quantum doubles from a class of noncocommutative weak hopf algebras the concept of biperfect noncocommutative weak hopf algebras is introduced and their properties are discussed a new type of quasi bicrossed products is constructed by means of weak hopf skew pairs of the weak hopf algebras which are generalizations of the hopf pairs introduced by takeuchi as a special case the quantum double of a finite dimensional biperfect noncocommutative weak hopf algebra is built examples of quantum doubles from a clifford monoid as well as a noncommutative and noncocommutative weak hopf algebra are given generalizing quantum doubles from a group and a noncommutative and noncocommutative hopf algebra respectively moreover some characterizations of quantum doubles of finite dimensional biperfect weak hopf algebras are obtained

classification of irreducible weight modules over higher rank virasoro algebras let g be a rank n additive subgroup of c and vir g the corresponding virasoro algebra of rank n in the present paper irreducible weight modules with finite dimensional weight spaces over vir g are completely determined there are two different classes of them one class consists of simple modules of intermediate series whose weight spaces are all 1 dimensional the other is constructed by using intermediate series modules over a virasoro subalgebra of rank n 1 the classification of such modules over the classical virasoro algebra was obtained by o mathieu in 1992 using a completely different approach

nonreduced extended affine weyl groups abstract in this paper we study the structure of the weyl groups of nonreduced extended affine root systems we show that similar to the case of reduced types an extended affine weyl group w of type bcl is semidirect product of a finite weyl group w of type bl and a heisenberg like normal subgroup h which is also a characteristic subgroup of w moreover h is of the form h h h 0 where both h and h 0 are normal subgroups of h with h h 0 1 h is naturally isomorphic to the root lattice of a finite root system of type bcl furthermore the semidirect product of w and h is isomorphic to the weyl group of a kac moody affine subroot system of r of type bcl

integrable representations of toroidal lie algebras co ordinatized by rational quantum tori for any positive integers d n with d 1 and n 1 we fix an n by n complex matrix q qij satisfying qii 1 and qij qji 1 with all qij roots of unity let d q be the universal central extension of the lie subalgebra sld cq of gld cq with trace in cq cq where cq is the rational quantum torus associated to q and let d q be the lie algebra by adding the n degree derivations to d q with respect to the n non commuting variables in cq the lie algebra d q called the toroidal lie algebra co ordinatized by the rational quantum torus cq has an n dimensional center c in this paper we obtain a classification of irreducible integrable modules with finite dimensional weight spaces and with non zero center action over the toroidal lie algebra d q

new generalized simple lie algebras of cartan type over a field with characteristic 0 abstract we construct four new series of generalized simple lie algebras of cartan type using the mixtures of grading operators and down grading operators our results in this paper are further generalizations of those in osborn s work j algebra 185 1996 820 835

states that are far from being stabilizer states stabilizer states are eigenvectors of maximal commuting sets of operators in a finite heisenberg group states that are far from being stabilizer states include magic states in quantum computation mub balanced states and sic vectors in prime dimensions the latter two fall under the umbrella of minimum uncertainty states muss in the sense of wootters and sussman we study the correlation between two ways in which the notion of far from being a stabilizer state can be quantified two theorems valid for all prime dimensions are given as well as detailed results for low dimensions in dimension 7 we identify the mub balanced states as being antipodal to the sic vectors within the set of mus in a sense that we make definite in dimension 4 we show that the states that come closest to being mus with respect to all of the six stabilizer mubs are the fiducial vectors for alltop mubs

galois quantum systems a finite quantum system in which the position and momentum take values in the galois field gf p is constructed from a smaller quantum system in which the position and momentum take values in using field extension the galois trace is used in the definition of the fourier transform the heisenberg weyl group of displacements and the sp 2 gf p group of symplectic transformations are studied a class of transformations inspired by the frobenius maps in galois fields is introduced the relationship of this galois quantum system with its subsystems in which the position and momentum take values in subfields of gf p is discussed

semilattice graded weak hopf algebra and its related quantum g double the major purpose of this paper is to construct a weak hopf algebra with grading from a family of hopf algebras and then to gain a related quantum g double with regular r matrix first over a field k we introduce a so called semilattice graded weak hopf algebra h yh then the quantum g double d h of h is obtained in case that h is commutative moreover it is shown that d h is semisimple respectively von neumann regular if and only if h is a semisimple respectively von neumann regular hopf algebra at last a nontrivial example of semilattice graded weak hopf algebras is obtained

