periodic and partially periodic representations of su n q the gelfand zetlin basis is adapted tosu n q forq a root of unit extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center a crucial identity is derived and proved which guarantees the periodicity of the action of the generators full periodicity is relaxed by stages some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation in the extreme case of highest weight representations all the raising and lowering operators are nilpotent as an alternative approach an auxiliary algebra giving all the periodic representations is presented an explicit solution of this system forn 3 while fully equivalent to the g z basis turns out to be much simpler

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

the coexistence problem for the discrete mathieu operator we solve the coexistence problem for the periodic discrete mathieu operator in all parametric cases the main tool in the proof will be bezout s theorem for projective plane curves as an additional result we obtain the gap opening and gap growth powers for this operator

holder continuity of the spectral measures for one dimensional schrodinger operator in exponential regime avila and jitomirskaya prove that the spectral measure v xf of quasi periodic schrodinger operator is 1 2 holder continuous with appropriate initial vector f if satisfies diophantine condition and is small in the present paper the conclusion is extended to that for all with the spectral measure v xf is 1 2 holder continuous with small if v is real analytic in a neighbor of x c where c is a large absolute constant in particular the spectral measure xf of almost mathieu operator is 1 2 holder continuous if e c with c a large absolute constant

identities ore extensions and character skew derivations following 17 a derivation is called quasi algebraic if there exist a ai q such that k x i 1k 1ai i x ax k x a 0 for x q let k be the minimal such integer and g0 g1 mutually outer automorphisms of q let f zij be a generalized polynomial in variables zij where i 0 and 0 j k it is shown in 6 that if f gi j x or f jgi x is an identity then so is f zij we generalize this to sets of skew derivations in theorems 1 and 2 in theorem 3 we apply these results to the ore extensions of several variables of 2 and in theorem 4 to the character hopf algebras of the character skew derivations of 12

on the structure of semi invariant polynomials in ore extensions let d be an x outer s derivation of a prime ring r where s is an automorphism of r the following is proved among other things the degree of the minimal semi invariant polynomial of the ore extension r x s d is if charr 0 and is pk for some k 0 if charr p 2 where is the least integer 1 such that s ds d is x inner a similar result holds for cv polynomials these are done by introducing the new notion of k basic polynomials for each integer k 0 which enable us to analyze semi invariant polynomials inductively

kotani last problem and hardy spaces on surfaces of widom type this is a small theory of non almost periodic ergodic families of jacobi matrices with purely however absolutely continuous spectrum the reason why this effect may happen is that under our axioms we found an analytic condition on the resolvent set that is responsible for exactly equivalent to this effect

identities with a single skew derivation abstract let r be a prime ring with extended centroid c and a continuous skew derivation of r we define the notion of k polynomials which in the case that is an ordinary derivation reduces to polynomials of the form x p n 1 x p n 1 n x where i c it is shown that all generalized identities with are consequences of gpis of r and an identity in the form x x m b b x where x is a k polynomial of minimal possible order m

zn n 1 generalization of the chiral potts model we show that ther matrix which intertwines twon by nn 1 state cyclicl operators related with a generalization ofuq sl n algebra can be considered as a boltzmann weight of four spin box for a lattice model with two spin interaction just as ther matrix of the checkerboard chiral potts model the rapidity variables lie on the algebraic curve of the genusg n2 n 1 n 1 n n 1 defined by 2n 3 independent moduli this curve is a natural generalization of the curve which appeared in the chiral potts model factorization properties of thel operator and its connection to the sos models are also discussed

minimal quasi algebraic relations of skew derivations in characteristic p 2 abstract let r be a prime ring of char r p 2 the monic minimal quasi algebraic relation of an ordinary derivation d of r takes the form d p s i 0 s 1 i d p i ad a where i are in the extended centroid of r and where a is in the symmetric martindale quotient ring of r among other things we extend this result to an arbitrary skew derivation d this is explained in details in 0 below

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

asymptotics of determinants for finite sections of operators with almost periodic diagonals abstract let a a j k j k be a bounded linear operator on l 2 z whose diagonals d n a a j j n j l z are almost periodic sequences for certain classes of such operators and under certain conditions we are going to determine the asymptotics of the determinants det a n 1 n 2 of the finite sections a n 1 n 2 a j k j k n 1 n 2 1 as their size n 2 n 1 tends to infinity examples of such operators include block toeplitz operators and the almost mathieu operator

finite dimensional representations of quantum affine algebras at roots of unity we describe explicitly the canonical map x spec ue b spec ze where u b is a quantum loop algebra at an odd root of unity e here ze is the center of ue g and spec r stands for the set of all finite dimensional irreducible representations of an algebra r we show that spec z is a poisson proalgebraic group which is essentially the group of points of g over the regular adeles concentrated at 0 and oo our main result is that the image under x of spec u b is the subgroup of principal adeles department of mathematics harvard university cambridge massachusetts 02138 e mail address beckfmath harvard edu department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address kacimath mit edu this content downloaded from 207 46 13 124 on wed 22 jun 2016 06 11 29 utc all use subject to http about jstor org terms

