hidden sl n symmetry in conformal field theories we prove that an irreducible representation of the virasoro algebra can be extracted from an irreducible representation space of thesl 2 current algebra by putting a constraint on the latter using the becchi rouet stora tyutin formalism thus there is asl 2 symmetry in the virasoro algebra but it is gauged and hidden this construction of the virasoro algebra is the quantum analogue of the hamiltonian reduction we then are naturally lead to consider a constrainedsl 2 wess zumino witten model this system is also related to quantum field theory of coadjoint orbit of the virasoro group based on this result we present a canonical derivation of thesl 2 current algebra in polyakov s theory of two dimensional gravity it is a manifestation of thesl 2 symmetry in conformal field theory hidden by the quantum hamiltonian reduction we also discuss the quantum hamiltonian reduction of thesl 2 current algebra and its relation to thew n algebra of zamolodchikov this makes it possible to define a natural generalization of the geometric action for thew n algebra despite its non lie algebraic nature

irreducible unitary representations of su 2 2 we give an explicit classification of the irreducible unitary representations of the simple lie group su 2 2

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

polynomial realization of the uq sl 3 gel fand weyl zetlin basis an explicit realization of the u uq sl 3 gel fand weyl zetlin gwz basis as polynomial functions in three variables real or complex is given this realization is obtained in two complementary ways first a known correspondence is used between the abstract gwz basis and explicit polynomials in the quantum subgroup u of the raising generators then an explicit construction is used of arbitrary lowest weight holomorphic representations of u in terms of three variables on which the generators of u are realized as q difference operators the application of the gwz corresponding polynomials in this realization of the lowest weight vector the function 1 produces the first realization of this gwz basis another realization of the gwz polynomial basis is found by the explicit diagonalization of the operators of isospin i2 third component of isospin iz and hypercharge in the same realization as q difference operators the result is that the eigenvectors can be written in terms of q hypergeometric

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

singular dimensions of the n 2 superconformal algebras ii the twisted n 2 algebra we introduce a suitable adapted ordering for the twisted n 2 superconformal algebra i e with mixed boundary conditions for the fermionic fields we show that the ordering kernels for complete verma modules have two elements and the ordering kernels for g closed verma modules just one therefore spaces of singular vectors may be two dimensional for complete verma modules whilst for g closed verma modules they can only be one dimensional we give all singular vectors for the levels 1 and for both complete verma modules and g closed verma modules we also give explicit examples of degenerate cases with two dimensional singular vector spaces in complete verma modules general expressions are conjectured for the relevant terms of all primitive singular vectors i e for the coefficients with respect to the ordering kernel these expressions allow to identify all degenerate cases as well as all g closed singular vectors they also lead to the discovery of subsingular vectors for the twisted n 2 superconformal algebra explicit examples of these subsingular vectors are given for the levels 1 and finally the multiplication rules for singular vector operators are derived using the ordering kernel coefficients this sets the basis for the analysis of the twisted n 2 embedding diagrams

representation theory approach to the polynomial solutions of q difference equations uq sl 3 and beyond a new approach to the theory of polynomial solutions of q difference equations is proposed the approach is based on the representation theory of simple lie algebras g and their q deformations and is presented here for uq sl n first a q difference realization of uq sl n in terms of n n 1 2 commuting variables and depending on n 1 complex representation parameters ri is constructed from this realization lowest weight modules lwm are obtained which are studied in detail for the case n 3 the well known n 2 case is also recovered all reducible lwm are found and the polynomial bases of their invariant irreducible subrepresentations are explicitly given this also gives a classification of the quasi exactly solvable operators in the present setting the invariant subspaces are obtained as solutions of certain invariant q difference equations i e these are kernels of invariant q difference operators which are also explicitly given such operators were not used until now in the theory of polynom