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

vertex operator representation of some quantum tori lie algebras we are defining the trigonometric lie subalgebras in bar x infty bar a infty bar b infty bar c infty bar d infty which are the natural generalization of the well known sin lie algebra the embedding formulas into bar x infty are introduced these algebras can be considered as some lie algebras of quantum tori an irreducible representation ofa b series of trigonometric lie algebras is constructed special cases of the trigonometric lie factor algebras which can be considered as a quantum preserving lie algebra structure deformation of the kac moody algebras are considered

harish chandra modules of the intermediate series over the high rank virasoro algebras and high rank super virasoro algebras indecomposable modules with weight multiplicities 1 over the high rank virasoro algebras are constructed and classified a notion of the high rank super virasoro algebras is introduced the classification of all indecomposable modules with weight multiplicities 1 over the high rank super virasoro algebras is obtained

integrable representations for generalized virasoro toroidal lie algebras abstract in this paper we give the structure of irreducible integrable modules of generalized virasoro toroidal lie algebras with finite dimensional weight spaces and non zero central charge

unbiased bases hadamards for six level systems four ways from fourier in quantum mechanics some properties are maximally incompatible such as the position and momentum of a particle or the vertical and horizontal projections of a two level spin given any definite state of one property the other property is completely random or unbiased for n level systems the six level ones are the smallest for which a tomographically efficient set of n 1 mutually unbiased bases has not been found to facilitate the search we numerically extend the classification of unbiased bases or hadamards by incrementally adjusting relative phases in a standard basis we consider the nonunitarity caused by small adjustments with a second order taylor expansion and choose incremental steps within the four dimensional null space of the curvature in this way we prescribe a numerical integration of a four parameter set of hadamards of order of 6

complete sets of cyclic mutually unbiased bases in even prime power dimensions we present a construction method for complete sets of cyclic mutually unbiased bases mubs in hilbert spaces of even prime power dimensions in comparison to the usual complete sets of mubs complete cyclic sets possess the additional property of being generated by a single unitary operator the construction method is based on the idea of obtaining a partition of multi qubit pauli operators into maximal commuting sets of operators with the help of a suitable element of the clifford group as a consequence we explicitly obtain complete sets of cyclic mubs generated by a single element of the clifford group in dimensions 2 m for m 1 2 24

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

a class of integrable modules for the core of eala coordinatized by quantum tori extended affine lie algebras eala are higher dimensional generalizations of affine kac moody lie algebras introduced in 19 they have been further studied in 1 2 5 toroidal lie algebras which are universal central extensions of g c t 1 1 t 1 n g simple finite dimensional lie algebra and are prime examples of cores of ealas which are studied in 4 7 14 16 22 24 25 there are many ealas which allow not only laurent polynomial algebra c t 1 1 t 1 n as co ordinate algebra but also quantum tori jordan tori and the octonians tori as co ordinated algebras depending on type of the lie algebra see 1 3 5 6 26 the structure theory of the eala of type ad 1 is tied up with lie algebra gld c cq where cq is the quantum torus see section 1 quantum tori defined in 21 are non commutative analogue of laurent polynomial algebras in this paper we study integrable modules for the core of the eala corresponding to gld c cq which we call we will also add derivations d1 d2 to and call it here cq is two variable quantum torus defined as the ring of non commutating laurent polynomials c t 1 1 t 1 2 with relation to t1t2 qt2t1 note that if q is generic then is an eala we classify theorem 5 2 irreducible integrable modules see definition 1 9 for where the zero degree center acts non trivially we prove that any such module is isomorphic to a highest weight module or a lowest module up to an automorphism see section 5 let and aff be finite and affine root system of sld c and its affinization consider aff z 2 and consider the subalgebra b corresponding to see section 1 by highest weight modules we mean a module with a weight vector killed by b and the highest weight h c t t 1 given as in theorem 4 1 for any such we define a highest weight module

simple lie algebras of witt type abstract letkbe a field letabe an associative commutativek algebra and let be a nonzerok vector space of commutingk derivations ofa then with a rather natural definition a k a becomes a lie algebra and we obtain necessary and sufficient conditions here for this lie algebra to be simple with one minor exception in characteristic 2 simplicity occurs if and only ifais simple anda a acts faithfully as derivations ona

