanged when this does occur we show that the number of hamiltonian roots becomes strictly less

contributions to the classification of simple modular lie algebras we develop results directed towards the problem of classifying the finite dimensional simple lie algebras over an algebraically closed field of characteristic p 7 a 1 section of such a lie algebra relative to a torus t of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple in this paper we show that the sum q of those compositionally classical subalgebras is a subalgebra this extends to the general case a crucial step in the classification by block and wilson of the restricted simple lie algebras we derive properties of the filtration which can be constructed using q and obtain structural information about the 1 sections and 2 sections of q relative to t we further classify all those algebras in which q is solvable

c sections of lie algebras let m be a maximal subalgebra of a lie algebra l and a b a chief factor of l such that b m and a 6 m we call the factor algebra m a b a c section of m all such c sections are isomorphic and this concept is related those of c ideals and ideal index previously introduced by the author properties of c sections are studied and some new characterizations of solvable lie algebras are obtained mathematics subject classification 2000 17b05 17b20 17b30 17b50

toral rank one lie algebras let l be a finite dimensional lie algebra over an algebraically closed field f of characteristic p 0 assume that h is a cartan subalgebra of l we say that l has toral rank m relative to h if dim pa m where a is the set of roots with respect to h p is the prime field and pa is the p vector space spanned by a in this work we study lie algebras of toral rank one the motivation to investigate this class of algebras comes from the problem of classifying simple lie algebras if a is a finite dimensional simple lie algebra with cartan subalgebra h and if ci is a root of h then the subalgebra a ciep a has toral rank one relative to h in 12 133 wilson has shown that if a simple algebra a has characteristic p 7 then the restricted closure w of h in der a has the form r t z where i is the nil radical of b and t is the maximal torus which is central in j it follows that ad r on each root space a can be simultaneously upper triangularized indeed i is a lie algebra of nilpotent transformations on a and thus by 8 theorem 3 31 it annihilates a nonzero vector the space of such vectors is t invariant and so t i has a common eigenvector u on a one can proceed inductively on ajfu to argue that each root space a has a basis relative to which ad r is upper triangular with that in mind we say in what follows that a cartan subalgebra his triangularizable if there is a basis for each root space relative to which ad h is upper triangular we show in section 1 that if l is a lie algebra having a toral rank one cartan subalgebra h and if h is triangularizable then l modulo its maximal solvable ideal either is 0 or consists of a simple toral rank one lie algebra s together with derivations of s in h in the first case we say that the core of l is 0 and in the second case that the core of l is s wilson 141 has classified the simple toral rank one lie algebras using

on graded lie algebras of characteristic three with classical reductive null component abstract we consider finite dimensional irreducible transitive graded lie algebras l i q r l i over algebraically closed fields of characteristic three we assume that the null component l 0 is classical and reductive the adjoint representation of l on itself induces a representation of the commutator subalgebra l 0 of the null component on the minus one component l 1 we show that if the depth q and height r of l are both greater than one then this representation must be restricted

the classification of the simple modular lie algebras ii the toral structure abstract let l be a simple lie algebra over an algebraically closed field of characteristic p 7 and t an optimal torus in some p envelope l p we determine the action of t on the two sections of l which have been described in st4 we also give some new and noncomputational proofs to determine the conjugacy classes of the tori in w n 1 and of the cartan subalgebras of w 1 n

simple lie algebras of characteristic p with dependent roots we investigate finite dimensional simple lie algebras over an algebraically closed field f of characteristic p 7 having a cartan subalgebra h whose roots are dependent over f we show that h must be one dimensional or for some root a relative to h there is a 1 section l a such that the core of l a is a simple lie algebra of cartan type h 2 mr d 2 or w 1 n for some n 1 the results we obtain have applications to studying the local behavior of simple lie algebras and to classifying simple lie algebras which have a cartan subalgebra of dimension less than p 2

symmetric lie algebras lie rootsysrems r are introduced with axioms which reflect properties of the rootset of a lie algebra l as structured by representations of compatible simple restricted rank i subquotients of l the rank 1 lie rootsystems and the rank 2 lie rootsystems defined over h are classified up to isomorphism base closure and core are discussed the rootsystems of collapse on passage from r to core r are shown to be of type s given any lie rootsystem r its independent root pairs are shown to fall into eleven classes where the eleventh anomoly pair tz never occurs it is shown that r is contained in r s not always equal where r is a witr rootsystem and s is a classical roolsystem this result is of major importance to two papers d j winter generalized classicallalberttzassenhaus lie algebras to appear rootsystems of simple lie algebras to appear since it implies that the rootsystems of the simple nonclassical lie algebras considered there are witt rootsystems toral lie algebras and swmetric lie algehrus are introduced and studied as generalizations of classical albert zassenhaus lie algebras it is shown that their rootsystems are lie rootsystems the cores of toral lie algebras are shown to be classical albert zassenhaus lie algebras these results form the basis for the abovementioned papers on rootsystems of simple lie algebras and the classification of the rootsystems of two larger classes of lie algebras the generalized classical albert zassenhaus lie algebras and the classical albertpzassenhaus kaplansky lie algebras symmetric lie algebras are introduced as generalizations of classicallalbertzassenhaus lie algebras it is shown that their rootsets r are lie rootsystems consequently symmetric lie algebras can be studied locally using the classification of rank 2 lie rootsystems this is done in detail for oral lie algebras i 1985 academsc

modular representations as a possible basis of finite physics the approach 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 is discussed the mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large the above correspondence is considered in the examples of finite analogs of representations of so 2 3 so 1 4 and osp 1 4 algebras finite analogs of representations of infinite dimensional algebras are briefly discussed

taylor coefficients of anderson thakur series and explicit formulae for each positive characteristic multiple zeta value defined by thakur in function field arithmetic world scientific publishing river edge 2004 the first and third authors in chang and mishiba invent math 2020 https doi org 10 1007 s00222 020 00988 1 constructed a t module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value the objective of this paper is to use the taylor coefficients of anderson thakur series and t motivic carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

the classification of the simple modular lie algebras iii solution of the classical case we investigate the structure of simple modular lie algebras over an algebraically closed field of characteristic p 7 if the distinguished subalgebra q l t with respect to an optimal torus t in some p envelope lp of l equals l then either l has a two section of type h 2 1 z or l is of classical type

ext 1 quivers for the witt algebra w 1 1 let g be the finite dimensional witt algebra w 1 1 over an algebraically closed field of characteristic p 3 it is well known that all simple w 1 1 modules are finite dimensional each simple module admits a character g given g one can form the finite dimensional reduced enveloping algebra u g the simple modules for u g are precisely those simple w 1 1 modules admitting the character in this paper the authors compute ext1 between pairs of simple modules for u g