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

induced representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n we construct the induced holomorphic representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n the first superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup gl m n the second superalgebra is a hopf subalgebra of the first for a special choice of the parameters the representations are labelled by m n integer numbers respectively m n 1 complex numbers and act in the space of formal power series of m n m n 1 2 non commuting variables of which mn are odd and the rest are even these variables generate a q deformation of a flag supermanifold of the supergroup gl m n respectively sl m n

the general uq sl 2 invariant xxz integrable quantum spin chain the boundary conditions are constructed for the general xxz quantum integrable spin chain such that the resulting system is invariant under the quantum algebra uq sl 2 action the result generalizes that due to pasquier and saleur 1990 obtained for spin 1 2 and spin 1

the unitary irreducible representations of overline rm so _ 0 4 2 an exhaustive classification of all irreducible harish chandra mathfrak s mathfrak o 4 2 modules integrable to unitarizable projective representations of the conformal group is established by infinitesimal methods the classification is based 1 on the reduction upon the maximal compact subalgebra associated with a lattice of points in 3 and 2 on a set of additional parameters upon which the eigenvalues of central elements of the enveloping algebra depend polynomially

extremal results in sparse pseudorandom graphs abstract szemeredi s regularity lemma is a fundamental tool in extremal combinatorics however the original version is only helpful in studying dense graphs in the 1990s kohayakawa and rodl proved an analogue of szemeredi s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs many of the key applications of szemeredi s regularity lemma use an associated counting lemma in order to prove extensions of these results which also apply to sparse graphs it remained a well known open problem to prove a counting lemma in sparse graphs the main advance of this paper lies in a new counting lemma proved following the functional approach of gowers which complements the sparse regularity lemma of kohayakawa and rodl allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph we use this to prove sparse extensions of several well known combinatorial theorems including the removal lemmas for graphs and groups the erd s stone simonovits theorem and ramsey s theorem these results extend and improve upon a substantial body of previous work

fusion and singular vectors in a 1 1 highest weight cyclic modules we show how the interplay between the fusion formalism of conformal field theory and the knizhnik zamolodchikov equation leads to explicit formulae for the singular vectors in the highest weight representations ofa1 1

representations of uq sl n at roots of unity the gelfand zetlin basis for representations of uq sl n is improved to better fit the case when q is a root of unity the usual q deformed representations as well as the nilpotent periodic cyclic semi periodic semi cyclic and also some atypical representations are now described with the same formalism

group rings in which every element is uniquely the sum of a unit and an idempotent a ring r is called clean if every element is the sum of an idempotent and a unit and r is called uniquely clean if this representation is unique these rings are related to the boolean rings r is uniquely clean if and only if r j r is boolean idempotents lift modulo j r and idempotents in r are central it is shown that if the group ring rg is uniquely clean then r is uniquely clean and g is a 2 group the converse holds if g is locally finite in particular if g is solvable

analytic expressions for singular vectors of the n 2 superconformal algebra using explicit expressions for a class of singular vectors of then 2 untwisted algebra and following the approach of malikov feigin fuchs and kent we show that the analytically extended verma modules contain two linearly independent neutral singular vectors at the same grade we construct this two dimensional space and we identify the singular vectors of the original verma modules we show that in some verma modules these expressions lead to two linearly independent singular vectors which are at the same grade and have the same charge

minimal models of u 1 conformal current algebra a detailed study of the degenerate representations of the u 1 u 1 conformal current algebra and field theoretical models based on it is presented the main peculiarity of these models is that they contain an infinite set of fields with unfixed anomalous dimensions simple fusion rules for these fields are found and their four point functions are calculated explicitly the case of thirring like two dimensional models is discussed in detail

an algorithmic regularity lemma for hypergraphs in this paper we will consider the problem of designing an efficient algorithm that finds an epsilon regular partition of an l uniform hypergraph

positive energy representations holomorphic discrete series and finite dimensional irreps let g be a semi simple non compact lie group with unitary lowest highest weight representations we consider explicitly the relation among three types of representations of g positive energy unitary lowest weight representations holomorphic discrete series representations and non unitary finite dimensional irreps we consider mainly the conformal groups soo n 2 treating in full detail the cases n 1 3 4