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

classification of indecomposable modules of the intermediate series over the twisted n 2 superconformal algebra in this paper a classification of indecomposable modules of the intermediate series over the twisted n 2 superconformal algebra is obtained

mutually unbiased triplets from non affine families of complex hadamard matrices in dimension 6 we study the problem of constructing mutually unbiased bases in dimension 6 this approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions our technique suggests the existence of previously unknown symmetries in karlsson s non affine family k 2 6 which we confirm analytically also we obtain strong evidence that not more than three mutually unbiased bases can be constructed from pairs which contain members of some non affine families of complex hadamard matrices

weak hopf algebras and some new solutions of the quantum yang baxter equation abstract in this paper the concepts of a weak hopf algebra and a quasi braided almost bialgebra are introduced it is shown that the quantum quasi doubles of some weak hopf algebras are quasi braided almost bialgebras this fact implies that some new solutions of the quantum yang baxter equation can be constructed from some weak hopf algebras in particular when the weak hopf algebra is a finite clifford monoid algebra

γ conformal algebras conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points we classify these algebras and their representations in terms of lie algebras and their representations with an action of the group to every conformal algebra and a character of we associate a lie algebra generated by fields with the ope with simple poles examples include twisted affine kac moody algebras the sin algebra which is a conformal analogue of the general linear algebra and its analogues the algebra of pseudodifferential operators on the circle etc

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

braided version of shirshov witt theorem the shirshov witt theorem claims that every subalgebra of a free lie algebra is free in character istic zero this theorem can be restated in terms of a free associative algebra every hopf subalgebra of a free algebra kxiwith the coproduct xi xi 1 1 xi is free and it is freely generated by primitive elements our aim is to extend this result to free algebras with a braided coproduct as far as possible by means of p m cohn theory we show that if a subalgebra is a right categorical right coideal then it is free we consider more thoroughly involutive braidings 2 id over a field of zero characteristic in this case every braided hopf subalgebra is generated by primitive elements moreover the space of all primitive elements forms a free lie algebra in the context of this re sult we discuss the situation that arises around the problem of embedding of a lie algebra in its associative universal enveloping algebra 2005 elsevier inc all rights reserved

quasifinite highest weight modules over the lie algebra of matrix differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t tm 1 the unitary ones are classified as well similar results are obtained for the sin algebras

weak hopf algebras corresponding to uq sln we investigate the weak hopf algebras of li based on uq sln and sweedler s finite dimensional example we give weak hopf algebra isomorphisms between the weak generalizations of uq sln which are upgraded automorphisms of uq sln and hence give a classification of these structures as weak hopf algebras we also show how to decompose these examples into a direct sum which leads to unexpected isomorphisms between their algebraic structure

weak hopf algebras with projection and weak smash bialgebra structures abstract in this paper we study weak hopf algebras with projection if f h b g b h are morphisms of weak hopf algebras such that g f idh we prove that it is possible to find an object bh in the new category of weak yetter drinfeld modules that verifies similar conditions to the ones include in the definition of weak hopf algebra finally we define weak smash bialgebra structures and prove that under central and cocentral conditions bh and h determine an example of them

weak quantum borcherds superalgebras and their representations we define a weak quantum borcherds superalgebra nqd g which is a weak hopf superalgebra we also discuss the basis and the highest weight modules of nqd g then we study the weak a form and the classical limit of nqd g

twisted γ lie algebras and their vertex operator representations let be a generic subgroup of the multiplicative group c of nonzero complex numbers we define a class of lie algebras associated to called twisted lie algebras which are natural generalizations of the twisted affine lie algebras starting from an arbitrary even sublattice q of zn and an arbitrary finite order isometry of zn preserving q we construct a family of twisted vertex operators acting on generalized fock spaces which afford irreducible representations for certain twisted lie algebras as an application this recovers a number of known vertex operator realizations for infinite dimensional lie algebras such as twisted affine lie algebras extended affine lie algebras of type a trigonometric lie algebras of series a and b unitary lie algebras and bc graded lie algebras

