non semi regular quantum groups coming from number theory in this paper we study c algebraic quantum groups obtained through the bicrossed product construction examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi regular the crossed product of the quantum group acting on itself by translations does not contain any compact operator we describe all corepresentations of these quantum groups and the associated universal c algebras on the way we provide several remarks on c algebraic properties of quantum groups and their actions

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

counterexamples concerning the diagonal elements of normal matrices abstract an infinite class of conditions known to be satisfied by the diagonal elements of a normal matrix with prescribed spectrum is shown to be independent of other known conditions satisfied by the diagonal elements but nevertheless insufficient to characterize the diagonal

cross product bialgebras part ii abstract this is the central article of a series of three papers on cross product bialgebras we present a universal theory of bialgebra factorizations or cross product bialgebras with cocycles and dual cocycles we also provide an equivalent co modular co cyclic formulation all known examples as for instance bi or smash doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory furthermore our construction yields various novel types of cross product bialgebras

a trace inequality with a subtracted term abstract for fixed real or complex matrices a and b the well known von neumann trace inequality identifies the maximum of tr uavb z sfnc as u and v range over the unitary group the maximum being a bilinear expression in the singular values of a and b this paper establishes the analogue of this inequality for real matrices a and b when u and v range over the proper real orthogonal group the maximum is again a bilinear expression in the singular values but there is a subtracted term when a and b have determinants of opposite sign

2 cocycles and twisting of kac algebras we describe the twisting construction with the help of 2 cocycles on hopf von neumann and george kac algebras we show that twisted kac algebras are again kac algebras using this construction we give a wide class of new quantizations of the heisenberg group and describe several series of non trivial finite dimensional hopf algebras kac algebras of dimensions 4n and as twisting of finite groups

bicrossproduct structure of the quantum weyl group abstract the quantum weyl group formula associated to a complex simple lie algebra g consists of the quantum group uq g with certain quantum simple reflections w i adjoined let kw be the group algebra of the standard covering w of the weyl group of g here k c we show that formula has the structure of a cocycle bicrossproduct formula kw uq g it consists as an algebra of a cocycle semidirect product by a cocycle action of kw on uq g defined with respect to a certain non abelian cocycle it consists as a coalgebra of an extension by a non abelian dual cocycle the dual of formula is also a bicrossproduct and consists as an algebra of an extension of the dual of uq g by the commutative algebra of functions on w via a cocycle

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

optimal path cover problem on block graphs abstract let g v e be a block graph first we show that an algorithm for finding the path partition number p g by j h yan and g j chang gives wrong answers to some block graphs then we present an efficient algorithm for finding a minimum path partition of g not just the path partition number p g the complexity of this algorithm is o m where m e

on the k path partition of graphs the k path partition problem is to partition a graph into the minimum number of paths so that none of them has length more than k for a given positive integer k the problem is a generalization of the hamiltonian path problem and the problem of partitioning a graph into the minimum number of paths the k path partition problem remains np complete on the class of chordal bipartite graphs if k is part of the input and we show that it is np complete on the class of comparability graphs even for k 3 on the positive side we present a polynomial time solution for the problem with any k on bipartite permutation graphs which form a subclass of chordal bipartite graphs

singular values diagonal elements and extreme matrices abstract for complex matrices a and b there are inequalities related to the diagonal elements of ab and the singular values of a and b we study the conditions on the matrices for which those inequalities become equalities in all cases the conditions are both necessary and sufficient

deformation of a kac algebra by an abelian subgroup some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2 cocycle we make this construction in the framework of kac algebras we show that these deformations are still kac algebras using this construction we give new quantizations of the heisenberg group from this point of view we analyse the dimension 8 non trivial example of kac and paljutkin and give a new example of non trivial dimension 12 semi simple hopf algebras a dimension 12 kac algebra

on lei miranda and thompson s result on singular values and diagonal elements abstract thomspon and sing s result on the singular values and the diagonal elements of a complex n n matrix has been recently extended to arbitrary m matrices by lei and by miranda and thompson independently the real case in which the determinant of the product of the m real matrices is nonnegative or nonpositive was raised by miranda and thompson in this note we provide an answer to the question and we also extend some other results of lei