on the composition of q skew derivations in banach algebras abstract let a be a banach algebra let be an automorphism of a and let d be q skew derivations of a we show that if d a is quasi nilpotent for any a a then d a 3 lies in the radical of a for all a a this result simultaneously generalizes the previous results obtained by bresar and semrl 5 chebotar et al 7 and abdelali 1

quasitriangularity of quantum groups at roots of 1 an important property of a hopf algebra is its quasitriangularity and it is useful for various applications this property is investigated for quantum groupssl2 at roots of 1 it is shown that different forms of the quantum groupsl2 at roots of 1 are either quasitriangular or have similar structure which will be called braiding in the most interesting cases this property means that braiding automorphism is a combination of some poisson transformation and an adjoint transformation with a certain element of the tensor square of the algebra

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

on the universalr matrix of u_q widehat sl _2 at roots of unityat roots of unity we show that the action of the universalr matrix of the affine u q widehat sl 2 quantum algebra whenq is a root of unity can be renormalized by some scalar factor to give a well defined nonsingular expression satisfying the yang baxter equation it can be reduced to intertwining operators of representations corresponding to chiral potts if the parameters of these representations lie on the well known algebraic curve

inhomogeneous diophantine approximation in the coprime setting abstract given n n and x r let n x min n x m m z gcd n m 1 two conjectures in the coprime inhomogeneous diophantine approximation state that for any irrational number and almost every r lim inf n n n 0 and that there exists c 0 such that for all r q and 0 1 lim inf n n n c we prove the first conjecture and disprove the second one

anderson localization for the completely resonant phases abstract for the almost mathieu operator h u n u n 1 u n 1 2 cos 2 n u n avila and jitomirskaya conjecture that for every phase r r 2 z z h satisfies anderson localization if e 2 in the present paper we verify the conjecture for e 7

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

the extended centroid and x inner automorphisms of ore extensions for the ore extension r t s d where r is a prime ring we determine the center the extended centroid and the x inner automorphisms the results depend on the structure of ideals of r t s d

spectral theory of extended harper s model and a question by erdős and szekeres the extended harper s model proposed by d j thouless in 1983 generalizes the famous almost mathieu operator allowing for a wider range of lattice geometries parametrized by three coupling parameters by permitting 2d electrons to hop to both nearest and next nearest neighboring nnn lattice sites while still exhibiting its characteristic symmetry aubry andre duality previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space one of which is characterized by the positivity of the lyapunov exponent in this paper we complete the picture with a description of the spectral measures over the entire remaining self dual region for all irrational values of the frequency parameter the magnetic flux in the model most notably we prove that in the entire interior of this regime the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the nnn interaction becomes symmetric in physics literature extensive numerical analysis had indicated such spectral collapse however so far not even a heuristic argument for this phenomenon could be provided on the other hand in the remaining part of the self dual region the spectral measures are singular continuous irrespective of such symmetry the analysis requires some rather delicate number theoretic estimates which ultimately depend on the solution of a problem posed by erd s and szekeres on the product prod k 1 n 1 z a k publ de l institut mathematique paris 1950

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

algebraic q skew derivations abstract let r be a prime ring c its extended centroid and r f resp q its left resp symmetric martindale quotient ring let be a derivation of r where is an automorphism of r we show the equivalence of k polynomials resp k identities of and cv polynomials resp semi invariant polynomials in the ore extension q x we prove the existence of k polynomials of in certain rather general family of maps as applications the following are proved among other things consider the expression x i 0 n a i i x where a i r f and a n 0 1 if dim c r c then either r is a gpi ring or x 0 for all x r f 2 if r c then either r is commutative or x 0 for all x r f

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

flat periodic representations ofu q g we give explicit expression of flat periodic representations when they exist of the quantum analogues of simple lie algebras and their affine extensions for a parameter of deformationq equal to a root of unity by flat periodic we mean that these representations have no highest weight and that all the weights have multiplicity 1

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

abundance of mode locking for quasiperiodically forced circle maps we study the phenomenon of mode locking in the context of quasiperiodically forced non linear circle maps as a main result we show that under certain mathcal c 1 open condition on the geometry of twist parameter families of such systems the closure of the union of mode locking plateaus has positive measure in particular this implies the existence of infinitely many mode locking plateaus open arnold tongues the proof builds on multiscale analysis and parameter exclusion methods in the spirit of benedicks and carleson which were previously developed for quasiperiodic rm sl 2 mathbb r cocycles by young and bjerklov the methods apply to a variety of examples including a forced version of the classical arnold circle map