supports of weight modules over witt algebras as the first step towards a classification of simple weight modules with finite dimensional weight spaces over witt algebras w n we explicitly describe the supports of such modules we also obtain

superspace geometry the exact uncertainty relationship between complementary aspects after reviewing the standard uncertainty relations due to heisenberg 1927 robertson 1929 and schrodinger 1930 as well as the relations of deutsch and of maassen and uffink including the so called entropic relations the author presents a complete account of the uncertainty relationship between complementary aspects in terms of superspace geometry an approach not hitherto employed two incompatible properties a sigma alpha aalpha mod alpha alpha mod and n sigma nnn mod n n mod belong to a pair of complementary aspects defined by two orthonormal bases mod alpha and mod n in the hilbert space h if the state is mod psi then p alpha mod alpha mod psi mod 2 is the probability of obtaining the value aalpha in a measurement of a and p n mod n mod psi mod 2 is the probability of obtaining the value nn in a measurement of n the two aspects are characterised relative to mod psi by the numbers so called purities pi sigma alpha p alpha 2 and pi sigma n p n 2 both or 1 he gives a complete characterisation of the uncertainty relationship between a and n more precisely between their aspects in terms of the range of joint values of pi pi for arbitrary initial states pure as well as mixed a theorem of lenard is given an alternative proof employing only elementary superspace geometry the results depend on two angles phi m minimal angle and phi m maximal angle between the two aspects exact expressions for phi m and phi m are obtained in terms of the overlap matrix lambda lambda alpha n mod alpha mod n mod 2 as a corollary he finds the uncertainty relation for a pure state mod psi pi pi or 1 1 g 1 1 g cos phi m where g dim h and a sharper one for mixed states pi pi 2 is obtainable if and only if the intersection of the aspects holds a pure state if phi m pi 2 maximal incompatibility then pi pi or 1 1 g is a special case of a stronger relation sigma mu g o pi mu 2 which one obtains for g 1 maximally incompatible aspects by means of a thereom of ivanovic

dynamical systems on quantum tori lie algebras we use quantum tori lie algebras qtla which are a one parameter family of sub algebras ofgl to describe local and non local versions of the toda systems it turns out that the central charge of qtla is responsible for the non locality there are two regimes in the local systems conformal for irrational values of the parameter and non conformal and integrable for its rational values we also consider infinite dimensional analogs of rigid tops some of these systems give rise to quantized magneto hydrodynamic equations of an ideal fluid on a torus we also consider infinite dimensional versions of the integrable euler and clebsch cases

constants of coordinate differential calculi defined by yang baxter operators we investigate in details a first order differential calculus with right partial derivatives set up by a not necessarily invertible yang baxter operator the optimal algebra for this calculus has a natural structure of a braided hopf algebra and it is isomorphic to the quantum symmetric algebra the induced to the optimal algebra and to the free cover algebra calculi are right covariant they are bicovariant if and only if the related braiding is involutive by means of the p m cohn theory we show that the subalgebra of constants for the cover free differential algebra is a free algebra and an ad invariant left coideal if the given algebra is finitely generated then every differential left ideal is generated by constants a noncommutative taylor series decomposition formula is valid and the category of locally nilpotent modules over the operator algebra is semisimple with the only simple object that is isomorphic to the optimal algebra as a module we find a necessary and sufficient condition for a 1 form to be a complete differential

classification of harish chandra modules over the higher rank virasoro algebras we classify the harish chandra modules over the higher rank virasoro and super virasoro algebras it is proved that a harish chandra module i e an irreducible weight module with finite weight multiplicities over a higher rank virasoro or super virasoro algebra is either a module of the intermediate series or a finitely dense module as an application it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized witt algebra is either a uniformly bounded module i e a module with weight multiplicities uniformly bounded with all nonzero weights having the same multiplicity or a finitely dense module as long as the generalized witt algebra satisfies one minor condition

