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

symmetries of clebsch gordan coefficients of the quantum group uq n permutation symmetries for clebsch gordan coefficients of the quantum group uq n and for uq n 1 reduced clebsch gordan coefficients of uq n are derived in particular the formula relating clebsch gordan coefficients for the tensor product tm1 tm2 with clebsch gordan coefficients for the tensor product with permuted representations tm1 and tm is obtained where tm is one of the irreducible representations in the decomposition of tm1 tm2 contrary to the classical case coefficients connecting these clebsch gordan coefficients depend on gel fand tsetlin patterns of one of irreducible representations this dependence is explicitly determined

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

matrix elements and wigner coefficients for uq gl n in this series of papers the fundamental wigner coefficients and the matrix elements of the generators for the quantum group uq gl n are derived the uq gl n uq gl n 1 reduced wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain uq gl n 1 invariants in the quantum group uq gl n the matrix elements of the elementary quantum group generators in the gel fand tsetlin basis are derived below while all fundamental wigner coefficients are derived in the second paper of the series

on semigroups of matrices with traces in a subfield abstract it is shown that an irreducible semigroup of n n complex matrices with real spectra is simultaneously similar to a semigroup of real matrices weaker results are obtained for semigroups of matrices over a general field with traces in a subfield

racah wigner algebra for q deformed algebras the concepts of vector coupling coefficients and recoupling coefficients are generalized to q deformed algebras their properties under complex conjugation and permutation of irreps are derived relations between the coefficients for example the racah backcoupling rule are proved it is shown that these properties may be used to recursively calculate the coupling and recoupling coefficients for all q algebras the 3 jm and 6 j symbols for su 2 q are used to illustrate the building up method

the number of lifts of a brauer character with a normal vertex in this paper we examine the behavior of lifts of brauer characters in p solvable groups in the main result we show that if ibr g has a normal vertex q and either p is odd or q is abelian then the number of lifts of is at most q q as a corollary we prove that if ibr g has an abelian vertex subgroup q then the number of lifts of in irr g is at most q

multiplicity free uq n coupling coefficients the coupling wigner clebsch gordan coefficients of the unitary quantum algebras uq n are considered the tensorial properties of the generator powers and their ordered products used in the explicit projectors and weight lowering procedures are established different expressions for the multiplicity free isofactors of uq n coupling coefficients those coupling an arbitrary and symmetric representations are derived explicit expression of the arbitrary isofactors in terms of their boundary values are proposed proportionality of the semistretched isofactors to the stretched q recoupling coefficients of uq n 1 q analogues of 9j symbols is demonstrated the stretched isofactors of uq 3 are expressed in terms of clebsch gordan coefficients of uq 2

on fully ramified brauer characters let z be a normal subgroup of a finite group let p 5 be a prime and let ibr z be an irreducible g invariant p brauer character of z suppose that g e for some ibr g then g z is solvable in other words a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group

vertices for characters of p solvable groups suppose that g is a finite p solvable group we associate to every irreducible complex character x irr g of g a canonical pair q where q is a p subgroup of g and irr q uniquely determined by x up to g conjugacy this pair behaves as a green vertex and partitions irr g into families of characters using the pair q we give a canonical choice of a certain p radical subgroup r of g and a character irr r associated to x which was predicted by some conjecture of g r robinson

on clebsch gordan coefficients and matrix elements of representations of the quantum algebra uq su2 clebsch gordan coefficients and matrix elements of irreducible representations of the quantum algebra uq su2 were considered in several papers in particular a few expressions for them were derived an approach to clebsch gordan coefficients and to matrix elements of representations of uq su2 on the base of the theory of basic hypergeometric functions is given this approach allows one to obtain q analogs of all well known classical expressions for clebsch gordan coefficients most of them were absent new symmetry relations generating functions and recurrence formulas for clebsch gordan coefficients of uq su2 are obtained unlike other papers clebsch gordan coefficients and matrix elements are considered on the base of minimal theoretical constructions in fact without using the notion of a c algebra and of a hopf algebra

on racah coefficients of the quantum algebra uq su2 a new expression for the racah coefficients of the quantum algebra uq su2 is derived new formulae for these coefficients can be obtained with the help of the results of the theory of basic hypergeometric functions it is shown that the clebsch gordan coefficients are recovered as the asymptotic limit of the racah coefficients recurrence relations for the racah and clebsch gordan coefficients are derived