canonical q deformations of noncompact lie super algebras the author proposes a procedure for q deformations of the real forms g of complex lie super algebras associated with generalized cartan matrices the procedure gives different q deformations for the nonconjugate cartan subalgebras of g the author gives several illustrations e g q deformed lorentz and conformal super algebras the q deformed conformal algebra contains as a subalgebra a q deformed poincare algebra and as hopf subalgebras two conjugate 11 generator q deformed weyl algebras the q deformed lorentz algebra is a hopf subalgebra of both weyl algebras

a fast approximation algorithm for computing thefrequencies of subgraphs in a given graph in this paper we give an algorithm which given a labeled graph on n vertices and a list of all labeled graphs on k vertices provides for each graph h of this list an approximation to the number of induced copies of h in g with total error small this algorithm has running time o n 1 over log log n cdot m n where m n is the time needed to square a n by n matrix with 0 1 entries over the integers the main tool in designing this algorithm is a variant of the regularity lemma of szemeredi

periodic and flat irreducible representations ofsu 3 q we construct all the periodic irreducible representations ofu su 3 q forq am root of unity their dimensions arek 2m 2 fork 1 m onlyk 1 m 2 for evenm their interest is that they could be a tool to generalize the chiral potts model by truncation of these representations we construct flat representations ofu su 3 q in which all the multiplicities of the weights are set to 1

singular vectors of representations of quantum groups the author gives explicit formulae for singular vectors of verma modules over uq g they give the general formula for g an for some cases when g dn and for some rank two subalgebras of g not an dn for this he uses a special basis of uq g where g is the negative root subalgebra of g which was introduced in his earlier work on the case q 1 this basis seems more economical than the poincare birkhoff witt type of basis used by malikov feigin and fuchs 1986 for the construction of singular vectors of verma modules in the case q 1 furthermore this basis turns out to be part of a general basis introduced recently for other reasons by lusztig 1988 for uq b where b is a borel subalgebra of g

null vectors in integrable field theory the form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine gordon model this space has to be isomorphic to that of the corresponding minimal model of conformal field theory we describe the subspaces which correspond to the verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion neveu schwarz or ramond depending on the particular primary field the description of null vectors relies on the relation between form factors and deformed hyper elliptic integrals the null vectors correspond to the deformed exact forms and to the deformed riemann bilinear identity in the operator language the null vectors are created by the action of two operators linear in the fermion and quadratic in the fermion we show that by factorizing out the null vectors one gets the space of operators with the correct character in the classical limit using the operators and we obtain a new very compact description of the kdv hierarchy we also discuss a beautiful relation with the method of whitham

clean general rings an associative ring with unity is called clean respectively uniquely clean if every element is uniquely the sum of an idempotent and a unit in this paper we define clean general rings with or without a unity and extend many of the basic results to the wider class in particular a clean general ring is an exchange ring in the sense of ara we then study the general analogue of the uniquely clean rings and their relationship to the boolean rings finally we introduce semiboolean rings as a natural generalization of boolean rings 2005 elsevier inc all rights reserved

quantum group uq dl singular vectors in the poincare birkhoff witt basis we give explicit expressions for the singular vectors of uq dl in terms of the poincare birkhoff witt basis we relate these expressions to those in terms of the simple root vectors

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

fock representations and brst cohomology insl 2 current algebra we investigate the structure of the fock modules overa 1 1 introduced by wakimoto we show that irreducible highest weight modules arise as degree zero cohomology groups in a brst like complex of fock modules chiral primary fields are constructed as brst invariant operators acting on fock modules as a result we obtain a free field representation of correlation functions of thesu 2 wzw model on the plane and on the torus we also consider representations of fractional level arising in polyakov s 2d quantum gravity finally we give a geometrical borel weil like interpretation of the wakimoto construction