classification of quasifinite representations of a lie algebra related to block type abstract a well known theorem of mathieu s states that a harish chandra module over the virasoro algebra is either a highest weight module a lowest weight module or a module of the intermediate series it is proved in this paper that an analogous result also holds for the lie algebra b related to block type with basis l i c i z i 0 and relations l i l j i 1 j 1 l i j 0 i j 0 3 12 c namely an irreducible quasifinite b module is either a highest weight module a lowest weight module or a module of the intermediate series

representations of some quantum tori lie subalgebras in this paper we define the q analog virasoro like lie subalgebras in x a b c d the embedding formulas into x are introduced irreducible highest weight representations of aq bq and cq series of the q analog virasoro like lie algebras in terms of vertex operators are constructed we also construct the polynomial representations of the aq bq cq and dq series of the q analog virasoro like lie algebras

weak hopf algebras corresponding to cartan matrices we replace the group of grouplike elements of the quantized enveloping algebra uq g of a finite dimensional semisimple lie algebra g by some regular monoid and get the weak hopf algebra wqd g it is a subclass of weak hopf algebras but not hopf algebras then we devote to constructing a basis of wqd g and determine the group of weak hopf algebra automorphisms of wqd g when q is not a root of unity

automorphisms and verma modules for generalized schrodinger virasoro algebras abstract let f be a field of characteristic 0 g an additive subgroup of f f satisfying g and 2 g we define a class of infinite dimensional lie algebras which are called generalized schrodinger virasoro algebras and denoted by gsv g in this paper the automorphism group and irreducibility of verma modules for gsv g are completely determined

integrable representations for toroidal extended affine lie algebras abstract let g be any untwisted affine kac moody algebra any fixed complex number and g the corresponding toroidal extended affine lie algebra of nullity two for any k tuple 1 k of weights of g and k tuple a a 1 a k of distinct non zero complex numbers we construct a class of modules v a for the extended affine lie algebra g we prove that the g module v a is completely reducible we also prove that the g module v a is integrable when all weights i in are dominant thus we obtain a new class of irreducible integrable weight modules for the toroidal extended affine lie algebra g

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

regular obstructed categories and topological quantum field theory a proposal of the concept of n regular obstructed categories is given the corresponding regularity conditions for mappings morphisms and related structures are considered an n regular topological quantum field theory is introduced the connection of time nonivertibility with the regularity is shown

an extension of the korteweg de vries hierarchy arising from a representation of a toroidal lie algebra abstract in this article we show how to construct hierarchies of partial differential equations from the vertex operator representations of toroidal lie algebras in the smallest example rank 2 toroidal cover of sl2 we obtain an extension of the kdv hierarchy we use the action of the corresponding infinite dimensional group to construct solutions for these non linear pdes

classification of quasifinite modules over the lie algebras of weyl type abstract for a nondegenerate additive subgroup of the n dimensional vector space f n over an algebraically closed field f of characteristic zero there is an associative algebra and a lie algebra of weyl type w n spanned by all differential operators ud1m1 dnmn for u f the group algebra and m1 mn 0 where d1 dn are degree operators in this paper it is proved that an irreducible quasifinite w z 1 module is either a highest or lowest weight module or else a module of the intermediate series furthermore a classification of uniformly bounded w z 1 modules is completely given it is also proved that an irreducible quasifinite w n module is a module of the intermediate series and a complete classification of quasifinite w n modules is also given if is not isomorphic to z

hudson s theorem for finite dimensional quantum systems we show that on a hilbert space of odd dimension the only pure states to possess a non negative wigner function are stabilizer states the clifford group is identified as the set of unitary operations which preserve positivity the result can be seen as a discrete version of hudson s theorem hudson established that for continuous variable systems the wigner function of a pure state has no negative values if and only if the state is gaussian turning to mixed states it might be surmised that only convex combinations of stabilizer states give rise to non negative wigner distributions we refute this conjecture by means of a counterexample further we give an axiomatic characterization which completely fixes the definition of the wigner function and compare two approaches to stabilizer states for hilbert spaces of prime power dimensions in the course of the discussion we derive explicit formulas for the number of stabilizer codes defined on such systems