a polynomial solution to the k fixed endpoint path cover problem on proper interval graphs we study a variant of the path cover problem namely the k fixed endpoint path cover problem or kpc for short given a graph g and a subset t of k vertices of v g a k fixed endpoint path cover of g with respect to t is a set of vertex disjoint paths p that covers the vertices of g such that the k vertices of t are all endpoints of the paths in p the kpc problem is to find a k fixed endpoint path cover of g of minimum cardinality note that if t is empty or equivalently k 0 the stated problem coincides with the classical path cover problem the kpc problem generalizes some path cover related problems such as the 1hp and 2hp problems which have been proved to be np complete note that the complexity status for both 1hp and 2hp problems on interval graphs remains an open question damaschke 1993 9 in this paper we show that the kpc problem can be solved in linear time on the class of proper interval graphs that is in o n m time on a proper interval graph on n vertices and m edges the proposed algorithm is simple requires linear space and also enables us to solve the 1hp and 2hp problems on proper interval graphs within the same time and space complexity

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

optimal covering of cacti by vertex disjoint paths moran s and y wolfstahl optimal covering of cacti by vertex disjoint paths theoretical computer science 84 1991 179 197 a path couer or in short cover of a graph g is a set of vertex disjoint paths which cover all the vertices of g an optimal couer of g is a cover of the smallest possible cardinality notable applications of graph covering are code optimization and mapping parallel programs to parallel architectures the optimal covering problem is known to be np complete even for cubic 3connected planar graphs motivated by the intractability of this problem we develop an efficient optimal covering algorithm for cacti i e graphs where no edge lies on more than one cycle in doing so we generalize the results of boesch gimpel mchugh 1974 and of pinter wolfstahl 1987 where optimal covering algorithms for trees and graphs where no two cycles share a vertex were presented

on the k path cover problem for cacti in this paper we investigate the k path cover problem for graphs which is to find the minimum number of vertex disjoint k paths that cover all the vertices of a graph the k path cover problem for general graphs is np complete though notable applications of this problem to database design network vlsi design ring protocols and code optimization efficient algorithms are known for only few special classes of graphs in order to solve this problem for cacti i e graphs where no edge lies on more than one cycle we introduce the so called steiner version of the k path cover problem and develop an efficient algorithm for the steiner k path cover problem for cacti which finds an optimal k path cover for a given cactus in polynomial time

cross product bialgebras i abstract the subject of this article is bialgebra factorizations or cross product bialgebras without cocycles we establish a theory characterizing cross product bialgebras universally in terms of projections and injections especially all known types of biproduct double cross product and bicross product bialgebras can be described by this theory furthermore the theory provides new families of cocycle free cross product bialgebras besides the universal characterization we find an equivalent co modular description of certain types of cross product bialgebras in terms of so called hopf data with the help of hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras we are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of hopf bimodules over a hopf algebra majid s double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category this resembles the case of the drinfel d double which can be constructed as a twist of a specific cross product

the approximability of the weighted hamiltonian path completion problem on a tree given a graph the hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a hamiltonian path in this paper we show that the hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless np p this problem remains hard to approximate even when the given subgraph is a tree moreover if the edge weights are restricted to be either 1 or 2 the hamiltonian path completion problem on a tree is still np hard then it is observed that this problem is strongly np hard so it does not have any fully polynomial time approximation scheme fptas unless np p when the given tree is a k tree we give an approximation algorithm with performance ratio 1 5

extensions of hopf algebras and lie bialgebras let f g be finite dimensional lie algebras over a field of characteristic zero regard f and g the dual lie coalgebra of g as lie bialgebras with zero cobracket and zero bracket respectively suppose that a matched pair f g of lie bialgebras is given which has structure maps then it induces a matched pair u f ug of hopf algebras where u f is the universal envelope of f and ug is the hopf dual of ug we show that the group opext u f ug of cleft hopf algebra extensions associated with u f ug is naturally isomorphic to the group opext f g of lie bialgebra extensions associated with f g an exact sequence involving either of these groups is obtained which is a variation of the exact sequence due to g i kac if g g g there follows a bijection between the set ext u f ug of all cleft hopf algebra extensions of u f by ug and the set ext f g of all lie bialgebra extensions of f by g

matrices with some extremal properties abstract for a complex matrix a there are many inequalities related to its eigenvalues diagonal elements and three types of singular values we investigate the conditions on the matrices for which those inequalities become equalities it is shown that in many cases the conditions are both necessary and sufficient for the matrices to be unitarily similar to matrices of lower orders moreover we get alternative proofs of various characterization theorems of different classes of matrices