the massless representations of the conformal quantum algebra the massless representations of the conformal quantum algebra uq su 2 2 for complex q such that mod q mod 1 are studied in detail by factorizing out all singular vectors of the corresponding verma modules a simple basis for these representations is explicitly constructed this basis is new also for the usual conformal algebra su 2 2 this construction allows a straightforward treatment of the case q a root of unity when the representations are unitary and finite dimensional

elementary representations and intertwining operators for su 2 2 i the structure of the group su 2 2 and of its lie algebra is studied in detail the results will be applied in subsequent parts devoted to the explicit construction of elementary representations of su 2 2 induced from different parabolic subgroups and of the intertwining operators between these representations a summary of some results of parts ii and iii is given

approximate hypergraph partitioning and applications szemeredi s regularity lemma is a cornerstone result in extremal combinatorics it roughly asserts that any dense graph is composed of a finite number of pseudorandom graphs the regularity lemma has found many applications in theoretical computer science and thus a lot of attention was given to designing algorithmic versions of this lemma our main results in this paper are the following i we introduce a new approach to the problem of constructing regular partitions of graphs which results in a surprisingly simple o n time algorithmic version of the regularity lemma thus improving over the previous o n 2 time algorithms furthermore unlike all the previous approaches for this problem see n alon and a naor siam j comput 35 2006 pp 787 803 r a duke h lefmann and v rodl siam j comput 24 1995 pp 598 620 a frieze and r kannan electron j combin 6 1999 article 17 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 and y kohayakawa v rodl and l thoma siam j comput 32 2003 pp 1210 1235 which only guaranteed to find tower size partitions our algorithm will find a small regular partition if one exists in the graph ii for any constant r geq3 we give an o n time randomized algorithm for constructing regular partitions of r uniform hypergraphs thus improving the previous o n 2r 1 time deterministic algorithms a czygrinow and v rodl siam j comput 30 2000 pp 1041 1066 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 these two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here given a directed hypergraph with bounded edge arities a set of constraints on the set sizes and densities of a possible partition of its vertex set and an approximation parameter we provide in o n time a partition approximating the constraints if a partition satisfying them exists we can also test in o 1 time for the existence of such a partition given the approximation parameter this algorithm extends the result of goldreich goldwasser and ron for graph partition problems o goldreich s goldwasser and d ron j acm 45 1998 pp 653 750 and encompasses more recent hypergraph related results such as the maximal constraint satisfaction approximation of g andersson and l engebretsen random structures algorithms 21 2002 pp 14 32

hidden quantum group symmetry and integrable perturbations of conformal field theories the hidden quantum group symmetry in the quantum sine gordon model is found this symmetry provides the possibility to restrict the operator algebra of the model to subalgebras it is shown that these subalgebras are massive deformations of minimal conformal field theories

subsingular vectors and conditionally invariant q deformed equations we give a systematic discussion of the relation between subsingular vectors of verma modules over semisimple lie algebras g and differential equations which are conditionally g invariant this is extended to the drinfeld jimbo q deformation uq g of g we treat in detail the conformal algebra su 2 2 its complexification sl 4 and their q deformations the conditionally invariant equations are the d alembert equation and a new equation arising from a subsingular vector proposed by bernstein gel fand gel fand 1971 we also give the q difference analogues of these equations

singular dimensions of the n 2 superconformal algebras i verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1 following a method developed for the virasoro algebra by kent we introduce the concept of adapted orderings on superconformal algebras we prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space we apply this method to the topological n 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological verma modules 0 1 2 or 3 depending on the type of verma module and the type of singular vector as a consequence we prove the conjecture of gato rivera and rosado on the possible existing types of topological singular vectors 4 in chiral verma modules and 29 in complete verma modules interestingly we have found two dimensional spaces of singular vectors at level 1 finally by using the topological twists and the spectral flows we also obtain the maximal dimensions of the singular vector spaces for the neveu schwarz n 2 algebra 0 1 or 2 and for the ramond n 2 algebra 0 1 2 or 3