representations of the lie algebra of derivations for quantum torus abstract let der c q be the derivation lie algebra of the quantum torus c q we construct a functor from gld modules to der c q modules which generalizes the constructions obtained by shen scientia sinica a 29 1986 570 larsson internat j modern phys a 7 1992 6493 and rao j algebra 182 1996 401 we also give a complete description of the structure of the der c q modules when all the entries of the quantum torus matrix q qij d d are roots of unity

classification of harish chandra modules over some lie algebras related to the virasoro algebra abstract in this paper we provide a uniform method to thoroughly classify all harish chandra modules over some lie algebras related to the virasoro algebras we first classify such modules over the lie algebra w s for s 0 1 2 with this result and method we can also do such works for some lie algebras related to the virasoro algebra including the several kinds of schrodinger virasoro lie algebras which are open up to now

states that look the same with respect to every basis in a mutually unbiased set a complete set of mutually unbiased bases mubs in a hilbert space of dimension d defines a set of d 1 orthogonal measurements relative to such a set we define a mub balanced state to be a pure state for which the list of probabilities of the d outcomes of any of these measurements is independent of the choice of measurement up to permutations in this paper we explicitly construct a mub balanced state for each prime power dimension d for which d 3 mod 4 these states have already been constructed by appleby in unpublished notes but our presentation here is different in that both the expression for the states themselves and the proof of mub balancedness are given in terms of the discrete wigner function rather than the density matrix or state vector the discrete wigner functions of these states are rotationally symmetric in a sense roughly analogous to the rotational symmetry of the energy eigenstates of a harmonic oscillator in the continuous two dimensional phase space upon converting the wigner function to a density matrix we find that the states are expressible as real state vectors in the standard basis we observe numerically that when d is large and not a power of 3 a histogram of the components of such a state vector appears to form a semicircular distribution

extended affine weyl groups abstract in this paper we study the weyl groups of reduced extended affine root systems the root systems of extended affine lie algebras we start by describing the extended affine weyl group as a semidirect product of a finite weyl group and a heisenberg like normal subgroup this provides a unique expression for the weyl group elements in terms of some naturally arisen transformations which is crucial in the further study of extended affine weyl groups we use this to give a presentation called a presentation by conjugation for an important subclass of extended affine weyl groups using a new notion called the index which is an invariant of the extended affine root systems we show that one of the important features of finite and affine root systems related to weyl group holds for the class of extended affine root systems

locally extended affine lie algebras abstract we propose a new simplified definition of extended affine lie algebras ealas for short and also discuss a general version of extended affine lie algebras called locally extended affine lie algebras local ealas for short we prove a conjecture by v kac for local ealas it turns out that the root system of a local eala becomes a locally finite version of an extended affine root system several examples of new ealas and local ealas are introduced and finally we classify local ealas of nullity 0 and show the connection to locally finite split simple lie algebras

quasifinite representations of a lie algebra of block type abstract let b be the lie algebra of block type over c with basis l i c i z i 1 and relations l i l j i 1 j 1 l i j 0 i j 2c c l i 0 in this paper it is proved that a quasifinite irreducible b module is a highest or lowest weight module furthermore the quasifinite irreducible highest weight modules are classified and the unitary ones are proved to be trivial

representations of toroidal extended affine lie algebras abstract we show that the representation theory for the toroidal extended affine lie algebra is controlled by a voa which is a tensor product of four voas a sub voa v hyp of a hyperbolic lattice voa affine g and sl n voas and a virasoro voa a tensor product of irreducible modules for these voas admits the structure of an irreducible module for the toroidal extended affine lie algebra we also show that for n 12 v hyp becomes an exceptional irreducible module for the extended affine lie algebra of rank 0

highest weight modules over pre exp polynomial lie algebras abstract in this paper we introduce pre exp polynomial lie algebras which include loop algebras virasoro like algebras and some quantum torus lie algebras we study highest weight representations of these z n 1 graded lie algebras more precisely we show that non graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite dimensional weight spaces or not and they have all finite dimensional weight spaces if and only if the highest weight is an exp polynomial highest weight we also show that non graded and graded highest weight verma modules with the same highest weight are simultaneously irreducible or not and we determine necessary and sufficient conditions for a verma module to be irreducible