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

on a morita equivalence between the duals of quantum su 2 and quantum e 2 abstract let su q 2 and e q 2 be woronowicz s q deformations of respectively the compact lie group su 2 and the non trivial double cover of the lie group e 2 of euclidean transformations of the plane we prove that in some sense their duals are morita equivalent locally compact quantum groups in more concrete terms we prove that the von neumann algebraic quantum groups l su q 2 and l e q 2 are unitary cocycle deformations of each other

spectral inequalities and equalities involving products of matrices abstract let a 1 a k be n n matrices we studied inequalities and equalities involving eigenvalues diagonal entries and singular values of a 0 a 1 a k and those of a 1 a k it is shown that the matrices attaining equalities often have special reducible structure the results are then applied to study normality and reducibility of matrices extending some results and answering some questions of miranda wang and zhang

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

calculations of some groups of hopf algebra extensions w let us recall the theory of group extensions with abelian kernels mc x chap iv let g be a group and m a g module the equivalence classes of group extensions m a e a g giving rise to the g module action on m form a group with respect to the baer sum furthermore this group is 2 isomorphic to the cohomology group h g m the theory is extended to the theory of hopf algebra extensions which w x are abelian in some sense h s we restrict our interests to hopf algebra extensions over a field k which are of the form

an algebraic framework for group duality abstract a hopf algebra is a pair a whereais an associative algebra with identity and a homomorphism formatoa asatisfying certain conditions if we drop the assumption thatahas an identity and if we allow to have values in the so called multiplier algebram a a we get a natural extension of the notion of a hopf algebra we call this a multiplier hopf algebra the motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group also for these multiplier hopf algebras there is a natural notion of left and right invariance for linear functionals called integrals in hopf algebra theory we show that if such invariant functionals exist they are unique up to a scalar and faithful for a regular multiplier hopf algebra a i e with invertible antipode with invariant functionals we construct in a canonical way the dual it is again a regular multiplier hopf algebra with invariant functionals it is also shown that the dual of is canonically isomorphic with the original multiplier hopf algebra a it is possible to generalize many aspects of abstract harmonic analysis here one can define the fourier transform one can prove plancherel s formula because any finite dimensional hopf algebra is a regular multiplier hopf algebra and has invariant functionals our duality theorem applies to all finite dimensional hopf algebras then it coincides with the usual duality for such hopf algebras but our category of multiplier hopf algebras also includes in a certain way the discrete quantum groups and the compact quantum groups our duality includes the duality between discrete quantum groups and compact quantum groups in particular it includes the duality between compact abelian groups and discrete abelian groups one of the nice features of our theory is that we have an extension of this duality to the non abelian case but within one category this is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type we prove that also these are dual to each other we treat an example that is sufficiently general to illustrate most of the different features of our theory it is also possible to construct the quantum double of drinfel d within this category this provides a still wider class of examples so we obtain many more than just the compact and discrete quantum within this setting

a supplement to the von neumann trace inequality for singular values abstract the von neumann trace inequality for singular values is reexamined by determining the possible values for the diagonal elements contributing to the trace the possible values for the eigenvalues contributing to the trace are also found

cases of equality for certain inequalities involving a g doubly stochastic map abstract we study inner product inequalities for a g doubly stochastic map with respect to a group induced cone ordering we give necessary and sufficient conditions for the inequalities to become equalities a special condition is obtained if a birkhoff type theorem holds for a finite reflection group related to the ordering the conditions are applied to derive some classical and new results on equality cases for majorization inequalities connected with certain group actions

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

optimal path cover problem on block graphs and bipartite permutation graphs abstract the optimal path cover problem is to find a minimum number of vertex disjoint paths which together cover all the vertices of the graph in this paper we present linear time algorithms for the optimal path cover problem for the class of block graphs and bipartite permutation graphs