continuous modules are clean ar ingr is said to be clean if every element of r is a sum of an idempotent and a unit the class of clean rings is quite large and includes for instance semiperfect rings and thus finite rings and rings of linear transformations of vector spaces we prove that the endomorphism ring of every continuous or discrete module is clean

on the structure of unitary conformal field theory ii representation theoretic approach two dimensional unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras chiral algebras are equipped with a family of co multiplications which serve to define tensor product representations chiral vertices arise as clebsch gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra the algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by yang baxter braid matrices chiral fusion is analyzed it is shown that the braid and fusion matrices determine invariants of knots and links connections between the representation theories of chiral algebras and of quantum groups are sketched finally it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras

on the exchange matrix for wznw model the connection between the exchange algebra in thesu 2 wess zumino novikov witten model and the quantum groupsu 2 is discussed it is shown that on the quasiclassical level this connection has the simple interpretation in terms of the lie poisson action ofsu 2 on the chiral components of the fields in the wznw model

special representations of at the roots of unity we show how to adapt the gelfand zetlin basis for describing the special representations of when q is a root of unity the explicit construction of special representations is presented in detail for n 3

quantum group structure in the fock space resolutions of widehat sl n representations we describe a complex of wakimoto type fock space modules for the affine kac moody algebra widehat sl n the intertwining operators that build the complex are obtained from contour integrals of so called screening operators we show that a quantum group structure underlies the algebra of screening operators this observation greatly facilitates the explicit determination of the intertwiners we conjecture that the complex provides a resolution of an irreducible highest weight module in terms of fock spaces

analysis in space time bundles i general considerations and the scalar bundle abstract mathematical features involved in the systematic development of elementary particle theory on an alternative cosmos space time are presented bundles representative of physical fields are studied in the unique variant of minkowski space m 0 enjoying similar properties of causality and symmetry the universal cosmos m is the universal cover of the causal compactification of m 0 the bundles studied are induced from representations of the scale extended poincare group which forms the isotropy group in m of the universal cover g s u 2 2 of the connected component of the group of all causal transformations on m discrete symmetries and higher dimensional cases are also discussed the primary focus is on the temporal evolution especially stability involving positivity of the energy wave equations implicative of finite propagation velocity and the unitarity and or composition series of associated actions of g general spin bundles on m are treated parallelized and correlated with bundles on m 0 associated covariant wave equations and the spectral resolution of fundamental quantum numbers are studied in detail in the scalar case

quasiprimary composite fields and null vectors in critical ising type models operator product expansions ope s are studied in unitary minimal conformal models on the circle the presence of null vectors at a certain level signaled by the the vanishing of kac s determinant leads to linear relations among su 1 1 covariant quasiprimary fields at that level in the ising model this is shown to imply the proportionality of two composite tensor currents of dimension 6 and the vanishing of 4 among the first five composite fields expected to appear in the ope of the stress energy tensor and the canonical fermi field of weight 1 2 for the supersymmetric tricritical model it only implies the vanishing of the quasiprimary fermi current of dimension 9 2

systematic identification of statistically significant network measures we present a graph embedding space i e a set of measures on graphs for performing statistical analyses of networks key improvements over existing approaches include discovery of motif hubs multiple overlapping significant subgraphs computational efficiency relative to subgraph census and flexibility the method is easily generalizable to weighted and signed graphs the embedding space is based on scalars functionals of the adjacency matrix representing the network scalars are global involving all nodes although they can be related to subgraph enumeration there is not a one to one mapping between scalars and subgraphs improvements in network randomization and significance testing we learn the distribution rather than assuming gaussianity are also presented the resulting algorithm establishes a systematic approach to the identification of the most significant scalars and suggests machine learning techniques for network classification

the adapted ordering method for lie algebras and superalgebras and their generalizations in 1998 the adapted ordering method was developed for the representation theory of the superconformal algebras in two dimensions it allows us to determine maximal dimensions for a given type of space of singular vectors to identify all singular vectors by only a few coefficients to spot subsingular vectors and to set the basis for constructing embedding diagrams in this paper we present the adapted ordering method for general lie algebras and superalgebras and their generalizations provided they can be triangulated we also review briefly the results obtained for the virasoro algebra and for the n 2 and ramond n 1 superconformal algebras