trigonometric lie algebras affine lie algebras and vertex algebras abstract in this paper we explore natural connections among trigonometric lie algebras general affine lie algebras and vertex algebras among the main results we obtain a realization of trigonometric lie algebras as what were called the covariant algebras of the affine lie algebra a of lie algebra a gl gl with respect to certain automorphism groups we then prove that restricted modules of level l for trigonometric lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras v a l 0 or v a 2 l 0 furthermore we determine irreducible modules and equivariant quasi modules for the simple vertex algebra l a l 0 with l a positive integer in particular we prove that every quasi finite unitary highest weight irreducible module of level l for type a trigonometric lie algebra gives rise to an irreducible equivariant quasi l a l 0 module

harish chandra modules for the q analog virasoro like algebra abstract let l be the q analog virasoro like algebra in this paper we prove that any harish chandra module of l with nontrivial center is isomorphic to a generalized highest weight module moreover we show that a generalized highest weight harish chandra module of l is induced from an irreducible module of a heisenberg subalgebra of l

a classification of indecomposable sl2 c modules and a conjecture of kac on irreducible modules over the virasoro algebra in this paper we first give a classification of indecomposable harish chandra sl2 c modules then discuss a conjecture of kac on irreducible module over the virasoro algebra proving that the conjecture is true under a certain condition

irreducible representations of virasoro toroidal lie algebras toroidal lie algebras and their vertex operator representations were introduced in mey and a class of indecomposable modules were investigated in this work we extend the toroidal algebra by the virasoro algebra thus constructing a semi direct product algebra containing the toroidal algebra as an ideal and the virasoro algebra as a subalgebra with the use of vertex operators and certain oscillator representations of the virasoro algebra it is proved that the corresponding fock space gives rise to a class of irreducible modules for the virasoro toroidal algebra

lusztig symmetries and poincare birkhoff witt basis for wur sd osp 1 2n we investigate a new kind of two parameter weak quantized superalgebra wur sd osp 1 2n which is a weak hopf superalgebra it has a homomorphic image which is isomorphic to the usual two parameter quantum superalgebra ur s osp 1 2n of osp 1 2n we also discuss the basis of wur sd osp 1 2n by lusztig s symmetries

extended diffeomorphism algebras and trajectories in jet space let the dro diffeomorphism reparametrization observer algebra dro n be the extension of diff n diff 1 by its four inequivalent virasoro like cocycles here diff n is the diffeomorphism algebra in n dimensional spacetime and diff 1 describes reparametrizations of trajectories in the space of tensor valued p jets dro n has a fock module for each p and each representation of gl n analogous representations for gauge algebras higher dimensional kac moody algebras are also given the reparametrization symmetry can be eliminated by a gauge fixing procedure resulting in previously discovered modules in this process two dro n cocycles transmute into anisotropic cocycles for diff n thus the fock modules of toroidal lie algebras and their derivation algebras are geometrically explained

derivations isomorphisms and second cohomology of generalized witt algebras generalized witt algebras over a field f of characteristic 0 were defined by kawamoto about 12 years ago using different notations from kawamoto s we give an essentially equivalent definition of generalized witt algebras w w a t over f where the ingredients are an abelian group a a vector space t over f and a map t a k which is linear in the first variable and additive in the second one in this paper the derivations of any generalized witt algebra w w a t with the right kernel of being 0 are explicitly described the isomorphisms between any two simple generalized witt algebras are completely determined and the second cohomology group h2 w f for any simple generalized witt algebra is computed the derivations the automorphisms and the second cohomology groups of some special generalized witt algebras have been studied by several other authors as indicated in the references