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

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

the pattern calculus for tensor operators in quantum groups an explicit algebraic evaluation is given for all q tensor operators with unit norm belonging to the quantum group uq u n and having extremal operator shift patterns acting on arbitrary uq u n irreps these rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns a more conceptual derivation of these results is discussed using fundamental properties of the q 6j coefficients

towards the canonical tensor operators of uq 3 i the maximal null space case generalizing the su 3 canonical tensor operator concept biedenharn and louck to the quantum algebra uq 3 the wigner clebsch gordan coefficients of uq 3 with repeating irreducible representations are considered extremal projectors of the quantum algebra uq 3 in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq 3 canonical isoscalar factors explicit expressions of the uq 3 isofactors corresponding to the maximal null space case of the uq 3 unit canonical tensor operators and their normalization factors denominator functions are presented the transposition and conjugation phase factors for the su 3 and uq 3 canonical isofactors are correlated with phases and zeros of boundary isofactors invariance of the canonical isofactors or absence of such invariance under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance or numerical degeneracy of the usual splitting

raising and lowering operators for uq n after reviewing the defining relations for the gq algebra uq n the author constructs a cartan weyl basis and lists the q commutation relations of its generators he then uses the latter to explicitly construct sets of raising and lowering operators for the canonical chain of q algebras uq n uq n 1 generalizing those introduced by nagel and moshinsky 1965 for u m u n 1 finally he gives their normalization coefficients and shows how the normalized operators can be used to go from any gel fand tseitlin basis state that is of highest weight in uq n 1 to any other one and ultimately to construct the whole gel fand tseitlin basis from its highest weight state this work both generalizes a previous work of ueno et al 1989 on lowering operators and provides explicit expressions for their operators in terms of uq n cartan weyl generators

vertices and normal subgroups in solvable groups abstract for a finite group g and a fixed prime p one can attach to each irreducible brauer character of g a p subgroup q called the vertex of that is unique up to conjugacy in this paper we examine the behavior of the vertices of with respect to normal subgroups when g is assumed to be p solvable for arbitrary finite p solvable groups we develop a correspondence between the set of brauer characters of g with vertex q and the set of brauer characters of a certain subgroup of g with vertex q moreover in the case that g has odd order we extend a result of navarro regarding the behavior with respect to normal subgroups of a correspondence of brauer characters of p degree to a result regarding the behavior with respect to normal subgroups of a correspondence of brauer characters of arbitrary degree

reduced wigner coefficients for uq gl n this is the second paper in a series concerned with the racah wigner calculus for the quantum group uq gl n in this approach all reduced wigner coefficients are determined algebraically by the eigenvalues of certain generalized casimir invariants here the method is applied to evaluate all fundamental wigner coefficients and reduced wigner coefficients for uq gl n

bounds on the number of lifts of a brauer character in a p solvable group abstract the fong swan theorem shows that for a p solvable group g and brauer character ibr p g there is an ordinary character irr g such that 0 where 0 denotes restriction to the p regular elements of g this still holds in the generality of separable groups where ibr p g is replaced by i g for i g let l irr g 0 in this paper we give a lower bound for the size of l in terms of the structure of the normal nucleus of and if g is assumed to be odd and p we give an upper bound for l in terms of the vertex subgroup for

a generalization of brauer s theorem on splitting fields to semigroups abstract let f be either an algebraic number field or a p adic field and a a central simple algebra over f suppose a is spanned by a multiplicative semigroup a with the property that the minimal polynomial of every g splits over f then a represents the trivial class in the brauer group of f