an optimal algorithm for checking regularity we present a deterministic algorithm cal a that in o m2 time verifies whether a given m by m bipartite graph g is regular in the sense of szemeredi regular partitions of graphs in problemes combinatoires et theorie des graphes orsay 1976 colloques internationaux cnrs 260 cnrs paris 1978 pp 399 401 in the case in which g is not regular enough our algorithm outputs a witness to this irregularity algorithm cal a may be used as a subroutine in an algorithm that finds an varepsilon regular partition of a given n vertex graph gamma in time o n2 this time complexity is optimal up to a constant factor and improves upon the bound o m n proved by alon et al the algorithmic aspects of the regularity lemma j algorithms 16 1994 pp 80 109 where m n o n2 376 is the time required to square a 0 1 matrix over the integers our approach is elementary except that it makes use of linear sized expanders to accomplish a suitable form of deterministic sampling

representation theory of the affine lie superalgebra at fractional level n 2 noncritical strings are closely related to the wess zumino novikov witten model and there is much hope to further probe the former by using the algebraic apparatus provided by the latter an important ingredient is the precise knowledge of the representation theory at fractional level in this paper the embedding diagrams of singular vectors appearing in verma modules for fractional values of the level p and q coprime are derived analytically the nilpotency of the fermionic generators in hslc requires the introduction of a nontrivial generalisation of the mff construction to relate singular vectors among themselves the diagrams reveal a striking similarity with the degenerate representations of the n 2 superconformal algebra

structural analysis and elementary representations of sl 4 r and gl 4 r and their covering groups the structure of the groups sl 4 r and gl 4 r and their universal cover ing groups sl 4 r and gl 4 r respectively and lie algebras sl 4 r and gl 4 r respectively are studied the parabolic subgroups and subalgebras are identified and the cuspidal parabolic subgroups singled out the iwasawa and bruhat decompositions are given explicitly all elementary representations er of sl 4 r are explicitly given in two equivalent realizations using the preceding detailed structural analysis the sl 4 r constructions are used for the explicit realization of all er of sl 4 r gl 4 r and gl 4 r the results shall be applied among other things elsewhere for the construction of all irreducible representations of the above groups

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

q analogs of iu n and u n 1 the starting point is the nonsemisimple inhomogeneous lie algebra un i2n denoted also as iu n where i2n represents an abelian subalgebra in semidirect product with the homogeneous part u n this is realized by explicitly giving the matrix elements of the generators on a modified gelfand zetlin basis that allows representations of infinite dimensions the enveloping algebra is q lifted by introducing q brackets in the matrix elements giving uq iu n the deformation of the abelian structure of i2n is studied for q 1 some implications are pointed out the important invariants are constructed for arbitrary n the results are compared to the corresponding ones for jimbo s construction of uq u n 1 on a gelfeld zetlin basis finally the related construction of uq u n 1 is presented and discussed here uq su 1 1 the q analog of relativistic motion in a plane is analyzed in the context of this formalism

irregular uq sl 3 representations at roots of unity via gel fand weyl zetlin basis the gel fand weyl zetlin gwz description of the uq sl 3 irregular irreps at roots of unity is explicitly given those are irreps fixed by the same parameters as the unitary irreducible representations uirs of su 3 yet having dimensions smaller than their classical counterparts the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule this description is made geometrically transparent by an arrangement of the standard su 3 gwz basis in a hexagonal pyramid which is valid for any q and seems new even for q 1 the pyramid has as a base the standard hexagon which gives the weight space of the uirs of su 3 in the plane of third component of isospin iz and hypercharge y while third dimension of this pyramid is related to the isospin i algebraically this arrangement is related to a one to one correspondence between the abstract gwz states and monomials in the algebra of raising generators uq g however those monomials are not in the standard poi