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

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

irreducible representations of the lie algebra of the diffeomorphisms of ad dimensional torus in this paper we investigate the irreducibility of certain modules for the lie algebra of diffeomorphisms of torus t the lie algebra of diffeomorw 1 1 x phisms can be described as the derivations of a s c t t see 1 d w x w x rss we denote this lie algebra by der a larsson l1 constructed a functor from gl modules to der a modules in this paper we prove that d the image of a finite dimensional irreducible gl module is most often d irreducible the only exceptions are fundamental modules and the onedimensional modules in these cases we describe the sub modules and the quotients surprisingly the same class of modules image of finite dimen w x sional gl module are also constructed in e without any reference to d gl modules they are motivated by the vertex operator constructions in d w x em more details near the end of this introduction we will now describe our results in more detail let gl be the d lie algebras of d d matrices over complex numbers c let gl s sl d d c id where sl is the finite dimensional simple lie algebra of trace zero d matrices and id the identity matrix which is central it is well known that finite dimensional irreducible modules and dominant integral weights are in one one correspondence for a finite dimensional simple lie algebra so let v c be finite dimensional irreducible sl module corresponding to a d dominant integral weight c let id the central element act by a complex number b and denote the corresponding gl module by v c b d a d d larsson defined see 1 6 a functor f for a belonging to c rz from w x gl modules to der a modules in l1 these modules are weight modules d

integrable representations for the twisted full toroidal lie algebras the purpose of this work is to classify irreducible integrable modules of the twisted full toroidal lie algebra with finite dimensional weight spaces and non zero central charges there are three families of such modules two of which are classified

nonlocal integrable partners to generalized mkdv and two dimensional toda lattice equation in the formalism of a dressing method with quantized spectral parameter two new hierarchies milw2 and a two dimensional nonlocal toda lattice are constructed the characteristic property of the first one is the connection with the ilw2 hierarchy by means ofgl 2 miura transformation on the other hand milw2 equations turn out to be symmetry equations for a two dimensional nonlocal toda lattice a new version of the dressing technique with quantized spectral parameter is proposed

weight modules with a finite dimensional weight space over truncated virasoro algebras truncated virasoro algebras can be viewed as quotient algebras of the loop virasoro algebra in this paper we prove that any weight module m with a finite dimensional weight space over any truncated virasoro algebra l f vir c t 1 fc t 1 has a harish chandra submodule where vir is the virasoro algebra and f is a nonzero and not invertible laurent polynomial in c t 1 more precisely if m has a nontrivial upper bounded vir submodule then it has a highest weight l f submodule if m has a nontrivial lower bounded vir submodule then it has a lowest weight l f submodule if m is nontrivial over l f and does not contain any nontrivial upper bounded nor any nontrivial lower bounded vir submodules then it has a nontrivial harish chandra l f submodule s such that dims 1 for all supp s 0 as a corollary any irreducible weight module with a finite dimensional weight space over l f is a harish chandra module

unconditionally secure key distribution in higher dimensions by depolarization this paper presents a prepare and measure scheme using n dimensional quantum particles as information carriers where n is a prime power one of the key ingredients used to resist eavesdropping in this scheme is to depolarize all pauli errors introduced to the quantum information carriers using the shor preskill type argument we prove that this scheme is unconditionally secure against all attacks allowed by the laws of quantum physics for n 2n 2 each information carrier can be replaced by n entangled qubits in this case there is a family of eavesdropping attacks on which no unentangled qubit based prepare and measure pm quantum key distribution scheme known to date can generate a provably secure key in contrast under the same family of attacks our entangled qubit based scheme remains secure whenever 2nges4 this demonstrates the advantage of using entangled particles as information carriers and of using depolarization of pauli errors to combat eavesdropping attacks more drastic than those that can be handled by unentangled qubit based prepare and measure schemes

cyclic mutually unbiased bases fibonacci polynomials and wiedemann s conjecture we relate the construction of a complete set of cyclic mutually unbiased bases i e mutually unbiased bases generated by a single unitary operator in power of two dimensions to the problem of finding a symmetric matrix over f2 with an irreducible characteristic polynomial that has a given fibonacci index for dimensions of the form 22k we present a solution that shows an analogy to an open conjecture of wiedemann in finite field theory finally we discuss the equivalence of mutually unbiased bases

representations of the schrodinger virasoro algebras in this paper it is proved that an irreducible weight module with finite dimensional weight spaces over the schrodinger virasoro algebras is a highest lowest weight module or a uniformly bounded module furthermore indecomposable modules of the intermediate series over these algebras are completely determined